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Problems and SolutionsinReal and Complex Analysis,Integration,Functional EquationsandInequalities
byWilli-Hans SteebInternational School for Scientific ComputingatUniversity of Johannesburg, South Africa
Preface
The purpose of this book is to supply a collection of problems in analysis.Please submit your solution to one of th email addresses below.
e-mail addresses of the author:
[email protected][email protected]
Home page of the author:
http://issc.uj.ac.za
Prescribed books for problems.
1) Problems and Solutions in Theoretical and Mathematical Physics, ThirdEdition, Volume I: Introductory Level
by
Willi-Hans SteebWorld Scientific Publishing, Singapore 2009ISBN- 13 978 981 4282 14 7http://www.worldscibooks.com/physics/7416.html
2) Problems and Solutions in Theoretical and Mathematical Physics, ThirdEdition, Volume II: Advanced Level
by
Willi-Hans SteebWorld Scientific Publishing, Singapore 2009ISBN-13 978 981 4282 16 1http://www.worldscibooks.com/physics/7416.html
updated: January 16, 2018
v
vi
Contents
Preface v
Notation x
1 Sums and Products 11.1 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Supplementary Problems . . . . . . . . . . . . . . . . . . . . 12
2 Maps 262.1 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . 262.2 Supplementary Problems . . . . . . . . . . . . . . . . . . . . 35
3 Functions 413.1 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . 413.2 Supplementary Problems . . . . . . . . . . . . . . . . . . . . 60
4 Polynomial 734.1 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . 734.2 Supplementary Problems . . . . . . . . . . . . . . . . . . . . 86
5 Equations 915.1 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . 915.2 Supplementary Problems . . . . . . . . . . . . . . . . . . . . 98
6 Normed Spaces 1026.1 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . 1036.2 Supplementary Problems . . . . . . . . . . . . . . . . . . . . 106
7 Complex Numbers and Complex Functions 1077.1 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . 1077.2 Supplementary Problems . . . . . . . . . . . . . . . . . . . . 113
vii
8 Integration 1188.1 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . 1188.2 Programming Problems . . . . . . . . . . . . . . . . . . . . 1448.3 Supplementary Problems . . . . . . . . . . . . . . . . . . . . 145
9 Functional Equations 1529.1 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . 1529.2 Supplementary Problems . . . . . . . . . . . . . . . . . . . . 156
10 Inequalities 15710.1 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . 15710.2 Supplementary Problems . . . . . . . . . . . . . . . . . . . . 169
Bibliography 172
Index 177
viii
x
Notation
:= is defined as∈ belongs to (a set)/∈ does not belong to (a set)∩ intersection of sets∪ union of sets∅ empty setN set of natural numbersZ set of integersQ set of rational numbersR set of real numbersR+ set of nonnegative real numbersC set of complex numbersRn n-dimensional Euclidean space
space of column vectors with n real componentsCn n-dimensional complex linear space
space of column vectors with n complex componentsH Hilbert spacei
√−1
<z real part of the complex number z=z imaginary part of the complex number z|z| modulus of complex number z
|x+ iy| = (x2 + y2)1/2, x, y ∈ RT ⊂ S subset T of set SS ∩ T the intersection of the sets S and TS ∪ T the union of the sets S and Tf(S) image of set S under mapping ff g composition of two mappings (f g)(x) = f(g(x))v column vector in CnvT transpose of v (row vector)0 zero (column) vector‖ . ‖ normx · y ≡ x∗y scalar product (inner product) in Cnx× y vector product in R3
A,B,C m× n matricesdet(A) determinant of a square matrix Atr(A) trace of a square matrix Arank(A) rank of matrix AAT transpose of matrix A
xi
A conjugate of matrix AA∗ conjugate transpose of matrix AA† conjugate transpose of matrix A
(notation used in physics)A−1 inverse of square matrix A (if it exists)In n× n unit matrixI unit operator0n n× n zero matrixAB matrix product of m× n matrix A
and n× p matrix BA •B Hadamard product (entry-wise product)
of m× n matrices A and B[A,B] := AB −BA commutator for square matrices A and B[A,B]+ := AB +BA anticommutator for square matrices A and BA⊗B Kronecker product of matrices A and BA⊕B Direct sum of matrices A and Bδjk Kronecker delta with δjk = 1 for j = k
and δjk = 0 for j 6= kλ eigenvalueε real parametert time variableH Hamilton operator
Chapter 1
Sums and Products
1.1 Solved ProblemsProblem 1. The harmonic series can be approximated by
n∑j=1
1j≈ 0.5772 + ln(n) +
12n.
Calculate the left and rigt-hand side for n = 1 and n = 10.
Problem 2. The Bernoulli numbers B0, B1, B2, . . . are defined by thepower series expansion
x
ex − 1=∞∑j=0
Bjj!xj ≡ B0 +
B1
1!x+
B2
2!x2 + · · · .
One finds B0 = 1, B1 = −1/2, B2 = 1/6, B3 = 0, B4 = −1/30. One hasBj = 0 if j ≥ 3 and j odd. The Bernoulli numbers are utilized in the Eulersummation formulan∑j=1
f(j) =∫ n
1
f(t)dt+12
(f(n)+f(1))+n∑k=1
B2k
(2k)!(f (2k−1)(n)−f (2k−1)(1))+Rm(n)
where|Rm(n)| ≤ 4
(2π)2m
∫ m
1
|f (2m)(t)|dt.
1
2 Problems and Solutions
Calculate the sumn∑j=1
j2
using f(t) = t2.
Problem 3. Let z ∈ C and |z| < 1. Calculate the sum
(1− |z|2)2s∞∑n=0
n
(2s+ n− 1
n
)|z|2n.
Problem 4. Let x ∈ R and r ∈ N with r ≥ 1. Find the sum
r−1∑k=0
exp(−2πikx/r).
Problem 5. Let a1, a2, . . . , an ∈ [0, 1]. Show that there exists a real num-ber x in the unit interval such that the average of the (unsigned) distancesfrom x to the aj ’s is 1/2.
Problem 6. Let f : Rm → R be a differentiable function. Show thatthere exist m differentiable functions g1, g2, . . . , gm defined on Rm withthe properties
f(x) = f(0) +m∑j=1
xjgj(x)
and
gj(0) =∂f
∂xj(0)
where x = (x1, x2, . . . , xm).
Problem 7. Let n be a positive integer and f(j) = j(j − 1)(j − 2) withj = 1, 2, . . . , n. Let aj := f(j + 1)− f(j). By calculating the sum
∑nj=1 aj
show thatn∑j=1
j2 =n(n+ 1)(2n+ 1)
6.
Sums and Products 3
Problem 8. Show that N∑j=1
1u− vj
2
−N∑j=1
1(u− vj)2
≡ 2N∑i<jj=2i=1
1vi − vj
(1
u− vi− 1u− vj
).
This identity plays a role in the Bethe ansatz.
Problem 9. Show that the series
f(θ) =∞∑j=0
sin(3jθ)2j
is convergent. Is the series df/dθ convergent?
Problem 10. Find the radius of convergence of the power series
∞∑j=0
(j + k
j
)zj , k > 0.
Problem 11. Find the radius of convergence of the power series
∞∑j=1
j!jjzmj , m = 1, 2, . . . .
Problem 12. Let (s0, s1, ..., sn−1)T ∈ Rn, where n = 2k. This vector inRn can be associated with a piecewise constant function f defined on [0, 1)
f(x) =2k−1∑j=0
sjΘ[j2−k,(j+1)2−k)(x)
where Θ[j2−k,(j+1)2−k)(x) is the step function
Θ[j2−k,(j+1)2−k)(x) :=
1 x ∈ [j2−k, (j + 1)2−k)0 x /∈ [j2−k, (j + 1)2−k)
with the support [j2−k, (j + 1)2−k). Let xj+1 = 4xj(1 − xj) with j =0, 1, 2, . . . and x0 = 1/3. Then
x0 =13, x1 =
89, x2 =
3281, x3 =
62726561
.
4 Problems and Solutions
Find f(x) for this data set and then calculate∫ 1
0
f(x)dx.
Problem 13. Let a1, a2, . . . , an be a finite sequence of numbers. ItsCesaro sum is defined as
s1 + s2 + · · ·+ snn
wheresk = a1 + a2 + · · ·+ ak
for each k, 1 ≤ k ≤ n. Suppose that the Cesaro sum of the 99-term sequencea1, a2, . . . , a99 is 100. Find the Cesaro sum of the 100-term sequence 1,a1, a2, . . . , a99.
Problem 14. Each x ∈ [0, 1] can be written as
x =∞∑j=1
εj2j
with εj = 0 or εj = 1. Define the function f : [0, 1]→ [0, 1) as
f(x) =∞∑j=1
2εj3j.
The function f is known as Cantor function. Let x = 1/8. Find f(x).
Problem 15. Let s ≥ 2. Simplify the sum
∞∑j1=1
∞∑j2=1
1(j1 + j2)s
.
Problem 16. Consider
1 + x
1− x− x2=∞∑j=0
cjxj .
Find cj .
Sums and Products 5
Problem 17. The Cantor series approximation is defined as follows.For arbitrary chosen integers n1, n2, . . . (equal or larger than 2), we canapproximate any real number r0 as follows
xj = integer part(rj), j = 0, 1, 2, . . .rj+1 = (rj − xj)nj
and
r0 ≈ x0 +N∑j=1
xjn1n2 · · ·nj
.
The approximation error is
EN =1
n1n2 · · ·nN.
Apply this approximation to r0 = 2/3 and the golden mean number withnj = 2 for all j and N = 4.
Problem 18. Calculate the sum
S =√
2 +√
2−√
2√
2−√
2−√
2−√
2.
Problem 19. Let n1, n2, n3 ∈ Z. Calculate
1− eiπ(n1+n2+n3).
Problem 20. Show that 7 + 2√
10 has a square root the form√x+√y.
Problem 21. Calculate∞∑j=0
j3 + 1j!
xj .
Hint. Write j3 + 1 in the form
a+ bj + cj(j − 1) + dj(j − 1)(j − 2).
Problem 22. Let n ∈ N. Find the sum
(n−1)/2∑k=−(n−1)/2
k2
6 Problems and Solutions
where k runs in steps of 1.
Problem 23. Assume that the series
1, 14, 51, 124, 245, 426, . . .
is of the formak3 + bk2 + ck + d.
Find the integer coefficents a, b, c, d.
Problem 24. Find the sum
Sn =2
1 · 2 · 3+
22 · 3 · 4
+2
3 · 4 · 5+ · · ·+ 2
n · (n+ 1) · (n+ 2).
Calculate Sn for n→∞. Hint. Find a, b, c for
2n · (n+ 1) · (n+ 2)
=a
n+
b
n+ 1+
c
n+ 2.
Problem 25. Let a > 0. Show that
∞∑n=−∞
1z − na
=1z
+ 2z∞∑n=1
1z2 − n2a2
.
Problem 26. Let x 6= y, x 6= z, y 6= z. Find the sum
1(x− y)(x− z)
+1
(y − x)(y − z)+
1(z − x)(z − y)
.
Problem 27. Given the expansion
ln(1 + x) = x− x2
2+x3
3− x4
4+ · · · , −1 < x ≤ 1.
Calculate ln(11).
Problem 28. Let n ∈ N0 and p, a > 0. Show that
∞∑k=0
(k + n)!k!n!
exp(−2πkp/a) ≡ 1(1− exp(−2πp/a))n+1
.
Sums and Products 7
Hint: Start with
f(n) :=∞∑k=0
(k + n)!k!n!
exp(−2πkp/a)
and find f(n+ 1).
Problem 29. Let (s1, s2) ∈ Z2. Let P (φ1, φ2) be a probability densityand let θ1, θ2 ∈ R. We can express the characteristic double sequence as
χ(s1, s2) =∫ θ1+2π
θ1
∫ θ2+2π
θ2
exp(i(s1φ1 + s2φ2))P (φ1, φ2)dφ1dφ2.
Find P (φ1, φ2) as a function of χ(s1, s2). Note that χ(0, 0) = 1.
Problem 30. Show thatπ
4= 4 arctan(1/5)− arctan(1/239).
Hint. Let θ = arctan(1/5). Thus tan(θ) = 1/5. Applying the double angleformula for the tangent we have tan(2θ) = 5/12 and tan(4θ) = 120/119.Finally apply that tan(π/4) = 1.
Problem 31. Let n ∈ N0. Consider an infinite number of time variablet = (t1, t2, t3, . . .). Consider the sum
pn(x+ t1, t2, t3, . . .) =∑
k0+k1+2k2+3k3+···=nk0,k1,k2,k3,...≥0
xk0tk11 tk22 · · ·
k0!k1!k2! · · ·.
Find p0, p1, p2, p3.
Problem 32. Let n be an integer with n ≥ 1. Simplify the series
S =1
1 · 2+
12 · 3
+1
3 · 4+ · · ·+ 1
n(n+ 1).
Problem 33. Let N be a positive integer and a, b be positive integers orpositive half-integers. Find
f(N, a, b) =12
(1 + (−1)2a + (−1)2b + (−1)2a+2b+N ).
Problem 34. Let −1 < x < +1 and n ≥ 2.
8 Problems and Solutions
(i) Show that
Sn(x) =n−1∑j=0
jxj =nxn
x− 1+x(1− xn)(1− x)2
.
(ii) Show thatlimn→∞
Sn(x) =x
(1− x)2.
(iii) Let x = 1/2. Show that limn→∞ Sn(1/2) = 2.
Problem 35. Let x, y, z ∈ R and x 6= y, x 6= z, y 6= z. Show that
x
(z − x)(x− y)+
y
(x− y)(y − z)+
z
(y − z)(z − x)= 0.
Problem 36. Let
x 6= 0, x 6= ±2π, x 6= ±4π, . . .
and n ∈ N. Show that
sin(x) + sin(2x) + · · ·+ sin(nx) =cos(x/2)− cos((n+ 1/2)x)
2 sin(x/2).
For the values x = 0, x = ±2π, x = ±4π etc the sum is given by 0.
Problem 37. Let n = 0, 1, 2, . . .. The function
Kn(θ) :=1
n+ 1sin2
((n+1
2
)θ)
sin2(
12θ)
is called the Fejer kernel.(i) Show that
Kn(θ) :=j=+n∑j=−n
(1− |j|
n+ 1
)eijθ.
(ii) Show that Kn(θ) ≥ 0.(iii) Show that for any continuous 2π periodic function f on R one has
Kn ? f(θ) :=1
2π
∫ π
−πKn(θ − α)f(α)dα
=j=+n∑j=−n
(1− |j|
n+ 1
)(1
2π
∫ π
−πe−ijαf(α)dα
)eijθ.
Sums and Products 9
(iv) Show that Kn ? f(θ)→ f(θ) uniformely in θ as n→∞.
Problem 38. Let λ > 0.(i) Calculate
S1(λ) =∞∑j=0
je−λλj
j!.
(ii) Calculate
S2(λ) =∞∑j=0
j(j − 1)e−λλj
j!.
Problem 39. (i) Let α > 0. Show that
∞∑k=1
(−1)k cos(kx)k2 + α2
=π
2α2· cosh(αx)
sinh(αx)− 1
2α2
where −π ≤ x ≤ π.(ii) Let α > 0. Show that
∞∑k=1
(−1)k sin(kx)k2 + α2
= − π
2α2· sinh(αx)
sinh(απ)
for −π < x < π.
Problem 40. Let a > 0 and b > 0. Show that the continued fraction
a+1b+
1a+
1b+
1a+· · ·
can be written asa
2+
√a2
4+a
b.
Problem 41. Let M,N ≥ 1. Find the sum
M∑m=1
N∑n=1
(m+ n)!(2m+ n)!(m+ 2n)!
.
Problem 42. Study the sum∞∑
`,m,n=1
`
(`m+ `n+mn+ 1)2x`+m+n.
10 Problems and Solutions
Problem 43. Show that∞∑
n=−∞
1n2 + 3
4n+ 18
= 4π.
Problem 44. Let n ≥ 1. Find the sum
n∑j=1
3j(j + 3)
.
What happens for n→∞?
Problem 45. Let F1 = 1, F2 = 2, Fk+1 = Fk−1 + Fk be the Fibonaccinumbers. Calculate
∞∑k=1
log2(Fk+1)2k+1
.
Problem 46. Let Z be the set of integers. Simplify the sum∑k∈Z
sin(π(x− k)) sin(π(y − k))π(x− k)π(y − k)
.
Problem 47. Let n ≥ 1. One has
(1 + x)n =n∑k=0
(n
k
)xk. (1)
Show that
1 ·(n
1
)+2 ·
(n
2
)+3 ·
(n
3
)+ · · ·+(n−1) ·
(n
n− 1
)+n ·
(n
n
)= n2n−1. (2)
Problem 48. Let n > 2 and A, h > 0. Find Vn given by
Vn =Ah
n3
n−1∑j=1
j2.
Then find limn→∞ Vn. This concern the volume of a pyramide, h is theheight and A is the area of the base.
Sums and Products 11
Problem 49. Let n ≥ 1 and c1, . . . , cn be constants. Find the minimaof the function f : R→ R
f(x) =n∑j=1
(x− cj)2.
Problem 50. Let x ∈ R. The sequence of functions fk(x) is definedby f1(x) = cos(x/2) and for k > 1 by
fk(x) = fk−1(x) cos(x/2k).
Thusfk(x) = cos(x/2) cos(x/22) · · · cos(x/2k).
Obviously, we have fk(0) = 1 for every k. Calculate limk→∞ fk(x) as afunction of x for x 6= 0.
Programming Problems
Problem 1. Let m,n ≥ 2. Calculate the finite sum
Sn,m =m∑j=1
n∑k=1
|j − k|.
Give a C++ implementation.
Problem 2. Let n ∈ N. Calculate the finite sum
Sn =n∑
j=−n
n∑k=−n
|j + k|.
Problem 3. Let m ≥ 1 and n ≥ 1. Find the sum
min(m,n)∑k=0
(n+m− 2k + 1).
12 Problems and Solutions
1.2 Supplementary Problems
Problem 1. Show that
1 + 2 tanh2(λ) + 3 tanh4(λ) + · · ·+ (n+ 1) tanh2n(λ) + · · · ≡ cosh4(λ).
Note that tanh(0) = 0 and cosh(0) = 1.
Problem 2. Show that∞∑j=2
1j(2j − 1)
= 2 ln(2)− 1.
Problem 3. (i) Show that
14
=1
1 · 2 · 3+
12 · 3 · 4
+1
3 · 4 · 5+ · · ·
(ii) Let n ≥ 1. Show that
11 · 2
+1
2 · 3+ · · ·+ 1
n(n+ 1)=
n
n+ 1.
Problem 4. Let n,m ∈ N and a, b, c, d ∈ R. Show that
(a+ b)n(c+ d)m =n∑r=0
m∑s=0
(n
r
)(m
s
)arbn−rcsdm−s.
Problem 5. Let ` ≥ 1. Study the Gauss sum
G(k, `) =1√`
`−1∑r=0
e2πikr2/`.
Problem 6. Show that if |x| < 1 we have the expansion
1(1− x)2
= 1 + 2x+ 3x2 + 4x3 + · · · .
Let a > 0. Use this expansion to show that
1(a+ x)2
=1a2
(1− 2x
a+
3x2
a2− 4x3
a3+ · · ·
).
Sums and Products 13
Problem 7. Apply mathematical induction to show that
1 + 3 + 5 + · · ·+ (2n− 1) = n2
12 + 42 + 72 + · · ·+ (3n− 2)2 =12n(6n2 − 3n− 1)
13 + 33 + 53 + · · ·+ (2n− 1)3 = n2(2n2 − 1).
Problem 8. Show that
ln(2) =1
1 · 2+
13 · 4
+1
5 · 6+ · · ·
Problem 9. Let x > 0. Show that a complicated way to calculate 1/x isgiven by
1x+ 1
+1!
(x+ 1)(x+ 2)+
2!(x+ 1)(x+ 2)(x+ 3)
+ · · ·
Problem 10. Let y ∈ R and byc be the integer part of y. Let ω =(1 +
√5)/2 be the golden mean number (which is an irrational number).
We definexj := j + ((j + 1)/ω)(λ− 1), j = 0, 1, . . .
where λ > 0 and
µk = xk − xk−1, k = 1, 2, . . .
Show that the sequence µ1, µ2, . . . is non-periodic.
Problem 11. Let s > 0 and
f(s) =∫ ∞t=0
ln(1 + st)1 + t2
dt.
Show that
f(s) = (1− ln(s))s+πs2
4+(
ln(s)3− 1
9
)s3 +O(s4).
Problem 12. Let n be an integer and n > 1. Show that
sn = 1 +12
+13
+ · · ·+ 1n
14 Problems and Solutions
cannot be integer for all n > 1.
Problem 13. Let x ∈ R. Show that the series
x
1 + x2+
x2
1 + x4+
x3
1 + x6+ · · ·
converges absolutely for all values of x, except for +1 and −1.
Problem 14. Show that the series∞∑j=0
zj =1
1− z,
∞∑j=0
jzj =z
(1− z)2,
∞∑j=0
j2zj =z(1 + z)(1− z)3
converge for all |z| < 1.
Problem 15. Let n be an integer with n ≥ 2. Find∑1≤i<j≤n
∣∣∣∣ 12i− 1
3j
∣∣∣∣ .Problem 16. Let ` ≥ 1. Show that∑
j=1
14j2 − 1
=`
2`+ 1
and ∑j=2
1j(j2 − 1)
=14
(1− 2
`(`+ 1)
).
Problem 17. Let 0 ≤ r < 1. Calculate the sum∞∑j=1
rj cos(jφ)
utilizing the identity
cos(jφ) ≡ 12
(eijφ + eijφ).
Problem 18. Calculate the sum∞∑j=1
1j2 + 1
Sums and Products 15
with the knowledge that∞∑j=1
1j2 + 1
≡∞∑j=1
1j2−∞∑j=1
1j2(j2 + 1)
,
∞∑j=1
1j2
=π2
6.
Problem 19. Show that
cos(π/4) =12
√2, cos(π/8) =
12
√2 +√
2, cos(π/16) =12
√2 +
√2 +√
2
sin(π/4) =12
√2, sin(π/8) =
12
√2−√
2, sin(π/16) =12
√2−
√2 +√
2.
Problem 20. Let n ≥ 0 and x 6= 0. Find
Sn(x) =n∑k=0
kxk.
Utilize that
Sn+1(x) = Sn(x) + (n+ 1)xn+1 =n∑k=0
(k + 1)xk+1 =n∑k=0
kxk+1 +n∑k=0
xk+1
= xSn(x) +n∑k=0
xk+1
and a case study with x = 1 and x 6= 1.
Problem 21. The Fibonacci numbers are given by xt+2 = xt+1 +xt witht = 0, 1, . . . and x0 = x1 = 1. Consider
F =1x0
+1x1
+1x2
+1x3
+1x4
+1x5
+ · · · ≡ 11
+11
+12
+13
+15
+18
+ · · · .
Show that the series converges. Is F < 4?
Problem 22. Calculate the sum
S =∞∑j=1
2n+ 1n2(n+ 1)
.
Problem 23. Show that the Cantor series
c1 +c22!
+c33!
+c44!
+ · · ·
16 Problems and Solutions
with 0 ≤ cn ≤ n− 1 is convergent.
Problem 24. Let N ≥ 1 be an integer and k = −4N,−4N + 1, . . . , 4N .We define
f(k) =1N
N∑j=1
cos((
j − 12
)kπ
N
).
Show that f(0) = 1, f(±2N) = −1, f(±4N) = 1 and 0 otherwise.
Problem 25. Show that∑k∈Z
1(x− k)
= π cot(πx).
Note that cot(0) =∞ and cot(π/2) = 0.
Problem 26. Let c > 0. Apply the Poisson summation formula
+∞∑n=−∞
f(n+ a) =+∞∑
k=−∞
exp(2πika)∫ +∞
−∞f(x) exp(−2πikx)dx
for the function
f(x) :=x exp(−cx2) for x ≥ 0
0 for x < 0
Problem 27. Let k,m = 1, 2, . . .. Write a C++ program that implementsthe sum
E(k,m) =1
2k+m−1
m−1∑i=0
(k +m− 1
i
)for a given k and m. The sum plays a role for the arithmetic triangle.
Problem 28. Let L ∈ N0 and κ be a nonnegative real number.(i) Show that
Z(κ) =
(L∑n=0
(L
n
)e−κLn
)2
= (1 + e−κL)2L.
(ii) Show that for κ > 0 one has for large L that Z ≈ 1 + 2Le−κL. Showthat at κ = 0 one has Z(κ = 0) = 22L.
Sums and Products 17
Problem 29. Let n > 2. Show thatn∑j=2
1j≈ ln(n).
Problem 30. Let n ≥ 1 and ω be the primitive nth root of 1. Show thatn−1∑j=0
ωj = 0.
Problem 31. Show that
ex + e−x + 2 cos(x) = 4∞∑n=0
x4n
(4n)!.
Problem 32. Show that ∞∑j=0
1j!
( ∞∑k=0
(−1)k
k!
)= 1
Problem 33. Let n ∈ N. Applying the principle of mathematical induc-tion show that
A(n) =n∑k=0
k(k + 1) =n(n+ 1)(n+ 2)
3.
Note that A(0) = 0.
Problem 34. Let N ≥ 2. Find the sumN∑
j1,j2,j3=1j3≥j2≥j1
(j1j2j3 + (j1 + j2 + j3)).
Problem 35. Let h, r > 0 and fixed and m,n ∈ N. Consider
Sm,n =mnr sin(π/n)
√(h
m
)2
+ (r − r cos(π/n))2
= πrsin(π/n)π/n
√h2 +
14π4r2
m2
n4
(sin(π/n)π/n
)4
.
18 Problems and Solutions
(i) Consider limm,n→∞ Sm,n with m = n2.(ii) Consider limm,n→∞ Sm,n with m = n3.(iii) Consider limm,n→∞ Sm,n with limm→∞,n→∞(m2/n4) = 0 and showthat limm,n→∞ Sm,n = πrh which is a surface area of a half cylinder withradius r and length h.
Problem 36. (i) Let a > 0. Show that
∞∑k=1
cos(kx)k2 + a2
=π
2acosh(a(π − x))
sinh(aπ)− 1
2a.
(ii) Let a > 0. Show that
∞∑k=−∞
1(2k + 1)2 + a2
=π
2atanh
(aπ2
).
Problem 37. Let k ≥ 2. Consider
f(k) = k!(
12!− 1
3!+
14!− · · ·+ (−1)k
k!
)≡ k!
k∑j=2
(−1)j
j!.
Thus f(2) = 1. Find f(3), f(4), f(5), f(6), ... . Which of these numbersis a prime number?
Problem 38. Let M be an integer with M ≥ 2 and =(φ) < 0. Show that
S(k) =M−1∑m=1
e(2πimk)/M
sin2(π(m+ φ)/M)= −4kMe−2πikφ/M
1− e−2πiφ+M2e−2πikφ/M
sin2(πφ).
Note that1
sin2(π(m+ φ)/M)= −4
∞∑p=0
pe−2πi(m+φ)p/M .
Problem 39. The q-exponential function is defined by
exq :=∞∑j=0
xn
[j]!
where
[j] :=qj − q−j
q − q−1.
Sums and Products 19
Find exq for x = 1 and q = 1/2.
Problem 40. Let a, b, c ∈ R. Factorize
(b− c)3 + (c− a)3 + (a− b)3.
The following Maxima may be helpful
T: (b-c)^3 + (c-a)^3 + (a-b)^3;T: expand(T);T: ratsimp(T);
Problem 41. Let n ≥ 0. Starting from
n∑j=0
cjxj = (1 + x)n
show that
c0 + c1 + · · ·+ cn = 2n
c0 − c1 + c2 − c3 + · · ·+ (−1)ncn = 0c1 + 2c2 + 3c3 + · · ·ncn = n2n−1
c20 + c21 + · · ·+ c2n =(2n)!(n!)2
.
Problem 42. Let n ≥ 1. Show that
n∑j=1
ln(1 + 1/j) = ln(n+ 1).
Problem 43. Let x ∈ (−1, 1). Show that
∞∑j=0
jxj =x
1− x)2.
Problem 44. (i) Assuming that
1− 12
+13− 1
4+
15− 1
6+
17− 1
8+ · · · = ln(2).
20 Problems and Solutions
Show that
1− 12− 1
4+
13− 1
6− 1
8+
15− 1
10− 1
12+ · · · = 1
2ln(2).
Note that the convergence of the first series is not absolute.(ii) Show that
1 +13
+15
+ · · ·+ 12n+ 1
− 12
ln(n)
tends to a finite value as n→∞. Find this value.
Problem 45. Let β > 0. Show that
∞∑n=0
e−βn =1
1− e−β.
Problem 46. Let n be a positive integer and f(θ1, . . . , θN ) be a periodicfunction, i.e. periodic 2π for each θj (j = 1, . . . , N). Show for large n wehave (
N∏k=1
∫ 2π
0
dθk
) N∑j=1
cos(θj)
n
f(θ1, . . . , θN )
≈ Nn
(N∏k=1
∫ ∞−∞
dθke−n(θk)2/(2N)
)f(θ1, . . . , θN ).
Problem 47. Let A, B be finite sets and n(A), n(B) the numbers ofelements in A and B, respectively. Is
n(A) + n(B) = n(A ∪B) + n(A ∩B) ?
Prove or disprove.
Problem 48. Let f : [0, 1] → [0, 1] be a differentiable function, forexample the logistic map f(x) = 4x(1− x). Let f (k) be the k-th iterate ofthe function f . Let
bk := sup0≤x≤1
∣∣∣∣ ddxf (k)(x)∣∣∣∣ .
Show thatlimk→∞
(bk)1/k
exist. Apply the chain rule.
Sums and Products 21
Problem 49. Let n ≥ 2. Show that
n∑k=1
k
(n
k
)= n2n−1.
Problem 50. Is the series
S =∞∑k=1
k
1 + 2k3
convergent?
Problem 51. Give a non-trival infinite sequence (x0, x1, x2, . . .) such that
∞∑j=0
|xj |1 + |xj |
is finite.
Problem 52. Let a > 0. Show that
∞∑j=−∞
(−1)j1
j2 + a2=
π
a sinh(πa).
Problem 53. Let n = 0, 1, 2, . . .. The Fermat numbers are given by
Fn = 2(2n) + 1.
Show that the Fermat numbers satisfy the recurrence relation
Fn+1 = (Fn − 1)2 + 1
with F0 = 3
Problem 54. Let N ≥ 1. Consider the Hilbert space L2([−1, 1]). TheChebyhev-Gauss-Lobatto points are given by
xj = cos(jπ/N), j = 0, 1, . . . , N.
Find the four points for N = 3.
22 Problems and Solutions
Problem 55. Let |z| < 1. Show that∞∑j=1
jzj = (1− z)−1∞∑j=1
zj .
Show that∞∑j=1
j2zj = (1 + z)(1− z)−2∞∑j=1
zj
Problem 56. Show that
13
+29
+ · · ·+ 2n
3n+1+ · · · = 1
3
∞∑n=0
(23
)n=
13
11− 2/3
= 1.
Problem 57. (i) Let a > 0. Show that∞∑n=0
1(a+ n)(a+ n+ 1)
=1a.
(ii) Show that∞∑n=1
1n(n+ 1)(n+ 2)
=14.
(iii) Let n ≥ 2. Show that
n−1∑j=0
1(j + 1)(j + 2)
= 1− 1n+ 1
and thus show that∞∑j=0
1(j + 1)(j + 2)
= 1.
Problem 58. Let n be a positive integer. Show that
sin(2nα)2n sin(α)
= cos(α) cos(2α) cos(22α) · · · cos(2n−1α).
Problem 59. Let Γ(x) be the gamma function. Show thatn∑k=0
(−1)k(n
k
)(α+ 2k)Γ(α+ k)Γ(α+ k + n+ 1)
= δn,0.
Sums and Products 23
Problem 60. Let a > 0 and a /∈ N. Show that
cos(ax) =2a sin(ax)
π
(1
2a2− cos(x)a2 − 12
+cos(2x)a2 − 22
− cos(3x)a2 − 32
+ · · ·)
for all x ∈ [−π, π].
Problem 61. (i) Let n ∈ N. Show that
14 + 24 + · · ·+ n4 =130n(n+ 1)(2n+ 1)(3n2 + 3n− 1).
(ii) Let n ≥ 1. Show that
n∑k=0
k4 =n5
5+n4
2+n3
3− n
30.
Problem 62. Let N ≥ 1. Find the sums
vk =N−1∑j=0
(exp
(2πNjk
)− 1), k = 0, 1, . . . , N − 1.
Problem 63. (i) Let ` be a positive integer and φ ∈ R. Show that
+∑m=−`
eimφ =sin((`+ 1/2)φ)
sin(φ/2).
(ii) The function
Dn(θ) :=sin((n+ 1
2 )θ)sin( 1
2θ)
is called the Dirichlet kernel. Show that
Dn(θ) =k=+n∑k=−n
eikx.
Problem 64. Let k be a positive integer and k ≥ 2. Show that
1 + x− 2xk ≡ (1− x) + 2x(1− xk−1) ≡ (1− x)
1 + 2k−1∑j=1
xj
.
24 Problems and Solutions
Problem 65. Let n ≥ 2. Let Sn be the standard n-simplex embedded inRn
Sn :=
x ∈ Rn :n∑j=1
xj = 1, for xk ≥ 0, for k = 1, . . . , n
.
We denote by Sk,n−1 the kth face of the boundary of Sn. They are (n−1)-simplexes
Sk,n−1 := x ∈ Rn : xk = 0, x ∈ Sn for k = 1, . . . , n.
Show that the boundary ∂Sn of Sn is the union
∂Sn = ∪nj=1Sj,n−1
of the faces.
Problem 66. Show that∞∏j=1
cos( x
2j)
=sin(x)x
for x ∈ R.
Problem 67. The triple sum
∞∑k1=−∞
∞∑k2=−∞
∞∑k3=−∞
′ (−1)k1+k2+k3√k2
1 + k22 + k2
3
plays a role in solid state physics (Madelung constant). Here ′ indicatesthat the term (k1, k2, k3) = (0, 0, 0) is omitted. Given a positive integer N .Write a C++ program that implements the sum
N∑k1=−N
N∑k2=−N
N∑k3=−N
′ (−1)k1+k2+k3√k2
1 + k22 + k2
3
.
Run the program for different N and compare with the exact value.
Problem 68. Find the sum
SL =L∑n=1
exp(iπ
2Ln2
)for L = 1, 2, 3.
Sums and Products 25
Problem 69. Show that the series∞∑j=1
sin(jx)√j
converges for all x ∈ R.
Problem 70. Let k ≥ 0 and s > 0. Show that
k∏`=0
s+ `+ 1s+ `
=s+ k + 1
s.
Problem 71. Let n be a positive integer. Give a C++ implementationof the sum
n∑k=0
n∑`=0
(k
`
)using templates so that the Verylong class of SymbolicC++ can be used.
Problem 72. Let n,m ≥ 1. Implement the sum
Sn,m :=(2m)!22mm!
m∑k=0
2k(n
k
)(m
k
)using the Verylong class and Rational class of SymbolicC++.
Problem 73. Let n ∈ N and m1,m2, . . . ,mn ∈ N0. Consider the function
f(m1,m2, . . . ,mn) :=δm1,0 for n = 1
δm2,0δm1,1 for n = 2δmn,0δ(m1+···+mn),n−1
∏n−2j=1 H(j −
∑jk=1mn−k) for n ≥ 3
where H is the step function
H(x) :=
1 for x ≥ 00 for x < 0
Give an implementation of this function in SymbolicC++ for a given nusing the Verylong class.
Chapter 2
Maps
2.1 Solved ProblemsProblem 1. Newton’s sequence takes the form of a difference equation
xt+1 = xt −f(xt)f ′(xt)
where t = 0, 1, 2, . . . and x0 is the initial value at t = 0. Let f : R → R begiven by
f(x) = x2 − 1
and x0 6= 0. Find the fixed points of f . Find the fixed points of thedifference equation. Let x0 = 1/2. Find x1, x2, x3. Find Newton’s sequencefor this function. Obtain the exact solution of the difference equation.
Problem 2. (i) Solve the nonlinear recurrence relation
xn+1 = x2n, n = 0, 1, . . .
where x0 = 2.(ii) Solve the linear recurrence relation
xn+1 = xn + xn−1 + xn−2, n = 2, 3, . . .
and the initial values x0 = x1 = x2 = 1.
26
Maps 27
(iii) Solve the linear recurrence relation
xn+1 = 1 +n−1∑j=0
xj , x0 = 1.
Problem 3. Let x > 0 and p > 0. Consider the map
f(x) = xep−x.
(i) Find the fixed points. Study the stability of the fixed points.(ii) Show that f has a least one periodic point x∗ with x∗ 6= 0 or p.
Problem 4. Let f : R → R+ be a positive, continously differentiablefunction, defined for all real numbers and whose derivative is always neg-ative. Show that for any real number x0 (initial value) the sequence (xk)obtained by Newton’s method
xk+1 = xk −f(xk)f ′(xk)
has always limit ∞.
Problem 5. Let f : R→ R be a continuosly differentiable map. Let f (n)
be the n-th iterate of f . Calculate the derivative of f (n) at x0.
Problem 6. (i) Solve the second order linear difference equation
xt+2 = xt+1 + xt t = 0, 1, 2, . . .
where x0 = 0 and x1 = 1.(ii) Give the definition of the golden mean number and derive this number.(iii) Calculate
limt→∞
xt+1
xt.
Problem 7. The recursion relation
Fn+2 = Fn + Fn+1
with the initial values F0 = F1 = 1 provides the Fibonacci sequence1, 1, 2, 3, 5, 8, 13, 21, 34, 55, . . .. A generalization of the Fibonacci sequenceis the q-analogue of the sequence defined by the recursion relation
Fn+2(q) = Fn(q) + qFn+1(q)
28 Problems and Solutions
and the initial condition F0(q) = 1 and F1(q) = q. Here, q is a real orcomplex number.(i) Give the first five terms of the sequence.(ii) Find a generating function of Fn(q).(iii) Find an explicit expression for Fn(q).
Problem 8. (i) Find the linear map f : 0, 1 → −1, 1 such that
f(0) = −1, f(1) = 1. (1)
(ii) Find a linear map g : −1, 1 → 0, 1 such that
g(−1) = 0, g(1) = 1. (2)
This is obviously the inverse map of f .
Problem 9. Consider the differentiable function f : R→ R
f(x) =1− cos(2πx)
x
where using L’Hospital f(0) = 0.(i) Find the zeros of f .(ii) Find the maxima and minima of f .
Problem 10. Given two manifolds M and N , a bijective map φ from Mto N is called a diffeomorphism if both φ : M → N and its inverse φ−1 aredifferentiable. Let f : R→ R be given by the analytic function
f(x) = 4x(1− x)
and the analytic function g : R→ R be given by
g(x) = 1− 2x2.
(i) Can one find a diffeomorphism φ : R→ R such that
g = φ f φ−1 ?
(ii) Consider the diffeomorphism ψ : R→ R
ψ(x) = sinh(x).
Calculate ψ g ψ−1.
Maps 29
Problem 11. Consider the polynomial p(x) = x3−3x+3. Show that forany positive integer N , there is an initial value x0 such that the sequencex0, x1, x2, . . . obtained from Newton’s method
xn+1 = xn −p(xn)p′(xn)
=2x3
n − 33(x2
n − 1), n = 0, 1, 2, . . .
has period N .
Problem 12. Consider the linear recursion equation for F (n) with n =0, 1, 2, . . .
(n+ 1)F (n+ 1)− nF (n− 1) + F (n) = 0
where F (0) = 0, F (1) = 1. We define
f(z) :=∞∑n=1
F (n)zn
where z is an undeterminate. Find the differential equation for f with theinitial condition. Solve the differential equation.
Problem 13. Let f : Rn → Rn be an analytic function. Consider themap
xj = f(xj−1) = · · · = f(x0).
To study the evolution of phase-space distributions, we can introduce theevolution operator U(x′,x, j) such that any initial phase-space distributionρ(x, 0) evolves into
ρ(x′; j) =∫
Ω
U(x′,x; j)ρ(x, 0)dx
where Ω is the phase space area. Find U(x′,x; j).
Problem 14. Find the solution of the recursion relation
aj+2 =23aj+1 +
13aj , j = 1, 2, . . .
with a1 = 5, a2 = 1. Calculate limj→∞ aj .
Problem 15. Solve the linear difference equation
xt+1 = 2xt − t, t = 0, 1, 2, . . .
with the initial value x0 = 1.
30 Problems and Solutions
Problem 16. LetN0 := 0, 1, 2, . . .
Find an invertible map f : N0 × N0 × N0 → N0 with
f(0, 0, 0) = 0, f(0, 0, 1) = 1, f(0, 1, 0) = 2, f(0, 1, 1) = 3,
f(1, 0, 0) = 4, f(1, 0, 1) = 5, f(1, 1, 0) = 6, f(1, 1, 1) = 7
etc. Find the inverse function.
Problem 17. Consider the second order difference equation
xt+2 = xt+1xt
with the initial values x0 = a, x1 = b. Give the solution. What are thefixed points?
Problem 18. Let A be a set. Suppose that it is possible to define subsetsA1, A2, . . . of A which have the properties that(i) the sets are pairwise disjoint; that is Ai ∩ Aj = ∅ for all i, j = 1, 2, . . .and j 6= i
(ii) A1 ∪A2 ∪ · · · = A.
Then the family of sets A1, . . . is called a partition of A.
Let S = 1, 2, 3, . . . , 9 and A = 1, 4, 7. B = 2, 3, 5, 6, C = 7, 8, 9. IsA,B,C a partition of S?
Problem 19. Let S be a finite set. Let P(S) be the power set of S. ThenP(S) has 2|S| elements.(i) Show that the composition
A B := (A ∪B) ∩ (A ∪ B)
defines a group, where A,B ∈ P(S). Here denotes the complement.(ii) Show that the composition
A B := (A ∩B) ∪ (A ∩ B)
defines a group, where A,B ∈ P(S). Here denotes the complement.
Problem 20. Find a function f : [0, 1]→ (0, 1] that is one-to-one.
Maps 31
Problem 21. Let n ∈ N. Consider the map
ut+1 =12
(ut +
n
ut
), t = 0, 1, 2, . . .
given the initial value u0 with u0 > 0. Show that
ut+1 −√n
ut+1 +√n
=(ut −
√n
ut +√n
)2
.
Show that ut →√n as t→∞. Show that
√n is a fixed point.
Problem 22. Let 0 ≤ α < π/4. Consider the transformation
X(x, y, α) =1√
cos(2α)(x cos(α) + iy sin(α))
Y (x, y, α) =1√
cos(2α)(−ix sin(α) + y cos(α)).
(i) Show that X2 + Y 2 = x2 + y2.(ii) Do the matrices
1√cos(2α)
(cos(α) i sin(α)−i sin(α) cos(α)
)form a group under matrix multiplication?
Problem 23. A smooth function f : Rn → R is called homogeneous ofdegree r if
f(εx1, . . . , εxn) = εrf(x1, . . . , xn). (1)
Show that (Euler’s identity)
n∑j=1
∂f
∂xjxj = rf.
Problem 24. Find the invariance group of the function f : R→ R
f(x) = sin(x).
Problem 25. Consider the analytic function f : R→ R
f(x) = cos(sin(x))− sin(cos(x)).
32 Problems and Solutions
Show that f admits at least one critical point. By calculating the secondorder derivative find out whether this critical point refers to a maxima orminima.
Problem 26. Let f : R2 → R, g : R2 → R be analytic functions. Wedefine the star product
f(x1, x2) ? g(x1, x2) :=
limx′1→x1,x′2→x2
exp(
∂
∂x1
∂
∂x′2− ∂
∂x′1
∂
∂x2
)f(x1, x2)g(x′1, x
′2).
Letf(x1, x2) = sin(x1 + x2), g(x1, x2) = sin(x1 − x2).
Find the star product.
Problem 27. LetN1, N2 be given positive integers. Let n1 = 0, 1, . . . , N1−1, n2 = 0, 1, . . . , N2 − 1. There are N1 ·N2 points. The points (n1, n2) area subset of N0 × N0 and can be mapped one-to-one onto a subset of N0
j(n1, n2) = n1N2 + n2
where j = 0, 1, . . . , (N1 − 1)(N2 − 1). Find the inverse of this map.
Problem 28. Let a ∈ R. Consider the transformation
t(t, x) =1aeax sinh(at), x(t, x) =
1a
(eax cosh(at)− 1)
with lima→0 t = t, lima→0 x = x. Find the inverse of the transformation.
Problem 29. Show that the map f : (0, 1)→ R
x 7→ f(x) =x− 1/2x(x− 1)
is bijective. Note that f(1/2) = 0 and f(1/4) = 4/3.
Problem 30. Consider the map
f : R→ (x1, x2) : x21 + (x2 − 1)2 = 1 ∧ x2 6= 2
defined by
f(t) =(
4tt2 + 4
,2t2
t2 + 4
).
Maps 33
Show that f is bijective. Show that f and f−1 are continuous.
Problem 31. Let N ≥ 2 and δ := 1/(N + 1). Solve the linear one-dimensional linear difference equation
xj+1 − 2xj + xj−1 = −12δ4(j + 1)2, j = 0, 1, . . . , N − 1
with the boundary conditions x−1 = xN = 0.
Problem 32. (i) Consider the analytic map f : R→ R
f(x) = x(1− x).
Find the fixed points of f . Are the fixed points stable? Prove or disprove.(ii) Let x0 = 1/2 and iterate
f(x0), f(f(x0)), f(f(f(x0))), . . .
Does this sequence tend to a fixed point? Prove or disprove.(iii) Let x0 = 2 and iterate
f(x0), f(f(x0)), f(f(f(x0))), . . .
Does this sequence tend to a fixed point? Prove or disprove.(iv) Find the critical points of f . Then find the extrema of f .(v) Find the roots of f , i.e. solve f(x) = 0.(vi) Find the minima of the function g(x) := |f(x)|.
Problem 33. Let c > 0 and 0 ≤ v1 < c, 0 ≤ v2 < c. We define thecomposition
v1 ? v2 :=v1 + v2
1 + v1v2/c2.
Is the composition associative?
Problem 34. Solve the initial value problem of the system of first orderdifference equations
x1,t+1 =−2x1,t
x2,t+1 =89x1,t − x2,t.
The first difference equation is independent of x2,t and we find the solutionof the initial value problem
x1,t+1 = (−2)tx1,0.
34 Problems and Solutions
Problem 35. Let mA, mB , RA, RB be the masses and centre-of masscoordinates of mass A and B, respectively. We set m = mA + mB . Findthe inverse of the transformation
r(RA,RB) = RA −RB , R(RA,RB) =1m
(mARA +mBRB).
Problem 36. Consider the map f : N0 × N0 → N0 × N0
f1(n1, n2) = |n1 − n2|, f2(n1, n2) = n1 + n2.
Is the map invertible?
Problem 37. Let n,m ≥ 0. Consider the differential operators
Kn :=n∑j=0
uj(x)dj
dxj, Lm :=
m∑j=0
vj(x)dj
dxj
where the uj(x)’s and vj(x)’s are smooth functions. If the two differentialoperators Kn and Lm commute, then there is a nonzero polynomial R(z, w)such that R(Kn, Lm) = 0. The curve Γ defined by R(z, w) = 0 is called thespectral curve. If we consider the eigenvalue problem
Knψ = zψ, Lmψ = wψ
then (z, w) ∈ Γ.(i) Let (α ∈ R)
K =(d2
dx2+ x3 + α
)2
+ 2x,
L =(d2
dx2+ x3 + α
)3
+ 3xd2
dx2+ 3
d
dx+ 3x(x2 + α).
Show that K and L commute with w2 = z3 − α.(ii) Show that
K =(d3
dx3+ x2 + α
)2
+ 2d
dx
and
L =(d3
dx3+ x2 + α
)3
+ 3d4
dx4+ 3(x2 + α)
d
dx+ 3x
commute.
Problem 38. Let A, B, C be nonempty sets. Let f : A → B andg : B → C are invertible maps. Then the composition of the functions
Maps 35
g f : A→ C in invertible and (g f)−1 = f−1 g−1. Function compositionis associative.
Problem 39. Consider the map f : (−1, 1)→ R
f(x) =x
1− |x|.
Show that the map f−1 : R→ (−1, 1) is given by
f−1(x) =x
1 + |x|.
Problem 40. Let τ ∈ R. Show that the curves
x(τ) =(x1(τ)x2(τ)
)=(
1τ
), y(τ) =
(y1(τ)y2(τ)
)=(ττ2
)are linearly independent.
Problem 41. Let r, k be positive integers with r ≤ k and M =x1, . . . , xr and N = y1, . . . , yk. Then there
k!(k − r)!
surjective maps f : M → N . Find all the maps for r = k = 2.
Problem 42. Consider the first order difference equation
xτ+1xτ + 2xτ + xτ+1 = 0, τ = 0, 1, 2, . . .
with x0 6= 0. Show that this difference equation can be linearized withxτ = 1/yτ where x0 6= 0. First show that x∗ = 0 and x∗ = −3 are solutionsof the nonlinear difference equation.
2.2 Supplementary Problems
Problem 1. Let x,y ∈ R3 and × be the vector product
x× y =
x2y3 − x3y2
x3y1 − x1y3
x1y2 − x2y1
.
36 Problems and Solutions
Hence the vector product is a map from R3 × R3 into R3. Show that themap is differentiable.
Problem 2. Study the map (Legendre map) f : R2 → R2
f1(x1, x2) = ex1 cos(x2), f2(x1, x2) = −ex1 sin(x2).
Find the functional determinant.
Problem 3. Consider the four symbols A,B,C,D. The Rudin-Shapirosubstitution is
A 7→ AC, B 7→ DC, C 7→ AB, D 7→ DB.
Find the sequence starting of with B.
Problem 4. Solve the difference equation
xt+1 = txt + t2 mod 2
with t = 0, 1, 2, . . . and x0 = 1.
Problem 5. Study the map f : Z× Z× Z→ Z× Z× Z
f1(x1, x2, x3) = x1 − x2x3, f2(x1, x2, x3) = −x2 + x1x3,
f3(x1, x2, x3) = x3 − x1x2.
(i) Show that the fixed points are
(0, 0, 0), (1, 0, 0), (0, 0, 1), (−1, 0, 0), (0, 0,−1).
(ii) Show that (x1 = 2, x2 = 2, x3 = 2) provides a periodic orbit.(iii) Show that (x1 = 1, x2 = 2, x3 = 1) provides an eventually periodicorbit.(iv) Is the map invertible?
Problem 6. Consider the map f : [0, 1]3 → [0, 1]3 given by
f1(x1, x2, x3) = 2x1, f2(x1, x2, x3) =12x2, f3(x1, x2, x3) =
12x3
for 0 ≤ x1 ≤ 1/2 and
f1(x1, x2, x3) = 2x1−1, f2(x1, x2, x3) =12
(x2+1), f3(x1, x2, x3) =12
(x3+1)
Maps 37
for 1/2 < le ≤ 1. Find the orbit for x1 = x2 = x3 = 1/3.
Problem 7. Consider the map f : R2 → R2
f1(x1, x2) = sinh(x1) sin(x2), f2(x1, x2) = cosh(x1) cos(x2).
Find the fixed points of the map and study their stability.
Problem 8. Study the system of difference equations
x0,t+1 = 2x0,t − x1,t − x2,t − x3,t − x4,t
x1,t+1 =−12x1,t + x0,t
x2,t+1 =−12x2,t + x0,t
x3,t+1 =−12x3,t + x0,t
x4,t+1 =−12x4,t + x0,t
with the initial conditions
x0,0 = 1, x1,0 = x2,0 = x3,0 = x4,0 = 0.
Problem 9. Consider the difference equation
xt+1 = 2xt − 2, t = 0, 1, 2, . . .
where x0 = 1. Find the general solution. Show that x∗ = 2 is a particularsolution (the fixed point).
Problem 10. The Fibonacci numbers can be defined by(xt+2
xt+1
)=(
1 11 0
)(xt+1
xt
)with x0 = x1 = 1. Find the eigenvalues of the 2 × 2 matrix. Study thesequence
xt+4
xt+3
xt+1
xt
= (A⊗A)
xt+3
xt+2
xt+1
xt
with x0 = x1 = x2 = x3 = 1 and ⊗ denotes the Kronecker product.
38 Problems and Solutions
Problem 11. Let x0 ∈ R+ and r ∈ R+. Show that
xj+1 =12
(xj +
r
xj
), j = 0, 1, 2, . . .
tends to√r.
Problem 12. Let A = (ajk) be a 3×3 matrix with det(A) 6= 0. Considerthe map f : R2 → R2 given by
f1(x1, x2) =a11x1 + a12x2 + a13
a31x1 + a32x2 + a33, f2(x1, x2) =
a21x1 + a22x2 + a23
a31x1 + a32x2 + a33,
Find the inverse of the map.
Problem 13. Consider the map f : R2 → R2
f1(x1, x2) = x1 + x2, f2(x1, x2) = x1x2
or written as difference equation
x1,τ+1 = x1,τ + x2,τ , x2,τ+1 = x1,τx2,τ .
Find the fixed points of the map. Then solve the difference equation forx1,0 = 1/4, x2,0 = 1/5.
Problem 14. Show that(∂F (x, y)∂x
)2
+(∂F (x, y)∂y
)2
expressed in polar coordinates x = r cos(θ), y = r sin(θ) takes the form(F (x(r, θ), y(r, θ)
∂r
)2
+1r
(∂F (x(r, θ), y(r, θ))
∂θ
)2
.
Problem 15. Let r > 1 and τ = 0, 1, 2, . . .. Study the system of differenceequation
x1,τ+1 =1rex2,τ , x2,τ+1 = re−x1,τ
with the initial conditions x1,0 = x2,0 = 0. Are there fixed points?
Problem 16. Let m ∈ N. Solve the recurrence relation
f(n) = 1 + f(n/2) for n ≥ 2, n = 2m
Maps 39
with the initial condition f(1) = 1.
Problem 17. Study the initial value problem second order differenceequation
xτ+2 = xτ+1 + xτ − xτ+1xτ
with x0 = 1, x1 = 1/2.
Problem 18. A set A is equilvalent to the set B (one writes A ∼ B) ifthere is a mapping f : A→ B which is (1, 1) and onto. Show that R2 ∼ R.Utilize that
(·a1a2a3 . . . , ·b1b2b3 . . .)↔ ·a1b1a2b2 . . .
defines a (1, 1) map between pairs of decimal expansions and single expres-sions of numbers in the interval (0, 1).
Problem 19. Let Ω = [0, 1), T (x) = 2x (mod 1), and µ the Lebesguemeasure. Show that T is ergodic.
Problem 20. (i) Let c > 0. Study the difference equation
xτ+1 =xτ
c+ xτ, τ = 0, 1, . . .
with x0 > 0. First find the fixed points.(ii) Study the difference equation
xτ+1 =6xτ
(1 + xτ )2
with x0 > 0. First find the fixed points.
Problem 21. Let x ∈ [0, 1] and bac be the integer part of a. Show thatthe sequence xt (t = 0, 1, . . .) given by
x0 = 0, and xt+1 = xt +1
2t+1b2t+1(x− xt)c
converges to x.
Problem 22. The Kustaanheimo-Stiefel transformation is defined by themap from R4 (coordinates u1, u2, u3, u4) to R3 (coordinates x1, x2, x3)
x1(u1, u2, u3, u4) = 2(u1u3 − u2u4)x2(u1, u2, u3, u4) = 2(u1u4 + u2u3)x3(u1, u2, u3, u4) = u2
1 + u22 − u2
3 − u24
40 Problems and Solutions
together with the constraint
u2du1 − u1du2 − u4du3 + u3du4 = 0.
(i) Show that
r2 = x21 + x2
2 + x23 = u2
1 + u22 + u2
3 + u24.
(ii) Show that
∆3 =14r
∆4 −1
4r2V 2
where
∆3 =∂2
∂x21
+∂2
∂x22
+∂2
∂x23
, ∆4 =∂2
∂u21
+∂2
∂u22
+∂2
∂u23
+∂2
∂u24
and V is the vector field
V = u2∂
∂u1− u1
∂
∂u2− u4
∂
∂u3+ u3
∂
∂u4
(iii) Consider the differential one form
α = u2du1 − u1du2 − u4du3 + u3du4.
Find dα. Find LV α, where LV (.) denotes the Lie derivative.(iv) Let g(x1(u1, u2, u3, u4), x2(u1, u2, u3, u4), x3(u1, u2, u3, u4)) be a smoothfunction. Show that LV g = 0.
Problem 23. The Kustaanheimo-Stiefel map R4 → R3 ((u1, u2, u3, u4)→(x1, x2, x3)) is given by
x1 = 2(u1u3 − u2u4), x2 = 2(u1u4 + u2u3), x3 = u21 + u2
2 − u23 − u2
4
together with the constraint
α ≡ u2du1 − u1du2 − u4du3 + u3du4 = 0.
Show that applying the Kustaanheimo-Stiefel map the Laplacian operator∆3 in R3 can be written as
∆3 =14r
∆4 −1
4r2V 2
where
r = (x21+x2
2+x23)1/2 = u2
1+u22+u2
3+u24, V = u2
∂
∂u1−u1
∂
∂u2−u4
∂
∂u3+u3
∂
∂u4.
Chapter 3
Functions
3.1 Solved ProblemsProblem 1. Let fj : Rn → R, j = 1, 2, . . . , n be real-valued functionswith continuous second-order partial derivatives everywhere on Rn. Sup-pose that there are constants cij such that
∂fi∂xj− ∂fj∂xi
= cij
for all i and j, 1 ≤ i ≤ n. Prove that there is a function g : Rn → Rsuch that fi + ∂g/∂xi is linear for all i = 1, 2, . . . , n. A linear functionp : Rn → R is of the form
p(x) = a0 + a1x+ a2x2 + · · ·+ anxn.
Problem 2. Let f : Rn → R be a differentiable function and x ∈ Rn.Consider the invertible n × n matrix R and the transformation x′ = Rx.The transformation operator PR associated with the invertible matrix R isdefined by
PRf(x) := f(R−1x).
Note that the operator acts upon the coordinates x and not on the argumentof f . This means PRPSf(x) = f(S−1R−1x), where S is another invertiblen× n matrix. Show that
PRPSf(x) = PRSf(x).
41
42 Problems and Solutions
Problem 3. A ratio list is a finite list of positive numbers, (r1, r2, . . . , rn).An iterated function system realizing a ratio list (r1, r2, . . . , rn) in a metricspace S is a list (f1, f2, . . . , fn), where fj : S → S is a similarity withratio rj . A nonempty compact set K ⊆ S is an invariant set for theiterated function system (f1, f2, . . . , fn) iff K = f1(K)∪f2(K)∪· · ·∪fn(K).The triadic Cantor set is an invariant set for an iterated function systemrealizing the ratio list (1/3, 1/3). The Sierpinski gasket is an invariant setfor an iterated system realizing the ratio list (1/2, 1/2, 1/2). The dimensionassociated with a ratio list (r1, r2, . . . , rn) is the positive number s such thatrs1 + rs2 + · · · + rsn = 1. Let (r1, r2, . . . , rn) be a ratio list. Suppose eachrj < 1. Show that there is a unique nonnegative number s satisfying
n∑j=1
rsj ≡n∑j=1
es ln(rj) = 1.
The number s is 0 iff n = 1.
Problem 4. The arc length of the equilateral hyperbola
h(t) =√t2 − 1, t ≥ 1
starting at t = 1 is given by
Lh(x) =∫ x
1
√2t2 − 1t2 − 1
dt
as a function of the terminal point t = x. The tangent line to the hyperbolaat t = x is
Th(t) =√x2 − 1 +
x√x2 − 1
(t− x)
whose intersection with the t-axis is t = 1/x (t ∈ (0, 1)). The line
Nh(t) = −√x2 − 1x
t
is perpendicular to Th passing through the origin.(i) Find the point Ph where the lines Th and Nh intersect.(ii) Calculate the distance from (x, h(x)) to the common point Ph.
Problem 5. Given a differentiable function f . The logarithmic derivativeof f is defined as
1f
df
dx.
Functions 43
When f is a real function of real variables and takes strictly positive valuesthen the chain rule provides
d
dxln(f(x)) =
1f
df
dx.
The logarithmic derivative has the following properties.1. The logarithmic derivative of the product of functions is the sum of theirlogarithmic derivatives
1fg
d
dx(fg) =
1f
df
dx+
1g
dg
dx
2. The logarithmic derivative of the quotient of functions is the differenceof their logarithmic derivatives
1f/g
d
dx(f/g) =
1f
df
dx− 1g
dg
dx
3. The logarithmic derivative of the α-th power of a function f is α timesthe logarithmic derivative of the function
1fα
d
dx(fα) = α
1f
df
dx
4. The logarithmic derivative of the exponential of a function equals thederivative of a function
1ef
d
dxef =
df
dx
(i) Find the logarithmic derivative of f : R→ R, f(x) = cosh(x).(ii) Let f be a meromorphic function in the open and conncted set D ⊆ C.Let G ⊆ D be a region such that its closure G ⊆ D and its boundary ∂G isa continuous curve not containing a zero or pole of f . Let N be the numberof zeros of f lying inside G and P be the number of poles of f lying insideG. Then
N − P =1
2πi
∫∂G
1f(z)
df(z)dz
dz
where ∂G is an oriented boundary of G. Calculate the left and right-handside for the function f(z) = 1/z and G = (x, y) : x2 + y2 ≤ 1 .
Problem 6. Let f be an analytic function. Let p ∈ N and α ∈ R. Showthat (
d
dx+ αx
)pf ≡ exp
(−αx
2
2
)dp
dxp
(exp
(αx2
2
)f
).
44 Problems and Solutions
Problem 7. Consider
K(t, s) =
1 if 0 ≤ t ≤ s0 if s ≤ t ≤ 1 .
Show that this kernel satisfies the functional equation
K(cu+ a, t) = K
(u,t− ac
)where c > 0.
Problem 8. Find a continuous differentiable function f : R → R whichhas no fixed points and no critical points.
Problem 9. Let a, b, p, q, r ∈ R, b > a, a < p, p < r, r < q, q < b and
p− a = r − p, q − r = b− q.
In fuzzy logic the following membership function plays an important role
f(x; a, b, r, p, q) =
0 x ≤ a2m−1((x− a)/(r − a))m a < x ≤ p1− 2m−1((r − x)/(r − a))m p < x ≤ r1− 2m−1((x− r)/(b− r))m r < x ≤ q2m−1((b− x)/(b− r))m q < x < b0 x ≥ b
where m is the fuzzifier. Im most cases m = 2. Where are the crossoverpoints? What is the value at the centre?
Problem 10. Let f : R2 → R2 be the mapping of (x, y) → (u, v) givenby
u(x, y) = ex cos(y), v(x, y) = ex sin(y).
What is the range of f? Show that the Jacobian determinant is not zero atany point of R2. Thus every pont of R2 has a neighbourhood in which f isone-to-one. Nevertheless f is not one-to-one on R2. What are the imagesunder f of lines parallel to the coordinate axes?
Problem 11. Let x, y ∈ R. Calculate the square root√x2 + y2 − 2xy.
Functions 45
Problem 12. A criterion for linearly independence of functions f0, f1, . . . , fn ∈Cn[a, b] is that the Wronski determinant is nonzero
det
f0 f1 . . . fnf ′0 f ′1 . . . f ′n...
.... . .
...f
(n)0 f
(n)1 . . . f
(n)n
6= 0.
Here ′ denotes derivative. Apply this criterion to the functions
f0(x) = cos(x), f1(x) = sin(x).
Problem 13. For a sphere of radius r and mass density ρ the mass thatmust be concentrated at its centre is (λ ≥ 0)
M(λ) =4πrρλ2
(cosh(λr)− sinh(λr)/(λr)).
Find limλ→0M(λ).
Problem 14. Let −1 < a < 1. Find the inverse of the transformation
λ(z) =z − a1− az
.
Problem 15. Consider the Jacobi elliptic functions
sn(x, k), cn(x, k), dn(x, k)
where x ∈ R and k2 ∈ [0, 1]. We have
sn(x, 0) = sin(x), cn(x, 0) = cos(x), dn(x, 0) = 1
and
sn(x, 1) =ex − e−x
ex + e−x= tanh(x), cn(x, 1) = dn(x, 1) =
2ex + e−x
≡ sech(x).
(i) Find an expression using Jacobi elliptic functions that interpolates be-tween sin(x) for k = 0 and sinh(x) for k = 1.(ii) Find an expression using Jacobi elliptic functions that interpolates be-tween cos(x) for k = 0 and cosh(x) for k = 1.(iii) Use this result to interpolate between the matrices(
cos(x) sin(x)− sin(x) cos(x)
)and
(cosh(x) sinh(x)sinh(x) cosh(x)
).
46 Problems and Solutions
Problem 16. Let f1, f2 be differentiable functions and f1(x) > 0, f2(x) >0 for all x. Let f(x) = f1(x)f2(x). Find
d
dx(ln(f(x)).
Problem 17. Let x ∈ R. The sequence of functions fk(x) is definedby f1(x) = cos(x/2) and for k > 1 by
fk(x) = fk−1(x) cos(x/2k).
Thusfk(x) = cos(x/2) cos(x/22) · · · cos(x/2k).
Obviously, we have fk(0) = 1 for every k. Calculate limk→∞ fk(x) as afunction of x for x 6= 0.
Problem 18. (i) Consider the transformation in R3
x0(a, θ1) = cosh(a)x1(a, θ1) = sinh(a) sin(θ1)x2(a, θ1) = sinh(a) cos(θ1)
where a ≥ 0 and 0 ≤ θ1 < 2π. Find
x20 − x2
1 − x22.
(ii) Consider the transformation in R4
x0(a, θ1, θ2) = cosh(a)x1(a, θ1, θ2) = sinh(a) sin(θ2) sin(θ1)x2(a, θ1, θ2) = sinh(a) sin(θ2) cos(θ1)x3(a, θ1, θ2) = sinh(a) cos(θ2)
where a ≥ 0, 0 ≤ θ1 < 2π and 0 ≤ θ2 ≤ π. Find
x20 − x2
1 − x22 − x2
3.
Extend the transformation to Rn.
Problem 19. A fixed charge Q is located on the z-axis with coordinatesra = (0, 0, d/2), where d is interfocal distance of the prolate spheroidal
Functions 47
coordinates
x(η, ξ, φ) =12d((1− η2)(ξ2 − 1))1/2 cosφ
y(η, ξ, φ) =12d((1− η2)(ξ2 − 1))1/2 sin(φ)
z(η, ξ, φ) =12dηξ
where −1 ≤ η ≤ +1, 1 ≤ ξ ≤ ∞, 0 ≤ φ ≤ 2π. Express the Coulombpotential
V =Q
|r− ra|in prolate spheroidal coordinates.
Problem 20. Toroidal coordinates are defined by
x1(α, β, φ) =c sinh(α) cos(φ)
cosh(α)− cos(β)
x2(α, β, φ) =c sinh(α) sin(φ)
cosh(α)− cos(β)
x3(α, β, φ) =c sin(β)
cosh(α)− cos(β).
Use the L’Hosptial rule to find x1, x2, x3 for α→ 0 and β → 0.
Problem 21. Let c, θ ∈ R and
f(θ) = c(eiθ + e−iθ).
Calculate exp(f(θ)).
Problem 22. A continuous function f : R2 → R is called an alternatingfunction if
f(x, y) = −f(y, x).
Give an example of an analytic alternating function. Find the minima andmaxima of the function.
Problem 23. Consider the functions
j1(x) =1x2
(sin(x)− x cos(x))
j2(x) =1x3
((3− x2) sin(x)− 3x cos(x)).
48 Problems and Solutions
Use the L’Hospital rule to find j1(0) and j2(0).
Problem 24. Find the invariance group of the function f : R→ R
f(x) = sin(x).
Problem 25. Let a > 0. Consider the transformation
u(x, y) =a sin(2ax)
2(sin2(ax) + sinh2(ay)), v(x, y) =
a sinh(2ay)2(sin2(ax) + sinh2(ay))
.
Find the inverse transformation.
Problem 26. (i) Find an analytic function f : R → R that has no fixedpoint and no critical point. Draw the function.(ii) Find an analytic function f : R→ R that has no fixed point and exactlyone critical point at x = 0. Draw the function.
Problem 27. Let f : R → R be an analytic function. Calculate thecommutator [
cos(x)d
dx, sin(x)
d
dx
]f.
Problem 28. (i) Consider the analytic function f : R2 → R2
f1(x1, x2) = sinh(x2), f2(x1, x2) = sinh(x1).
Show that this function admits the (only) fixed point (0, 0). Find thefunctional matrix at the fixed point(
∂f1/∂x1 ∂f1/∂x2
∂f2/∂x1 ∂f2/∂x2
)∣∣∣∣(0,0)
.
(ii) Consider the analytic function g : R2 → R2
g1(x1, x2) = sinh(x1), g2(x1, x2) = − sinh(x2).
Show that this function admits the (only) fixed point (0, 0). Find thefunctional matrix at the fixed point(
∂g1/∂x1 ∂g1/∂x2
∂g2/∂x1 ∂g2/∂x2
)∣∣∣∣(0,0)
.
(iii) Multiply the two matrices found in (i) and (ii).
Functions 49
(iv) Find the composite function h : R2 → R2
h(x) = (f g)(x) = f(g(x)).
Show that this function also admits the fixed point (0, 0). Find the func-tional matrix at this fixed point(
∂h1/∂x1 ∂h1/∂x2
∂h2/∂x1 ∂h2/∂x2
)∣∣∣∣(0,0)
.
Compare this matrix with the matrix found in (iii).
Problem 29. An approximation of e−x (x ≥ 0) as a rational polynomialusing a 3rd order Pade approximation is given by
e−x ≈ 1− x/2 + x2/10− x3/1201 + x/2 + x2/10 + x3/120
.
Note that e−x ≥ 0 for all x. For which value of x > 0 does the right-handside takes negative values?
Problem 30. Let x ∈ R. Consider the function
f(x) =x− ix+ i
.
Find f(0), f(1), f(−1) and f(x→∞). Is there an inverse?
Problem 31. Consider the function
f(x) = x3 − x2.
Find an approximation of the derivative of f by using
f ′(x) ≈ f(x− 2h)− 8f(x− h) + 8f(x+ h)− f(x+ 2h)12h
− 130h4f (5)(ξ)
f ′(x) ≈ −3f(x− h)− 10f(x) + 18f(x+ h)− 6f(x+ 2h) + f(x+ 3h)12h
+120h4f (5)(ξ)
f ′(x) ≈ −25f(x) + 48f(x+ h)− 36f(x+ 2h) + 16f(x+ 3h)− 3f(x+ 4h)12h
−15h4f (5)(ξ)
where x ≤ ξ ≤ x+ h.
Problem 32. Let z ∈ C. The Airy functions Ai(z) and Bi(z) are definedas sums of the power series
Ai(z) = c1f(z)− c2g(z)
Bi(z) =√
3(c1f(z) + c2g(z))
50 Problems and Solutions
where
f(z) = 1 +13!z3 +
1 · 46!
z6 +1 · 4 · 7
9!z9 + · · ·
g(z) = z +24!z4 +
2 · 57!
z7 +2 · 5 · 8
10!z10 + · · ·
with
c1 =1
2πΓ(
13
)3−1/6, c2 =
12π
Γ(
23
)31/6.
Show that the radius of convergence of these infinite series is infinite.
Problem 33. Find the inverse of the function y = tanh(x). Note thaty ∈ (−1, 1).
Problem 34. Consider the analytic function f : R→ R
f(x) =∞∑j=0
xj
(j + 1)!≡ ex − 1
x.
Find the fixed points and critical points of the function. Note that f(0) = 1.
Problem 35. Find non-negative analytic functions f : R→ R such that
f(0) = 0, f(1) = 1, f(2) = 0.
Problem 36. The Euler dilogarithm function Li2(x) is defined for x ∈(0, 1) as
Li2(x) =∞∑j=1
xj
j2= −
∫ x
0
ln(1− t)t
dt.
It can be analytically continued to the complex plane with the branch cutfrom 1 to ∞ along the real axis using the integral. Calculate Li2(1/2).
Problem 37. Let Ai be the Airy function and Jn, In are the Bessel andmodified Bessel functions of order n. Show that
d2Ai
dx2= xAi(x)
Ai(x) =13x1/2(I−1/3(z)− I1/3(z))
Ai(−x) =13x1/2(J1/3(z) + J−1/3(z))
Functions 51
where z := 2x3/2/3.
Problem 38. Find a continuous function f : R→ R such that
f(x) = f(−x), f(x) = f(x+ 2π), f(0) = 0.
Problem 39. Let N be an integer and N ≥ 2. The generalized hyperbolicfunction for a given N is defined as
f(N)j (x) :=
∞∑k=0
xj+kN
(j + kN)!, j = 0, 1, . . . , N − 1.
The functions f (N)j are analytic.
(i) Show thatf
(N)j (x) = f
(N)j+N (x)
i.e. f (N)j is periodic with respect to the index j.
(ii) Show thatd
dxf
(N)j (x) = f
(N)j−1(x).
(iii) Let ωN = 1, i.e. ω is the N -th primitive root of unity. Show that
f(N)j (x) =
1N
N−1∑k=0
ω−jk exp(ωkx).
Problem 40. Let n = 2, 3, 4, . . .. Find
Γ(n/2)Γ(n)
where Γ denotes the gamma function.
Problem 41. Consider the hyperspherical coordinates
x1(θ, φ, ψ) = cos(θ)x2(θ, φ, ψ) = sin(θ) cos(φ)x3(θ, φ, ψ) = sin(θ) sin(φ) cos(ψ)x4(θ, φ, ψ) = sin(θ) sin(φ) sin(ψ)
with x21+x2
2+x23+x2
4 = 1. Show that the angular distance can be calculatedas
cos(djk) =
52 Problems and Solutions
cos(θj) cos(θk)+sin(θj) sin(θk)(cos(φj) cos(φk)+sin(φj) sin(φk) cos(ψj−ψk)).
Problem 42. Consider the function
f(x) = 1 + x+ 2x2 + 3x3.
Find the second order derivative of f at a = 1 applying
f ′′(a) = limε→0
f(a+ ε)− 2f(a) + f(a− ε)ε2
.
Problem 43. Let −1 < x < 1 and∞∑j=0
ajxj =
1√1− x
.
Find the expansion coefficients aj .
Problem 44. Show that the function
f(x) = cos(x) cosh(x) + 1
has infinitely many roots, i.e. solutions of f(x) = 0. What happens for xlarge?
Problem 45. The Jacobi elliptic functions sn(x, k), cn(x, k), dn(x, k)with k ∈ [0, 1] and k2 + k′2 = 1 have the properties
sn(x, 0) = sin(x), cn(x, 0) = cos(x), dn(x, 0) = 1
and
sn(x, 1) =ex − e−x
ex + e−x, cn(x, 1) =
2ex + e−x
= dn(x, 1).
We define
u1(x, y, k, k′) = sn(x, k)dn(y, k′)u2(x, y, k, k′) = cn(x, k)cn(y, k′)u3(x, y, k, k′) = dn(x, k)sn(y, k′).
(i) Find u1(x, y, 0, 1), u2(x, y, 0, 1), u3(x, y, 0, 1) and calculate u21(x, y, 0, 1)+
u22(x, y, 0, 1) + u2
3(x, y, 0, 1).(ii) Find u1(x, y, 1, 0), u2(x, y, 1, 0), u3(x, y, 1, 0) and calculate u2
1(x, y, 1, 0)+u2
2(x, y, 1, 0) + u23(x, y, 1, 0).
Functions 53
Problem 46. Let m be an integer with m ≥ 1. Then the gamma functionΓ is given by
Γ(m) = (m− 1)!.
Thus Γ(1) = Γ(2) = 1, Γ(3) = 2, Γ(4) = 6, Γ(5) = 24. Furthermore
Γ(m+12
) =1 · 3 · 5 · · · · · (2m− 1)
2m√π.
Thus Γ(3/2) =√π/2, Γ(5/2) = 3
√π/4, Γ(7/2) = 15
√π/8, Γ(9/2) =
105√π/16. Calculate
Γ(n2 −
12
)Γ(n2
)for m = 5, m = 10, m = 20.
Problem 47. Let c1 > 0, c2 > 0. Consider the function f : R→ R
f(x) = x2 +c1x
2
1 + c2x2.
Find the minima and maxima. Find the fixed points.
Problem 48. Let c > 0. Consider the function f : R→ R
fc(x) =4x2
sin2(cx
2
).
Find fc(0). Show that fc has a maximum at x = 0.
Problem 49. (i) Find the minima and maxima of the analytic functionf : R→ R
f(x) = cosh(x) cos(x).
Find the fixed points.(ii) Find the minima and maxima of the analytic function f : R→ R
f(x) = sinh(x) sin(x).
Find the fixed points.
Problem 50. Let S1 and S2 be two finite sets with n1 and n2 elements,respectively. The number of functions f : S1 → S2 is given by
nn12 .
(i) Let n1 = n2 = 2. Find all possible functions.
54 Problems and Solutions
(ii) Let n1 = n2 = n. Then there are n! one-to-one functions f : S1 → S2.Let n = 3. Find all one-to-one functions.
Problem 51. Let µ > 0 and
R =
x1
x2
x3
, R′ =
x′1x′2x′3
.
(i) Show thatexp(−|R−R′|/µ)
|R−R′|=
∞∑n=0
εn cos(n(φ−φ′))∫ ∞
0
1k2 + 1/µ
Jn(kr)J(kr′) exp(−√k2 + 1/µ2|x3−x′3|)kdk
where εn = 1 for n = 0 and εn = 2 for n > 0 and Jm(kr) is ordinary Besselfunction of order m.(ii) Consider the functions
fs,k,n(R) =
√k
2πJn(kr)einφ+isx3
where 0 ≤ k <∞, −∞ < s <∞, and n = 0,±1,±2, . . .. Show that∫ 2π
0
∫ +∞
−∞dx3
∫ ∞0
fs,k,n(R)fs′,k′,n′(R)rdr = δnn′δ(s− s′)δ(k − k′).
Problem 52. Let f1, f2 : R→ R be analytic functions and
g1(x) =12
(f1(x) + f2(x)− |f1(x)− f2(x)|
g2(x) =12
(f1(x) + f2(x) + |f1(x)− f2(x)|.
Let f1(x) = sin(x) and f2(x) = cos(x). Find the maxima and minima ofg1. Find the maxima and minima of g2.
Problem 53. Let the function f be continuous on a closed interval [a, b]and is continuous differentiable in the open interval (a, b). Then there existsa point c in (a, b) such that (mean value theorem)
f(b)− f(a) = f ′(c)(b− a).
Apply the theorem to the function
f(x) = 1 + 2x+ 3x2
Functions 55
and the interval [−1, 1].
Problem 54. The plane (x1, x2, x3) : x1 + 12x2 + 1
3x3 = 1 and thecylinder (x1, x2, x3) : x2
1 + x22 = 4 intersect. Find the shortest distance
from (0, 0, 0) to this curve.
Problem 55. Consider the plane (x1, x2, x3) : 2x1 + 3x2 − x3 = 5 inthe three dimensional Euclidean space E3. Find the shortest distance from(0, 0, 0) to this plane.
Problem 56. We denotes by Q, R and C the rational, real and complexfields, respectively. Let B1 and B2 be Banach spaces. A map f : B1 → B2
is said to be additive if and only if
f(x1 + x2) = f(x1) + f(x2)
for all x1, x2 ∈ B1. Show that f is additive implies that f(qx) = qf(x) forall q ∈ Q. Show that f is additive and continuous implies that f(rx) =rf(x) for all r ∈ R.
Problem 57. Let a1d1 − b1c1 6= 0, a2d2 − b2c2 6= 0 and the functionsf1 : C→ C
f1(z) =a1z + b1c1z + d1
, f2(z) =a2z + b2c2z + d2
.
Find f2 f1, i.e. the function compositon.
Problem 58. Let x1, x2 ≥ 0. Consider
f(x1, x2) =√
1 + x1√1 + x1
√1 + x2
=∞∑j1=0
∞∑j2=0
F (j1, j2)xj11 xj22 .
Find the expansion coefficients.
Problem 59. Consider the function
f(x) =3x3
(sin(x)− x cos(x)).
Find f(0) using the L’Hospital rule.
Problem 60. Let
f(x) =sin(x)sinh(x)
.
56 Problems and Solutions
Find f(0).
Problem 61. Consider the function f : R→ R
f(x) = x3.
Show that f is not convex. Show that f is convex if we restrict to thedomain x ≥ 0.
Problem 62. Consider the convex functions f : R → R and g : R → R.Show that f + g is convex. Show that max f, g is convex.
Problem 63. Definition (Convex Set). A subset C of Rn is said tobe convex if for any a and b in C and any θ in R, 0 ≤ θ ≤ 1, the n-tupleθa + (1− θ)b also belongs to C. In other words, if a and b are in C then
θa + (1− θ)b : 0 ≤ θ ≤ 1 ⊂ C.
Definition (Convex Polyhedron). Let a1, . . . , ap be p points in Rn. Then-tuple
p∑j=1
θjaj , θj ≥ 0, j = 1, . . . , p,p∑j=1
θj = 1
is called a convex combination (or a convex sum) of a1, . . . ,ap. If X ⊂ Rnthen the set of all (finite) convex combination of points of X is called theconvex hull of X and is denoted by H(X). If X is finite, X = a1, . . . ,ap,then H(X) is called the convex polyhedron spanned by a1, . . . ,ap and isalso denoted by H(a1, . . . ,ap).
Defintion. Let S be a nonempty convex set in Rn, where Rn is the n-dimensional Euclidean space. The function f : S → R is said to be convexif
f(λx1 + (1− λ)x2) ≤ λf(x1) + (1− λ)f(x2)
for each x1,x2 ∈ S and for each λ ∈ [0, 1]. The function is said to bestrictly convex if the above inequality holds as a strict inequality for eachdistinct x1,x2 ∈ S and for each λ ∈ (0, 1).
Leta1 = (1,−1), a2 = (2, 2), a3 = (3, 1)
be points of R2. Find the convex polyhedron.
Problem 64. Show that the function φ : [0,∞)→ R defined by
φ(x) =
0 if x = 0x lnx ifx 6= 0 (1)
Functions 57
is strictly convex, i.e.,
φ(αx+ βy) ≤ αφ(x) + βφ(y) (2)
if x, y ∈ [0,∞), αβ ≥ 0, α+ β = 1 with equality only when x = y or α = 0or β = 0. By induction we get
φ
(k∑i=1
αixi
)≤
k∑i=1
αiφ(xi) (3)
if
xi ∈ [0,∞), αi ≥ 0,k∑i=1
αi = 1. (4)
Equality holds only when all the xi, corresponding to non-zero αi, are equal.
Problem 65. Let x ≥ 0 and
f(x) = x ln(x)− x+ 1. (1)
(i) Show thatf(x) ≥ 0. (2)
(ii) Show that from L’Hospital rule we find that f(0) = 1.(iii) Show that the function is convex.
Problem 66. Show that f : R→ R
f(x) = |x| (1)
is convex.
Problem 67. Let f : R2 → R be defined as
f(x, y) = ax2 + by2 + 2cxy + d
where a, b, c, d ∈ R. For what values of a, b, c, d is f concave?
Problem 68. Let r1, r2 ∈ R. Consider the map f : R2 → R2
f1(x1, x2) = r1x1(1− x1 − x2), f2(x1, x2) = r2x1x2.
Find df1, df2, df1 ∧ df2.
Problem 69. Find x = arctan(−1/√
3).
58 Problems and Solutions
Problem 70. Let f, g : R→ R be analytic functions. Find
ef(x) d2
dx2(e−f(x)g(x)).
Problem 71. Consider the map f : R2 → R2
f1(x1, x2) = sin(x21 + x2
2), f2(x1, x2) = cos(x21 + x2
2).
Find the Jacobian matrix and the Jacobian determinant.
Problem 72. Consider the map f : R→ R
f(x) = −12x2 − x+
12.
Then f(1) = −1, f(−1) = 1. Thus we have a periodic orbit with period 2.Is the orbit attracting or repelling? We set x1 = 1, x2 = −1. We have totest
|(df(x = x1)/dx)(df(x = x2)/dx)| < 1
for attracting and
|(df(x = x1)/dx)(df(x = x2)/dx)| > 1
for repelling.
Problem 73. Let α ∈ R and f(x) = eαx. Find
1h2
(f(x+ h)− 2f(x) + f(x− h)).
Then consider h→ 0.
Problem 74. Consider the analytic function f : R → R, f(x) = cos(x).The equation cos(x) = x has one solution, i.e. we have one fixed pointfor f . Consider f(f(x)) = cos(cos(x)). Does the equation cos(cos(x)) = xadmits other solutions besides the one of cos(x) = x.
Problem 75. Let a, b ∈ N. Consider the 2× 2 matrix
C =(
1 ab 1 + ab
)with det(C) = 1. Consider the map (generalized Arnold cat map)(
x1,τ+1
x2,τ+1
)= C
(x1,τ
x2,τ
)mod 1
Functions 59
where τ = 0, 1, 2, . . .. Find the eigenvalues of C which relates to the one-dimensional Liapunov exponents.
Problem 76. Consider the functions f : R→ R, g : R→ R
f(x) = 2x, g(x) = 3x2 − 1.
Find (f g)(x) = f(g(x)), (g f)(x) = g(f(x)) and the extrema of thesefunctions.
Problem 77. Let a ∈ R. Given the map f : R2 → R2
f(x1, x2) = (x1 + x2, 2x1 + ax2).
(i) Find the fixed points of the map.(ii) Find the matrix Df(x1, x2) and show that the matrix Df(x1, x2) isinvertible if and only if a 6= 2.
Problem 78. Let A, B be non-empty sets. If f : A → B is (1,1) andonto (i.e. bijective) one can define the inverse function f−1 : B → A as theunique function from B to A such that
(f f−1)(y) = y for all y ∈ B
(f−1 f)(x) = x for all x ∈ A.
The function f : (0, 1)→ R
x 7→ f(x) =x− 1/2x(x− 1)
is bijective. Find f−1. Note that f(1/2) = 0, f(1/4) = 4/3, f(3/4) = −4/3.
Problem 79. Give non-trivial functions f : R→ R which have no criticalpoints and satisfiy f(0) = 0. Trivial means f(x) = cx, where c is a constant.
Programming Problems
Problem 1. Let f1, f2, f3 : R3 → R be continuously differentiable func-tion. Find the determinant of the 3× 3 matrix A = (ajk)
ajk :=∂fj∂xk− ∂fk∂xj
.
60 Problems and Solutions
Apply computer algebra.
Problem 2. Consider the map f : R→ R given by
f(x) = x2 + x+ 1.
Find the minima and maxima of
g(x) = f(f(x))− f(x)f(x).
Apply computer algebra.
Problem 3. Consider the polynomials
f(x) = x2 + 2x+ 1, g(x) = x+ 4;
Find (f g)(x) = f(g(x)), (g f)(x) = g(f(x)) and f(g(x))− g(f(x)).
Problem 4. Let f : R → R be an analatic function. The Schwarzianderivative is defined by
S(f(x)) :=f ′′′(x)f ′(x)
− 32
(f ′′(x)f ′(x)
)2
where ′ denotes the derivative with respect to x. Let a, b, c, d ∈ R withad− bc 6= 0 and
g(x) =af(x) + b
cf(x) + d.
Show that S(g(x)) = S(f(x)). Apply computer algebra.
3.2 Supplementary Problems
Problem 1. Find
limx→0
√1 + x−
√1− x
x.
Problem 2. Consider the analytic function f : R2 → R
f(x, y) =xex(ey − 1)− yey(ex − 1)
xy(ex − ey).
Functions 61
(i) Show that f(0, 0) = 12 applying the L’Hospital rule.
(ii) Show that
f(x, x) =ex − x− 1
x2.
(iii) Show that
f(x,−x) =tanh(x/2)
x.
Problem 3. Consider the map from R4 to the vector space of 2 × 2matrices over R
f(x1, x2, x3, x4) =(x1 + x4 x2 + x3
−x2 + x3 x1 − x4
).
Find the inverse of the map. Note that the determinant of the matrix isgiven by x2
1 + x22 − x2
3 − x24.
Problem 4. Let
D = (x1, x2, x3) : x21 + x2
2 + x23 = 1, x1 ≥ 0, x2 ≥ 0, x3 ≥ 0 .
Let v be a normalized vector in R3 with nonnegative entries and A be a 3×3matrix over R with strictly positive entries. Show that the map f : D → D
f(v) =Av‖Av‖
has a fixed point, i.e. there is a normalized vector v0 such that
Av0
‖Av0‖= v0.
Problem 5. Let f, g : R→ R be analytic functions. Find
(ef(x) d2
dx2e−f(x))x.
Problem 6. Let Z be the set of integers and N0 be the set of naturalnumbers including 0. Consider the one-to-one map f : Z→ N0
f(j) =
0 if j = 02j − 1 if j is positive−2j if j is negative
.
62 Problems and Solutions
Give a C++ implementation with the class Verylong of the function f andits inverse.
Problem 7. Let f1, f2, . . . , fn be a set of convex functions from Rn →R. Show that the nonnegative linear combination
f(x) = α1f1(x) + α2f2(x) + · · ·+ αnfn(x), α1, α2, . . . , αn ≥ 0
is convex.
Problem 8. Study the map f : R2 → R2
f1(x1, x2) =√
1− sin(x2), f2(x1, x2) =√
1 + sin(x1).
First find the fixed points if there are any.
Problem 9. Let k ∈ N and x ≥ 0. Find the fixed points of the map
fk(x) =√k + x
and study their stability.
Problem 10. Consider the continuous function f : R+ → R+
f(x) =
2x x < 1max4− 2x, 1
4 x ≥ 1
Find the fixed points and study their stability.
Problem 11. Let
D = (x1, x2, x3) : x21 + x2
2 + x23 = 1, x1 ≥ 0, x2 ≥ 0, x3 ≥ 0.
Let v be a normalized vector in R3 with nonnegative entries and A be a3 × 3 matrix over R with strictely positive entries. Show that the mapf : D → D
f(v) =Av‖Av‖
has fixed point, i.e. there is a v0 such that
Av0
‖Av0‖= v0.
Problem 12. Let f, g : R → R be two analytic functions and S be theSchwarzian derivative. Show that if Sf < 0 and Sg < 0, then
S(f g) < 0.
Functions 63
Problem 13. Let x, y ∈ R+. Consider the function f : (R+)2 → R
f(x, y) =√
1 + x√1 + x+
√1 + x
.
Then f(0, 0) = 1/2 and f(x, y) + f(y, x) = 1. Set
√1 + x√
1 + x+√
1 + x=∞∑m=0
∞∑n=0
F (m.n)xmyn.
Show that F (m,n) = −F (n,m) unless m = n = 0 and F (m,m) = 0 form ≥ 1. Show that
F (0, 0) = 1/2, F (0, 1) = −F (1, 0) = −1/8, F (0, 2) = −F (2, 0) = 1/16,
F (0, 3) = −F (3, 0) = −5/128, F (1, 2) = −F (2, 1) = −1/128.
Problem 14. Consider the map f : R2 → R2
f1(x1, x2) = ex1 cos(x2), f2(x1, x2) = ex1 sin(x2).
Find the fixed points of the map.
Problem 15. Consider the function f : R3 → R3
f(x) =
f1(x1, x2, x3)f2(x1, x2, x3)f3(x1, x2, x3)
=
x1 cos(x2) sin(x3)x1 sin(x2) sin(x3)
x1 cos(x3)
.
Find the fixed points and study their stability.
Problem 16. Consider the function f : R2 → R2
f(x) =(f1(x1, x2)f2(x1, x2)
)=(x2
1 − x22
2x1x2
).
The Jacobian matrix is(∂f1/∂x1 ∂f1/∂x2
∂f2/∂x1 ∂f2/∂x2
)=(
2x1 −2x2
2x2 2x1
)with determinant equal to 4(x2
1 +x22). Find the fixed points and study their
stability.
64 Problems and Solutions
Problem 17. Consider the differential operators
D1 =d2
dx2+
c
x2+x2
16
D2 = sinh(α)(d2
dx2+
c
x2− x2
16
)+ cosh(α)
(− i
2xd
dx− i
4
)D3 = cosh(α)
(d2
dx2+
c
x2− x2
16
)+ sinh(α)
(− i
2xd
dx− i
4
).
(i) Show the differential operator satisfy (Lie algebra su(1, 1))
[D1, D2] = −iD3, [D2, D3] = iD1, [D3, D1] = iD2.
For the actual calculations let f : R → R be a smooth function and oneshows that [D1, D2]f(x) = −iD3f(x) etc.(ii) Show that
D23 −D2
1 −D22 = − 3
16− c
4.
Problem 18. Let
v(τ) =
v1(τ)v2(τ)v3(τ)
, w(τ) =
w1(τ)w2(τ)w3(τ)
where vj(τ), wj(τ) (j = 1, 2, 3) are smooth functions of τ . Calculate
d
dτ(v × (v ×w)).
Problem 19. Show that the (2, 2) Pade approximant of the cosine func-tion is given by
cos(x) ≈ 12− 5x2
12 + x2.
Problem 20. Let f(x) = x2 and g(x) = x ln(x). Show that
limx→∞
g(x)f(x)
= 0.
Problem 21. Let c > 0. Find
limx→0
sinh(cx)x
, limx→0
x
sinh(cx).
Functions 65
Problem 22. Show that the function f : (0, 1)→ R
f(x) =x− 1/2x(x− 1)
is bijective.
Problem 23. Show that the function
f(x) =π
sin(πx)
has poles at 0,±1,±2, . . ..
Problem 24. For any real number x one defines bxc (floor of x) as thelargest integer less than or equal to x. One defines dxe (ceiling of x) asthe smallest integer greater than or equal to x. Find the minima and themaxima of the function
f(x) = 2x− bxc.
Problem 25. Let 0 ≤ t < 2. Show that
ln(
2 + t
2− t
)= ln
(1 + t/21− t/2
)= 2arctanh(t/2).
Problem 26. Consider the continuous function f : R→ R
f(x) =12
(x
|x|+ 1+ 1).
Find the fixed points and study their stability.
Problem 27. Consider the function f : R2 → R2
f1(x1, x2) =12
(x21 − x2
2), f2(x1, x2) = x1x2.
Find the fixed points and study their stability. Is their an inverse f−1.
Problem 28. Explain
sin(α/2) = ±√
1− cos(α)2
, cos(α/2) = ±√
1 + cos(α)2
.
66 Problems and Solutions
Problem 29. Consider the function f : R→ R defined by
f(x) =e−1/x2
for x 6= 00 for x = 0
Show that the function is not analytic. Note that f(x) = f(−x) and
df
dx=
2x3f(x),
d2f
dx2=(
4x6− 6x4
)f(x)
Furthermore limx→±∞ f(x) = 1.
Problem 30. Let N > 0. Show that
limε→0
ε−Ne−(1/ε) = 0.
Problem 31. Consider the map f : R2 → R2 given by(f1(x1, x2)f2(x1, x2)
)=(x1 cos(x2)x1 sin(x2)
).
Find the fixed points and study their stability. Show that the functionalmatrix is given by(
∂f1/∂x1 ∂f1/∂x2
∂f2/∂x1 ∂f2/∂x1
)=(
cos(x2) −x1 sin(x2)sin(x2) x1 sin(x2)
).
Is the map f invertible?
Problem 32. Let a > 0. Show that
limx→0
sinh(2ax)sinh(x)
= 2a.
Problem 33. (i) Write down the first four terms of the Taylor expansionof cos(x). Then find (2, 2)[x] of the Pade approximant. Discuss the casex→∞ for both functions.(ii) Write down the first four terms of the Taylor expansion of cosh(x).Then find (2, 2)[2] of the Pade approximant. Discuss the case x → ∞ forboth functions.
Problem 34. Consider the Henon map (a > 0, b > 0)
x1,t+1 = a− bx2,t + x21,t, x2,t+1 = x1,t, t = 0, 1, 2, . . . .
Functions 67
Let |x1,0| < R, |x2,0| < R, where R is the larger root of λ2−(|b|+1)λ−a = 0.Show that all points (x1,0, x2,0) outside of this domain tend to ∞ or −∞for t→∞ or t→ −∞.
Problem 35. Consider the one-dimensional map (t = 0, 1, 2, . . .)
xt+1 =xt + 2xt + 1
, x0 ≥ 0.
Find the fixed points of the map. Let x0 = 0. Does limt→∞ xt tend to afixed point?
Problem 36. Consider the analytic functions f : R→ R, g : R→ R
f(x) = sinh(x), g(x) = 2x3.
Find f−1 g f , g−1 f g.
Problem 37. Let R > r > 0. Consider the function f : R2 → R2 (battleof the sexes)
f1(x1, x2) = (x1 1− x1 )(R 00 r
)(x2
1− x2
)= ((R+r)x2−r)x1+r(1−x2)
f2(x1, x2) = (x1 1− x1 )(r 00 R
)(x2
1− x2
)= ((R+r)x2−R)x2+R(1−x2).
Find the fixed points and study their stability.
Problem 38. Let c1, c2 ∈ R and f : R → R be an analytic function.Show that
exp(c1d
dx
)exp(c2x)f(x) ≡ exp(c2x) exp
(c1d
dx
)exp(c1c2)f(x).
Problem 39. Let n be a non-negative integer. Let k = 0, 1, . . . , n. Findthe derivative
∂n
∂xk∂xn−k(x+ y)n.
Show that∂n
∂xk∂xn−k(x+ y)n = n!.
68 Problems and Solutions
Problem 40. The content (n-dimensional volume) bounded by a hyper-sphere of radius r is known to be
Vn =2rnπn/2
nΓ(n/2)
where Γ is the gamma function. Let r = 1. Show that
limn→∞
Vn = 0.
Problem 41. Let n ≥ 1. Show that the vector space spanned by
xn, yxn−1, . . . , yn−1x, yn
is n+ 1 dimensional.
Problem 42. Let ε, x ∈ R. Show that
limε→0
sinh(εx)sinh(ε)
= x, limε→0
sin(εx)sin(ε)
= x.
Problem 43. The Hurwitz zeta-function ζH(s, a) is defined by
ζH(s, a) :=∞∑k=0
(n+ a)−s, 0 < a ≤ 1, <(s) > 1.
(i) Show that the Hurwitz zeta-function can be written in the form
ζH(s, a) = a−s +1
Γ(s)
∞∑n=1
∫ ∞0
ts−1e−(n+a)tdt
where Γ(s) is the Gamma function.(ii) Show that
∂
∂aζH(s, a) =−sζH(s+ 1, a)
∂
∂sζH(s, a)
∣∣∣∣s=0
= ln(Γ(a))− 12
ln(2π).
Problem 44. Let f : R→ R be an analytic function. Show that (differ-ential identity)
dn
dxnf(x) = x−n
dn
dεnf(εx)
∣∣∣∣ε=1
.
Functions 69
Problem 45. Find an analytic function f : R→ R such that
f(∞) = 1, f(−∞) = 0
andf(x1) ≤ f(x2) whenever x1 ≤ x2.
Find the derivative of the function f and determine the maxima of thefunction df/dx.
Problem 46. Let
Θ(x) =
1 for x > 00 otherwise
Show that
Θ(x1 − x3)−Θ(x2 − x3) =
1 for x1 > x3 > x2
−1 for x2 > x3 > x1
0 otherwise
Problem 47. (i) Let
f(θ) =cos(π2 cos(θ)
)sin(θ)
.
Find f(θ) for θ = nπ (n ∈ Z) using the L’Hospital rule. Plot f(θ) as afunction of θ.(ii) Apply the L’Hospital rule to show that
limx→0
ex − 1x
= 1.
(iii) Show that
limx→0
sinh(x)x
= 1, limx→0
x
sinh(x)= 1.
(iv) Consider the function f : R→ R
f(x) =x cosh(x)− sinh(x)
2x sinh2(x).
Findlimx→0
f(x), limx→∞
f(x).
70 Problems and Solutions
Problem 48. Consider the functions
f(x) =x
sinh(x), g(x) =
sinh(x)x
.
Findlimx→0
f(x), limx→0
g(x).
Problem 49. Let x ∈ R. Show that
limx→∞
1− e−x
x= 1.
Problem 50. (i) Show that
limx→0
1− cos(x)x
= 0.
(ii) Show that
limx→1
x− 1ln(x)
= 1.
(iii) Show that
limx→0
cosh(x)− cosh(2x)sinh(x)
= 0.
(iv) Let x > 0. Show that
limx→0
ln(1 + x)x
= 1.
Problem 51. Let α ∈ R. Find
limα→0
sin(α) cos(α)α
.
Problem 52. Let k > 0 and dimension 1/length. Find the extrema ofthe functions
f1(x) = ln(cosh(kx))f2(x) = cosh(kx) + α sinh(kx), −1 ≤ α ≤ +1
f3(x) =Asech(kx) ≡ A 1cosh(kx)
f4(x) = asech2(kx) ≡ a sinh2(kx)cosh2(kx)
.
Functions 71
Problem 53. Consider the analytic function f : R→ R,
f(x) = cos(x) + sin(x).
(i) Find the critical points of f and the minima and maxima of the function.(ii) Find the roots of f , i.e. solve f(x) = 0.(iii) Find the fixed points of f , i.e. solve f(x) = x.(iv) Find the differential equation together with the initial conditions thefunction f satisfies.
Problem 54. Consider the polynomial f : R→ R
f(x) = x3 − 32x2 + x− 3
2.
Show that there is a root, i.e. a solution of f(x) = 0 in the interval [1, 2].Apply the Newton method (say 3 steps) to find an approximate solutionfor the root. Apply the two initial condition x0 = 1 and x0 = 2. Afterfinding this root how would you proceed to find the other two roots?
Problem 55. A real valued function f defined on an open subset G ofa Banach space E is said to be Frechet differentiable at a point x ∈ G ifthere is f ′(x) ∈ E∗ such that
limv→0
|f(x+ v)− f(v)− 〈f ′(x), v〉|‖v‖
= 0.
Then f ′(x) is called the Frechet derivative of f at x. Let E = R. Letf(x) = x3 + 2x2 + 3x+ 4. Find the Frechet derivative.
Problem 56. Consider the function
f(x) =exp(−x2/2)
∫ x0
exp(τ2/2)dτx
.
Show thatlimx→0
f(x) = 1.
Problem 57. Consider the functions f : R→ [−1, 1], g : R→ [−1, 1]
f(x) = sin(x), g(x) = cos(x).
(i) Find the minima and maxima of the function
h1(x) = max(f(x), g(x)).
72 Problems and Solutions
Is the function h1 differentiable?(ii) Find the minima and maxima of the function
h2(x) = min(f(x), g(x)).
Is the function h2 differentiable?
Problem 58. Let a ∈ R. Show that
limx→∞
(1 + a/x)x = ea.
Problem 59. Let k ∈ R, x, y ∈ +1,−1 such that xy = ±1. Show that
ekxy = cosh(k) + xy sinh(k).
Problem 60. Consider the coordinate transformation
x1(r, α, β, γ) = r sin(β/2) sin((γ − α)/2)x2(r, α, β, γ) = r sin(β/2) cos((γ − α)/2)x3(r, α, β, γ) = r cos(β/2) sin((γ + α)/2)x4(r, α, β, γ) = r cos(β/2) sin((γ + α)/2).
Show that x21 + x2
2 + x23 + x2
4 = r2.
Chapter 4
Polynomial
4.1 Solved ProblemsProblem 1. The Chebyshev polynomials are defined by
Tk(x) = cos(k arccos(x)), k = 0, 1, . . . x ∈ [−1, 1]
Thus the first six polynomials are
T0(x) = 1T1(x) = x
T2(x) = 2x2 − 1T3(x) = 4x3 − 3xT4(x) = 8x4 − 8x2 + 1T5(x) = 16x5 − 20x3 + 5x.
Find 1, x, x2, x3, x4, x5 as functions of T0, T1, T2, T3, T4, T5.
Problem 2. Let Ln(x), Hn(x) be the Laguerre polynomials and Hermitepolynomials, where n = 0, 1, . . .. Let
L(α)n (x) :=
x−αex
n!dn
dxn(e−xxn+α)
73
74 Problems and Solutions
be the associated Laguerre polynomials with α > −1 and n = 0, 1, . . .. TheLaguerre polynomials are recovered by setting α = 0. We have
H2n(x) = (−4)nn!L(−1/2)n (x2) (1)
and the following addition formula for the associated Laguerre polynomialsLαn(x)
L(α+β+1)n (x+ y) =
n∑k=0
L(α)n−k(x)L(β)
k (y) . (2)
(i) Find a new sum rule by inserting (2) into (1).(ii) Consider the sum rule
1n!2n
Hn(√
2x)Hn(√
2y) =n∑k=1
(−1)kL(−1/2)n−k ((x+y)2)L(−1/2)
k ((x−y)2). (3)
Insert (1) into (3) to find a sum rule for Hermite polynomials.
Problem 3. The Hermite polynomial of degree n can be written as
Hn(x) =[n/2]∑k=0
(−1)kn!k!(n− 2k)!
(2x)n−2k.
Express xn using the Hermite polynomials.
Problem 4. Given the differentiable function f : R→ R with
f(x) = 4x(1− x).
(i) Find the fixed points of f(x) and f(f(x)).(ii) Find the critical points of f(x) and f(f(x)).
Problem 5. Let P (z) be a polynomial of degree n ≥ 2 with distinct zerosζ1, . . . , ζn. Show that
n∑j=1
1P ′(ζj)
= 0
where ′ denotes the derivative, i.e. P ′(ζ) ≡ dP (z = ζ)/dz.
Problem 6. Given a set of N real numbers x1, x2, . . . , xN . It is oftenuseful to express the sum of the j powers
sj = xj1 + xj2 + · · ·+ xjN , j = 0, 1, 2, . . .
Polynomial 75
in terms of the elementary symmetric functions
σ1 =N∑i=1
xi
σ2 =N∑i<j
xixj
σ3 =N∑
i<j<k
xixjxk
......
σN = x1x2 · · ·xN .
Consider the special case with three numbers x1, x2, x3. Then the elemen-tary symmetric functions are given by
σ1 = x1 + x2 + x3, σ2 = x1x2 + x1x3 + x2x3, σ3 = x1x2x3.
We know that the elementary symmetric functions are the coefficients (upto sign) of the polynomial with the roots x1, x2, x3. In other words thevalues of x1, x2, x3 each satisfy the polynomial equation
x3 − σ1x2 + σ2x− σ3 = 0. (1)
Find a recursion relation for
sj := xj1 + xj2 + xj3, j = 0, 1, 2, . . .
and give the initial values s0, s1, s2. Calculate s3 and s4.
Problem 7. The dominant tidal potential at position (r, φ, λ) due to themoon or sun is given by
U(r) =GM∗r2
r∗3P 0
2 (cos(ψ))
where M∗ is the mass of the moon or sun located at (r∗, φ∗, λ∗). Moreover,ψ is the angle between mass M∗ and the observation point at (r, φ, λ),where φ is the latitude and λ is the longitude. By the spherical cosinetheorem we have
cos(ψ) = sin(φ) sin(φ∗) + cos(φ) cos(φ∗) cos(λ− λ∗).
The Legendre polynomials are defined as
Pn(x) :=1
2nn!dn
dxn(x2 − 1)n
76 Problems and Solutions
with n = 0, 1, 2, . . . and the associated Legendre polynomials are defined as
Pmn (x) := (1− x2)m/2dm
dxmPn(x) =
(1− x2)m/2
2nn!dm+n
dxm+n(x2 − 1)n
with P 0n(x) = Pn(x) and Pmn = 0 if m > n.
(i) Show that U(r) can be written as
U(r) =GM∗r2
r∗3
(P 0
2 (sinφ)P 02 (sinφ∗) +
13P 1
2 (sinφ)P 12 (sinφ∗) cos(λ− λ∗)
+112P 2
2 (sinφ)P 22 (sinφ∗) cos(2(λ− λ∗))
).
(ii) Give an interpretation (maxima and nodes) of the terms in the paren-thesis.
Problem 8. Consider the cubic equation
y3 + py + q = 0, p, q ∈ R, pq 6= 0. (1)
Show that applying the nonlinear transformation
y(z) := z − p
3z
equation (1) can be reduced to
z6 + qz3 − p3
27= 0
and with u = z3 to a quadratic equation.
Problem 9. Find all integers c for which the cubic equation
x3 − x+ c = 0
has three integer roots.
Problem 10. Let Φ be an endomorphism of the space Cn[X] of polyno-mials of degree n with complex coefficients, which maps a polynomial p(X)to the polynomial p(X+1). Let Ψ be an endomorphism of the space Cn[X]which maps a polynomial p(X) to (1−X)np
(X
1−X
), which of course is also
a polynomial. Show that we have a braid-like relation
Φ Ψ Φ = Ψ Φ Ψ.
Polynomial 77
Problem 11. Let a ∈ R. Let r and s be the roots of the quadraticequation
x2 + ax+a2 − 1
2= 0.
Find r3 +s3 in terms of a, and express it as a polynomial in a with rationalcoefficients.
Problem 12. Consider the polynomial p(x) = x3 − x2 + x − 2. Doesthere exist a nontrivial polynomial q(x) with real coefficients such that thedegree of every term of the product p(x)q(x) is a multiple of 3? If so, findone. If not, show there is none.
Problem 13. (i) Let n be a positive integer. Let f : [a, b] → R bea continuous function. The Bernstein polynomials of degree n associatedwith the continuous function f are given by
Bn(f(x), x) :=1
(b− a)n
n∑j=0
(n
j
)(x− a)j(b− x)n−jf(xj)
wherexj = a+ j
b− an
, j = 0, 1, . . . , n.
Consider the function f : [0, 1] → R given by f(x) = sin(4x). Show thatB2(f, x) is not a “good approximation” for f . Consider x = π/8.(ii) The Bernstein basis polynomials are defined as
Bn,j(x) :=(n
j
)xj(1− x)n−j , j = 0, 1, 2, . . . , n, x ∈ [0, 1].
Show thatn∑j=0
Bn,j(x) = 1.
Show that Bn,j satisfies the recursion relations
Bn,j(x) = (1− x)Bn−1,j(x) + xBn−1,j−1(x), j = 1, 2, . . . , n− 1Bn,0(x) = (1− x)Bn−1,0(x)Bn,n(x) = xBn−1,n−1(x).
Problem 14. A one-dimensional map f is called an invariant of a two-dimensional map g if
g(x, f(x)) = f(f(x)).
78 Problems and Solutions
Letf(x) = 2x2 − 1.
Show that f is an invariant for
g(x, y) = y − 2x2 + 2y2 + d(1 + y − 2x2).
Problem 15. Consider the functions f(z) = z3 and h(z) = z+1/z. Finda function p such that
h(f(z)) = p(h(z)). (1)
Problem 16. Consider the map fc(z) = z2 + c, where c ∈ C. Find allcomplex c-values where the map fc has a fixed point z∗ with f ′c(z
∗) = −1.
Problem 17. Let p(x, y) be a real polynomial. Show that if p(x, y) = 0for infinitely many (x, y) on the unit circle x2 + y2 = 1, then p(x, y) = 0 onthe unit circle.
Problem 18. Show that the equation
zn = a (1)
where n is a positive integer and a is any nonzero complex number, hasexactly n roots. Hint. Set
a = ρ(cos(φ) + i sin(φ)). (2)
Problem 19. Let A be a 2 × 2 matrix over the real numbers R. Thetrace is defined as
tr(A) = a11 + a22.
It can be proved that the trace is the sum of the eigenvalues of A, i.e.
tr(A) = λ1 + λ2.
Thus we havetr(A2) = λ2
1 + λ22.
Let
A =(
1 11 1
).
Find the eigenvalues using the two equations.
Polynomial 79
Problem 20. Find the zeros of the cubic polynomial
α(x) := a0 + a1x+ a2x2 + x3 (1)
over C, where a0, a1, a2 ∈ R and a0 6= 0.
Problem 21. Show that the zeros of
z3 − 6z2 + 11z − 6 = 0 (1)
are given by 1, 2, and 3.
Problem 22. Find the zeros of the quartic polynomial
α(x) = a0 + a1x+ a2x2 + a3x
3 + x4 (1)
over C when a0 6= 0.
Problem 23. Find the zeros of
α(x) = 35− 16x− 4x3 + x4. (1)
Problem 24. The variational equation of the Lorenz model
dX
dt= −σX + σY (1a)
dY
dt= −XZ + τX − Y (1b)
dX
dt= XY − bZ (1c)
is given by dx0/dtdy0/dtdz0/dt
=
−σ σ 0(τ − Z) −1 −XY X −b
x0
y0
z0
. (2)
(i) Show that Lorenz model possess the steady-state solution X = Y =Z = 0, representing the state of no convection.(ii) Show that with this basic solution, the characteristic equation of thevariational matrix is
(λ+ b)(λ2 + (σ + 1)λ+ σ(1− τ)) = 0. (3)
80 Problems and Solutions
(iii) Show that this equation has three real roots when τ > 0; all are negativewhen τ < 1, but one is positive when τ > 1. The criterion for the onset ofconvection is therefore τ = 1.(iv) Show that when τ > 1, system (1) possess two additional steady statesolutions
X = Y = ±√b(τ − 1), Z = τ − 1. (4)
(v) Show that for either of these solutions, the characteristic equation ofthe matrix in (2) is
λ3 + (σ + b+ 1)λ2 + (τ + σ)bλ+ 2σb(τ − 1) = 0. (5)
(vi) Show that this equation possesses one real negative root and two com-plex conjugate roots when τ > 1. Show that the complex conjugate rootsare pure imaginary if the product of the coefficients of λ2 and λ equals theconstant term, or
τ = σ(σ + b+ 3)(σ − b− 1)−1. (6)
Problem 25. The variational equation of the Lotka Volterra model
du1
dt= u1 − u1u2,
du2
dt= −u2 + u1u2 (1)
is given by (dv1/dtdv2/dt
)=(
1− u2 −u1
u2 −1 + u1
)(v1
v2
)(2)
where u1 > 0 and u2 > 0.(i) Show that (1) possess the steady-state solution u1 = u2 = 1.(ii) Show that with this basic solution, the characteristic equation of thematrix in (2) is
λ2 + 1 = 0. (3)
(iii) Find the solution of the characteristic equation and discuss.
Problem 26. Show the following. Let P (z) be a polynomial. Then either
1. P (z) has a fixed point q with P ′(q) = 1,2. P (z) has a fixed point q with |P ′(q)| > 1.
Problem 27. Let Sn be the symmetric group. The symmetric groupSn acts naturally on polynomials in n variables. For a polynomial p in nvariables, define the symmetrized polynomial associated to p, Symn(p), by
Symn(p) :=∑σ∈Sn
σ(p).
Polynomial 81
Let n = 2 andp(x1, x2) = x2
1x2 + 2x2.
Find Sym2(p).
Problem 28. Consider the system of equations
z1 + z2 + · · ·+ zn−1 + zn = 0z1z2 + z2z3 + · · ·+ zn−1zn + znz1 = 0
...z1z2 · · · zn−1 + z2z3 · · · zn + · · ·+ zn−1zn · · · zn−3 + znz1 · · · zn−2 = 0
z1z2 · · · zn = 1.
Find the solutions for the case n = 2 and n = 3. This system of equationsarise as follows. Let p be a prime number. A vector x = (x0, x1, . . . , xp−1) ∈Cp viewed as a function Z/p → C has discrete Fourier transform x =(x0, x1, . . . , xp−1), where
xj =p−1∑k=0
ωjkxk, ω := e2πi/p.
The vector x is called equimodular if all its coordinates have the same abso-lute value, and x is called bi-equimodular if both x and x are equimodular.The question is: Which vectors are bi-equimodular?
Problem 29. Let α ∈ R. Find the roots of the characteristic equation
λ6 − 2 cos(3α)λ3 + 1 = 0.
Problem 30. Show that the n-th order polynomial
p(x) = a0 + a1x+ a2x2 + · · ·+ anx
n
which goes exactly through n+ 1 data points is unique.
Problem 31. Let p : R→ R be a polynomial that satisfies
p(1− x) + 2p(x) = 3x
for all x ∈ R. Find the values of p(0) and p(1). Give an example of apolynomial that satisfies this condition.
82 Problems and Solutions
Problem 32. (i) Let x1, x2 ∈ R and x1 6= x2. Find the solutions of thesystem of equations
x1 −1
(x1 − x2)2= 0, x2 +
1(x1 − x2)2
= 0.
(ii) Let x1, x2, x3 ∈ R and x1 6= x2, x1 6= x3, x2 6= x3. Find the solutionsof the system of equations
x1 −1
(x1 − x2)2− 1
(x1 − x3)2= 0
x2 +1
(x1 − x2)2− 1
(x2 − x3)2= 0
x3 +1
(x1 − x3)2+
1(x2 − x3)2
= 0.
Problem 33. Let z, w ∈ C. Find the solution of the system of equations
|z|2 + |w|2 = 1, z2 + w3 = 0.
Problem 34. Consider the two polynomials
p1(x) = a0 + a1x+ · · ·+ anxn, p2(x) = b0 + b1x+ · · ·+ bmx
m
where n = deg(p1) and m = deg(p2). Assume that n > m. Let r(x) =p2(x)/p1(x). We expand r(x) in powers of 1/x, i.e.
r(x) =c1x
+c2x2
+ · · ·
From the coefficients c1, c2, . . . , c2n−1 we can form an n×n Hankel matrix
Hn =
c1 c2 · · · cnc2 c3 · · · cn+1
......
. . ....
cn cn+1 · · · c2n−1
.
The determinant of this matrix is proportional to the resultant of the twopolynomials. If the resultant vanishes, then the two polynomials have anon-trivial greates common divisor. Apply this theorem to the polynomials
p1(x) = x3 + 6x2 + 11x+ 6, p2(x) = x2 + 4x+ 3.
Polynomial 83
Problem 35. Let p1 and p2 two polynomials with n = degree(p1), m =degree(p2) and n > m. We expand the rational function
r(x) =p2(x)p1(x)
with respect to powers of 1/x, i.e.,
r(x) = d1x−1 + d2x
−2 + · · ·
The coefficients d1, d2, . . . , d2n−1 are inserted into the n× n Hankel matrix
Hn =
d1 d2 . . . dnd2 d3 . . . dn+1
......
. . ....
dn dn+1 . . . d2n−1
.
A Hankel matrix is a diagonal matrix in which all the elements are thesame along any diagonal that slopes from northeast to southwest. If thedeterminant of this matrix is zero, then the two polynomials have a non-trival common divisor. Apply this algorithm to the polynomials
p1(x) = x3 + 6x2 + 11x+ 6, p2(x) = x2 + 6x+ 8
to test whether they have a common non-trivial divisor.
Problem 36. Let p be a polynomial with coefficients in R. If the equationp(x) = 0 has repeated roots, then p(x) and dp(x)/dx have a highest commonfactor. Apply this to the polynomial
p(x) = 32x4 − 64x3 + 24x2 + 8x− 3.
Problem 37. Let p be a polynomial with real coefficients. The equationp(x) = 0 cannot have more positive roots than there are changes of signfrom + to − and from − to + in the coefficients of the polynomial p(x),or more negative roots than there are changes of sign in p(−x) (Descartesrule of signs). Apply the rule to the polynomiol
p(x) = x6 + 7x3 + x− 2 = 0.
Problem 38. The Liouville-Riemann definition for the fractional integraloperator D−qx is given by
D−qx f(x) :=1
Γ(q)
∫ x
0
(x− y)q−1f(y)dy, q > 0.
84 Problems and Solutions
The fractional differential operator Dνx for ν > 0 is given by the definition
Dνxf(x) :=
dn
dxn(Dν−n
x f(x)), ν − n < 0.
Let f(x) = x2. Find D−qx f(x).
Problem 39. Let n ∈ N. Show by induction that xn − yn is divisiblewithout remainder by x− y for all values of n. We have
xn+1 − yn+1 ≡ x(xn − yn) + yn(x− y).
Problem 40. Consider the polynomial
Pn(x) =n∑j=0
(−1)jajxn−j , a0 = 1.
The homogeneous product sums symmetric functions hk(x1, . . . , xn) of thezeros of this polynomial are defined as follows
n∏j=1
(1− xjx) = (1 + h1x+ h2x2 + h3x
3 + · · ·)−1.
Show that the first few sums are given explicitly by
h1(x1, . . . , xn) =n∑j=1
xj
h2(x1, . . . , xn) =n∑j=1
x2j +
n∑j<k
xjxk
h3(x1, . . . , xn) =n∑j=1
x3j +
n∑j 6=k
+n∑
j<k<`
xjxkx`.
Problem 41. Consider the quintic equation
p(x) = x5 − 5x3 + 5x− 5 = 0.
Is p irreducible over the rational numbers? What is the Galois group of p?Look for solutions of the form r + 1/r.
Problem 42. The Bernoulli polynomials Bn(x) (n = 0, 1, . . .) can bedefined recursively by
dBn(x)dx
= nBn−1, n = 1, 2, . . .
Polynomial 85
with B0(x) = 1 and the condition∫ 1
0
Bn(x)dx = 0, n ≥ 1.
The Bernoulli numbers Bn are defined by Bn := Bn(x = 0).(i) Find the first four Bernoulli polynomials.(ii) Show that
Bm(x) =m∑n=0
1n+ 1
n∑k=0
(−1)k(n
k
)(x+ k)m.
(iii) Show thatBn(1− x) = (−1)nBn(x).
Problem 43. Consider the operators
D1 = x, D2 =d
dx, D3 = x
d
dx
which apply to the function of the Bargmann space
B := f(n) =xn√n!, n ∈ N0, x ∈ R .
Show that
D1f(n) =√n+ 1f(n+ 1), D2f(n) =
√nf(n− 1), D3f(n) = nf(n).
Find the commutators [D1, D2], [D1, D3], [D2, D3].
Problem 44. Express the polynomial
p(x) = 7x21 + 6x2
2 + 5x23 − 4x1x2 − 4x2x3 + 14x1 − 8x2 + 10x3 + 6
in matrix form
p(x) = (x1 x2 x3 )A
x1
x2
x3
+ vT
x1
x2
x3
+ c
Find the eigenvalues and normalized eigenvectors of the 3× 3 matrix.
Problem 45. Find the polynomials generated by
dpj+1(x)dx
= (j + 1)pj(x), j = 0, 1, 2, . . .
86 Problems and Solutions
with p0(x) = 1. The constants of integration we set to 0.
Problem 46. The complete Bell polynomials Bj(x1, x2, . . . , xj) are givenby the exponential generating function
exp
∞∑j=1
xjtj
j!
=∞∑n=0
Bn(x1, . . . , xn)tn
n!. (1)
Taking the n-th derivative with respect to t we obtain
dn
dtnexp
∞∑j=1
xjtj
j!
∣∣∣∣∣∣t=0
= Bn(x1, . . . , xn) (2)
with B0 = 1. Find the first four Bell polynomials.
Programming Problem
Problem 1. Consider the polynomials f1 : R→ R, f2 : R→ R
f1(x) = x2 + 2x+ 1, f2(x) = 2x.
Apply Maxima to find
h1(x) = f1(f2(x)), h2(x) = f2(f1(x))
and C = f1 − f2.
4.2 Supplementary Problems
Problem 1. Construct a polynomial
p(x) = x2 + ax+ b
that admits the roots16
(√
3 + 3), −16
(√
3− 3)
and the fixed points
16
(√
30 + 6), −16
(√
30− 6).
Polynomial 87
Problem 2. Let a, b ∈ R. Consider the quartic equation
(λ− 1)2(λ− b2)2 − λ2a2(1− b2)2 = 0
Show that the roots are given by
λ1(a, b) =12
(1− a+ (1 + a)b2 + (T+)1/2)
λ2(a, b) =12
(1− a+ (1 + a)b2 − (T+)1/2)
λ3(a, b) =12
(1 + a+ (1− a)b2 + (T−)1/2)
λ4(a, b) =12
(1 + a+ (1− a)b2 − (T−)1/2)
where T± = (1 + a2)(1− b2)2 ± 2a(b4 − 1).
Problem 3. Let n be a positive integer. Show that xn − y2 has x− y asa factor for all n.
Problem 4. Let s = 1/2, 1, 3/2, 2, . . . be the spin values. The Brillouinfunction is defined as
Bs(x) :=2s+ 1
2scoth
(2s+ 1
2sx
)− 1
2scoth
(12sx
)Find Bs(x = 0) applying L’Hospital.
Problem 5. Let n ∈ N0 and x,m ∈ R. The Sonine polynomials aredefined by
Snm(x) =n∑j=0
(−1)jΓ(m+ n+ 1)xj
Γ(m+ j + 1)(n− j)!j!
where Γ denotes the gamma function. Show that
S0m(x) = 1, S1
m(x) = m+ 1− x.
Show that the Sonine polynomials satisfy∫ ∞0
e−xxmSjm(x)Skm(x)dx =Γ(m+ j + 1)
j!= δj,k.
Problem 6. (i) Show that the cubic roots of unity z3 = 1 are
1, w = −12
+√
3i2, w2 = −1
2−√
3i2. (2)
88 Problems and Solutions
(ii) Show that the zeros of z3 + 1 = 0 are given by
−1,12
+i
2
√3,
12− i
2
√3. (1)
(iii) Show that the set
S = ω1 = −12
+i
2
√3, ω2 = −1
2− i
2
√3, ω3 = 1
of the cubic roots of 1, forms an abelian group with respect to multiplicationon the set of complex numbers C, i.e. show that
ω1ω2 = ω3, ω2ω1 = ω3, ω1ω3 = ω1,
ω3ω1 = ω1, ω2ω3 = ω2, ω3ω2 = ω2. (1)
Obviously, the neutral element is ω3. From (1) we see that each elementhas an inverse (which is unique), i.e.
(ω1)−1 = ω2, (ω2)−1 = ω1, (ω3)−1 = ω3.
The associative law is true for all complex numbers.
Problem 7. Let a, b ∈ R and z be denote a root of the quadratic equation
z2 + az + b = 0. (1)
Show that the sequence of powers of z, zn, n ≥ 2, satisfies the lineardifference equation
wn + awn−1 + bwn−2 = 0, n ≥ 2. (2)
Set wn = xn. Then wn−1 = xn−1.
Problem 8. A real number is said to be algebraic if it is a zero of apolynomial
cnxn + cn−1x
n−1 + · · ·+ c0
where the coefficients cj are integers. The height of such polynomials isdefined as the positive integer number
h = n+ |cn|+ |cn−1|+ · · ·+ |c0|.
Show that there are only finitely many polynomials of height h. Show thatthe set of all algebraic real numbers is enumerable.
Polynomial 89
Problem 9. Let p0 = 1, p1, p2, . . . be polynomials which satisfy therecursion relation
pn =1
2n
n∑k=1
22k
(n!
(n− k)!
)2
xkpn−k, n ∈ N.
Giva a SymbolicC++ and Maxima implementation of this recursion rela-tion. For example p1 = 2x1.
Problem 10. Show that every polynomial α(x) ∈ C[x] of degree m ≥ 1has precisely m zeros over C, where any zero of multiplicity is to be countedas n of the m zeros.
Problem 11. Show that if r ∈ C is a zero of any polynomial α(x) withreal coefficients, then r is also a zero of α(x), where r denotes the complexconjugate of r.
Problem 12. Let
P = fi(x), i = 1, . . . , k, x = (x1 . . . xn) (1)
be a set of multivariate polynomials in the ring Q[x1 . . . xn], with a solutionset
S = S(fi . . . fk) = x | fi(x) = 0, ∀i = 1, . . . , k (2)
All polynomialsu(x) =
∑j
gj(x)fj(x) (3)
for arbitrary polynomials gj will vanish in all points of S. The set of all uestablishes a polynomial ideal
I = I(f1 · · · fk) =
∑j
gj(x)fj(x)
(4)
and classical algebra tells that S is invariant if we replace the set f1, · · · , fkby any other basis for the ideal I(f1, · · · fk).
The Buchberger algorithm allows one to transform the set of polynomialsinto a canonical basis of the same ideal, the Grobner basis GB = GB(I).For the purpose of the equation solving especially Grobner bases computedunder lexicographical term ordering are important. They allow one to de-termine the set S directly. If I has dimension zero (S is a finite set ofisolated points, GB has in most cases the form
g1(x1, xk) = x1 + c1,m−1xm−1k + c1,m−2x
m−2k + · · ·+ c1,0
90 Problems and Solutions
g2(x2, xk) = x2 + c2,m−1xm−1k + c2,m−2x
m−2k + · · ·+ c2,0
· · ·gk−1(xk−1, xk) = xk−1 + ck−1,m−1x
m−1k + ck−1,m−2x
m−2k + · · ·+ ck−1,0
gk(xk) = xmk + ck,m−1xm−1k + ck,m−2x
m−2k + · · ·+ ck,0.
A basis in this form has the elimination property: the variable dependencyhas been reduced to a triangular form, just as with a Gaussian eliminationin the linear case. The last polynomial is univariate in xk. It can besolved with usual algebraic or numeric techniques; its zero xk then arepropagated into the remaining polynomials, which then immediately allowone to determine the corresponding coordinates (x1, . . . , xk−1).Consider the system
y2 − 6y, xy, 2x2 − 3y − 6x+ 18, 6z − y + 2x. (4)
(i) Show that that this system has for x, y, z the lexicographical Grobnerbasis
g1(x, z) = x−z2+2z−1, g2(y, z) = y−2z2−2z−2, g3(z) = z3−1. (5)
(iii) Show that the roots of the third polynomial are given byz = 1, z =
1−√
3i2
, z =√
3i− 12
. (6)
If we propagate one of them into the basis, we generate univariate polyno-mials of degree one which can be solved immediately; e.g. Selecting 1−
√3i
2for z the basis reduces to
x =3√
3i+ 32
, y, 0
(7)
such that the final solution for this branch isx =
3√
3i+ 32
, y = 0, z =1−√
3i2
. (8)
For zero dimensional problems the last polynomial will always be univariate.However, in degenerate cases the other polynomials can contain their lead-ing variable in a higher degree, then containing more mixed terms with thefollowing variables. And there can be additional polynomials with mixedleading terms (of lower degree) imposing some restrictions. But the varibledependency pattern will remain triangular (evenutally with more than krows). There is a special algorithm for decomposing such ideals using idealquotients.
Chapter 5
Equations
5.1 Solved ProblemsProblem 1. Solve the equation
13
=1x
+14.
Problem 2. Let x1, x2 be positive real numbers. Consider the equation
x2 sin(θ) = x1 cos(θ).
Find sin(θ).
Problem 3. Let L be a given positive real number. Solve the system oftwo coupled nonlinear equations
1 = x2L+ 2x2y + Lx2y2
0 =−x2 + 2x2y + Lx2y2.
Problem 4. Let c, y ∈ R. Solve the quadratic equation
x = cy + yx− x2
91
92 Problems and Solutions
with respect to x.
Problem 5. Consider a triangle in the plane. Consider a point withinthe triangle. We draw lines from this point to the three vertices, therebydividing the triangle into three triangles of area A1, A2, A3. The sides ofthe triangle are designated by the same number as the opposite vertex. Theareas are identified by the number of the adjacent side. The quantities Lj(j = 1, 2, 3)
L1 =A1
A, L2 =
A2
A, L3 =
A3
Awhere A is the area of the original triangle, are defined to be the triangularcoordinates. Show that
L1 + L2 + L3 = 1.
The relationship between the Cartesian coordinates x, y which are thecoordinates of the points in the elements, and the triangular coordinatesL1, L2, L3 are
x = L1x1 + L2x2 + L3x3, y = L1y1 + L2y2 + L3y3
where xj , yj (j = 1, 2, 3) are the coordinates of the nodes. The triangularcoordinates can be expressed in terms of the known locations of the vertices,i.e. 1
xy
=
1 1 1x1 x2 x3
y1 y2 y3
L1
L2
L3
.
Find L1, L2, L3 as function of xj , yj , x, y.
Problem 6. Let x, y 6= 0. Find all solutions of the equation
4x2y2
(x2 + y2)2= 1.
Problem 7. Solve the equation
71x2 = 133 (mod 11).
Problem 8. Let x, y ∈ N. Find all solutions of
16x+ 7y = 601.
Problem 9. Let x, y ∈ Z. Find all solutions of
18x+ 12y = 4.
Equations 93
Problem 10. Consider the two hyperplane (n ≥ 1)
x1 + x2 + · · ·+ xn = 2, x1 + x2 + · · ·+ xn = −2.
The hyperplanes do no intersect. Find the shortest distance between thehyperplanes. First consider the cases n = 1 and n = 2 and then the generalcase. What happens if n→∞?
Problem 11. Let d be a positive distance, v, V be velocities v 6= V andT1, T2 time-intervals. Assume that
d
V + v= T1,
d
V − v= T2.
Find d/V .
Problem 12. Solve the quadratic equation
x2 − ix− (1 + i) = 0.
Problem 13. Let z ∈ C. Solve(z + 1z − 1
)= i.
Problem 14. Solve the system of nonlinear equations
x2 − (y − z)2 = a2
y2 − (z − x)2 = b2
z2 − (x− y)2 = c2.
Problem 15. Show that the quartic equation
x4 + qx2 + rx+ s = 0
can be written as
(x2 − ex+ f)(x2 + ex+ g) = 0
where e2 is the root of the cubic equation
z3 + 2qz2 + (q2 − 4s)z − r2 = 0.
94 Problems and Solutions
Problem 16. Let
z =(z1
z2
), w =
(w1
w2
)be elements of C2. Solve the equation z∗w = w∗z.
Problem 17. A special set of coordinates on Sn called spheroconical (orelliptic spherical) coordinates are defined as follows: For a given set of realnumbers α1 < α2 < · · · < αn+1 and nonzero x1, . . ., xn+1 the coordinatesλj (j = 1, . . . , n) are the solutions of the equation
n+1∑j=1
x2j
λ− αj.
Find the solutions for n = 2.
Problem 18. Find all solutions of the system of equations
12
1111
=
cos(α) cos(β)cos(α) sin(β)sin(α) cos(β)sin(α) sin(β)
.
Problem 19. Let φ ∈ [0, 2π). Solve the cubic equation
4x3 − 3x− cos(φ) = 0
over the real numbers.
Problem 20. Let ε > 0. Find the solution of the coupled two non-linearequations
εx2 + x− y − 1 = 0, εy2 + x− y − 1 = 0.
Study ε→ 0 for these solutions.
Problem 21. Consider the six 3× 3 matrices
X12 =
a12 b12 0c12 d12 00 0 1
, X ′12 =
a′12 b′12 0c′12 d′12 00 0 1
,
X13 =
a13 0 b13
0 1 0c13 0 d13
, X ′13 =
a′13 0 b′13
0 1 0c′13 0 d′13
,
Equations 95
X23 =
1 0 00 a23 b23
0 c23 d23
, X ′23 =
1 0 00 a′23 b′23
0 c′23 d′23
.
Find the 9 conditions on the entries such that (local Yang-Baxter equation)
X12X13X23 = X ′23X′13X
′12.
Find solutions of these 9 equations for the 24 unkowns a12, . . . , d′23.
Problem 22. Solve the equations
2 arcsin(x) + arcsin(2x)− π/2 = 0
2 arcsin(x)− arccos(3x) = 0
arccos(x)− arctan(x) = 0
arccos(2x2 − 4x− 2)− 2 arcsin(x) = 0
2 arctan(x)− arctan(
2x1− x2
)= 0.
Problem 23. Let x ∈ R. Find all values of x that satisfy
|5x+ 7| = 3.
Then find the smallest and largest values.
Problem 24. Let θ ∈ [0, 2π). Can one find x ∈ R such that
sin(θ) =e−x/2√1 + e−x
?
Problem 25. Let x < 1. Find a solution of the equation
f2(x)− 2f(x) + x = 0.
Problem 26. What are the conditions on c1 > 0 and c2 > 0 such thatthe system of equations
x1 + x2 = c1, x1x2 = c2
has real solutions?
96 Problems and Solutions
Problem 27. Let m, n be positive integers. Find all solution of thesystem of equations
2mn+ (m2 + n2) = (m2 − n2)2, m− n = 1.
Problem 28. Let x ∈ R.(i) Find all solutions of 2x = |x|+ 1.(ii) Find all solutions of 2x = −|x|+ 1.
Problem 29. Let x ∈ Z. Solve the equation
x2 − 2x+ 2 = 0 mod 5.
Problem 30. Let a > b > 0. Find the points of intersections of the twoellipses
x21
a2+x2
2
b2= 1,
x21
b2+x2
2
a2= 1.
Problem 31. Let r1 ≥ 0, r2 ≥ 0, r3 ≥ 0. Find the solutions of thesystem of equations
r1 + r2 + r3 = 1, r21 + r2
2 + r23 = 1.
Problem 32. Consider the cubic equation
p(x) = x3 − 6x2 + 11x− 6 = 0.
Let x1, x2, x3 be the roots. Given the equations
x1 + x2 + x3 = −a1, x1x2 + x2x3 + x3x1 = a2, x1x2x3 = −a3
Find a1, a2, a3.
Problem 33. Let z ∈ C. Find all solutions of
z + z∗ + zz∗ = 0.
Set z = reiφ with r ≥ 0 and φ ∈ [0, 2π). An alternative would be settingz = x+ iy with x, y ∈ R.
Problem 34. We want to find the positive root of f(x) = 0 with
f(x) = x3 − x2 − x− 1.
Equations 97
We write x3 − x2 − x− 1 = 0 as x = 1 + 1/x+ 1/x2 and set
xt+1 = 1 + 1/xt + 1/x2t , t = 0, 1, . . .
with x0 = 1. Do we find the positive root?
Problem 35. Find all 2× 2 invertible matrices S over R with det(S) = 1such that
S
(0 10 1
)=(
0 10 1
)S S
(0 10 1
)S−1 =
(0 10 1
).
Thus we have to solve the three equations
s21 = 0, s11 + s12 = s22, s11s22 = 1.
Problem 36. Let z ∈ C. Find all solutions of
z + z∗ + zz∗ = 0.
Set z = reiφ with r ≥ 0.
Problem 37. Let a be a real constant. Solve the two equations
−a3 + x3 − 3ax2 + 3a2x1 = 0, x2 − 2ax1 + a2 = 0
with respect to x1, x2, x3.
Programming Problems
Problem 1. Let ε ∈ R. Consider the quadratic equation
x2 − εx+ ε− 1 = 0.
Find the roots x1(ε), x2(ε). Then find the mimimum of x21 +x2
2 with respectto ε.
A SymbolicC++ program to solve this problem is:
// quadratic.cpp
#include <iostream>
#include "symbolicc++.h"
using namespace std;
98 Problems and Solutions
int main(void)
Symbolic x("x"), eps("eps"), f = 0;
Equations soln = solve((x^2)-eps*x+eps-1==0,x);
cout << "Solutions: " << endl << soln << endl;
Equations::iterator i;
for(i=soln.begin();i!=soln.end();i++) f += (i->rhs^2);
cout << "f(eps) = " << f << endl;
Equations min = solve(df(f,eps)==0,eps);
for(i=min.begin();i!=min.end();i++)
if(double(df(f,eps,2)[*i]) > 0)
cout << "Minimum at " << *i << endl;
return 0;
/*
Solutions:
[ x == eps-1,
x == 1 ]
f(eps) = eps^(2)-2*eps+2
Minimum at eps == 1
*/
5.2 Supplementary Problems
Problem 1. Let x, y ∈ R. Find all solutions of
e−x−y = e−x + e−y.
Problem 2. Solve the system of nonlinear equations
x1x2 = 1, x2x3 = 2, x3x1 = 3.
Extend to the general case
x1x2 = c1, x2x3 = c2, . . . , xn−1xn = cn−1, xnx1 = cn
where cj (j = 1, . . . , n) are positive constants.
Problem 3. Find the solution of the equation
e−x − 1 = 2(√
2 + 1).
Equations 99
Problem 4. Let n be a positive integer. Solve the nonlinear equation
1− ne−x
1 + ne−x=x
2
for n = 1, 2, 3, 4, 5.
Problem 5. Let c1 > 0 and c2 > 0. What is the condition on c1, c2 suchthat
x1 + x2 = c1, x1x2 = c2
has a real solution?
Problem 6. Let n be a positive integer. Solve
1− ne−x
1 + ne−x=x
2
for n = 1, 2, 3, 4, 5.
Problem 7. Let x > 0. Show that solution of the equation
x(1 + x1/2) = (1− x1/2)
is given by
x =19
((17 + 3√
33)1/3 + (17− 3√
33)1/3 − 1)2 ≈ 0.2955977 . . .
Problem 8. Let x1, x2, x3 ∈ R. Find solutions of the equation
x21x
22 + x2
2x23 + x2
3x21 = x1x2x3
with x1 6= 0, x2 6= 0, x3 6= 0. Note that
x1 = x2 = x3 = 1/3, x1 = 1/3, x2 = x3 = −1/3, x1 = x2 = −1/3, x3 = 1/3, x1 = x3 = −1/3, x2 = 1/3
are solutions.
Problem 9. (i) Consider the function
f(z, z) = z + z + zz.
Find all solutions of f(z, z) = 0.(ii) Let A be an n× n matrix over C. Find all solutions of
A+A∗ +AA∗ = 0n.
100 Problems and Solutions
Find all 2× 2 matrices A such that
A+A∗ +AA∗ = I2.
Problem 10. (i) Let φ, θ ∈ [0, 2π). Consider the equation
eiφ sin(θ) =2η
|η|2 + 1.
Let η = 0, 1,−1, i,−i. Find φ and θ.(ii) Let θ ∈ [0, 2π). Consider the equation
cos(θ) =|η|2 − 1|η|2 + 1
.
Let η = 0, 1,−1, i,−i. Find φ and θ.
Problem 11. Show that roots of the polynomial f : R→ R
f(x) = x4 − x3 − 10x2 − x+ 1
are given by
x1 =12
(√
5− 3), x2 =12
(−√
5− 3), x3 = 2 +√
3, x4 = 2−√
3.
Problem 12. Find the real root of the polynomial f : R→ R
f(x) = x3 + x− 1.
The real root lie in the interval [0, 1]. Note that f(0) = −1 and f(1) = 1.
Problem 13. Find solution solutions of the equation (Scherk’s surface)
cos(x2)ex3 = cos(x1).
For example (x1, x2, x3) = (0, 0, 0) is a solution and (x1, x2, x3) = (π/3, π/3, 0)is a solution.
Problem 14. Let x > 0. Find all solutions of
x =√
1 +√
2 + x.
First show that x4 − 2x2 − x− 1 = 0.
Equations 101
Problem 15. Show that the system of equations
3x+ y − z + u2 = 0x− y + 2z + u= 0
2x+ 2y − 3z − 2u= 0
can be solved for x, y, u in terms of z; for x, z, u in terms of y; for y, z, uin terms of x; but not for x, y, z in terms of u.
Problem 16. Are there solutions of the equation (z ∈ C)
sin(z) = ez and cos(z) = ez ?
Problem 17. (i) Show that cos(π/7) is a real root of
8x4 + 4x3 − 8x2 − 3x+ 1 = 0.
(ii) Show that
cos(π/7) cos(2π/7) cos(4π/7) = −1/8.
Problem 18. Draw the curve in the plane given by
sinh(x) = exp(−2y).
Note that if y = 0, then sinh(2x) = 1 and therefore 2x = arcsinh(1).
Problem 19. Solve the system of integral equations∫ 2π
0
(f(φ))3 cos(φ)dφ = 0,∫ 2π
0
(f(φ))3 sin(φ)dφ = 0.
Problem 20. Consider the two circles in the plane
(x1 + 4)2 + (x2 + 5)2 = 194, (x1 − 3)2 + (x2 − 2)2 = 40.
Show that the two circles intersect and find the points of intersection. Showthat x1 + x2 = 9.
Chapter 6
Normed Spaces
Let C be the field of complex numbers. Let V be a vector space (linearspace) over C. Then V is a normed linear space if for every f ∈ V there isa real number ‖f‖ such that (c ∈ C)
(i) ‖f‖ ≥ 0
(ii) ‖f‖ = 0 if and only if f = 0
(iii) ‖cf‖ = |c| ‖f‖for every c ∈ C
(iv) ‖f + g‖ ≤ ‖f‖+ ‖g‖.
A normed linear space V which does have the property that all Cauchysequences are convergent is said to be complete. A complete normed linearspace is called a Banach space.
102
Normed Spaces 103
6.1 Solved Problems
Problem 1. Consider the Banach space R4 and the four vectors
v1 =
1x
x2/2x3/3
, v2 =
01xx2
, v3 =
001
2x
. v4 =
0002
where x ∈ R. Show that the vectors are linearly independent. Find thedistances
‖v1 − v2‖, ‖v2 − v3‖, ‖v3 − v4‖, ‖v4 − v1‖.
Problem 2. Consider the Hilbert space C2 and the vectors
v1 =(
cos(i)sin(i)
), v2 =
(− sin(i)cos(i)
).
Find the distance ‖v1 − v2‖.
Problem 3. (i) Let a, b ∈ R. Show that
d(a, b) := | arctan(a)− arctan(b)|
defines a distance in R.(ii) Show that xn = arctan(n), (n ∈ N) is a Cauchy sequence. Is the metricspace R , d complete?
Problem 4. Let n ≥ 1, 0 ≤ a < b and p ∈ Rn. Show that there existsa map k : C∞(Rn,R) such that k(x) = 0 for ‖x − p‖ ≥ b, k(x) = 1 for‖x− p‖ ≤ a, and 0 < k(x) ≤ 1 for ‖x− p‖ ≤ b.
Problem 5. Let x ∈ R2. Is
‖x‖ := |x1x2|1/2
a norm on R2?
Problem 6. Show that if a function f : R→ R satisfies
|f(x)− f(y)| ≤M(|x− y|)a
for some fixed M > 0 and a > 1, then f is a constant function, i.e., f isidentically equal to some real number b for all x ∈ R.
104 Problems and Solutions
Problem 7. Let f , g be continuously differentiable functions on theinterval [0, 1]. One defines
〈f, g〉 =∫ 1
0
(f(x)g(x) +
df
dx
dg
dx
)dx.
Show that this satisfies the properties of an inner product. Calculate 〈f, g〉for f(x) = sin(x), g(x) = cos(x). Extend it to
〈f, g〉 =∫ 1
0
n∑j=0
djf(x)dxj
djg(x)dxj
dx.
Problem 8. The p-norm of a vector (x1, . . . , xn) ∈ Rn is defined as
‖x‖p :=
n∑j=1
|xj |p1/p
where p ∈ R+. Find the norm for p→∞.
Problem 9. Let x ∈ Rn and ‖x‖ be the Euclidean norm of x. If B ⊂ Rmis a nonempty compect set and x ∈ Rn, then we define the distance
dist(x, B) := min ‖x− b‖ : b ∈ B .
If A,B ⊂ Rn are nonempty compact sets then we define the distance
dist(A,B) := max dist(a, B) : a ∈ A .
Show thatdist(A,B) = 0 ⇔ A ⊂ B
and dist(A,B) < ε means that A ⊂ Nε(B) where
Nε(B) := x : dist(x, B) < ε
is the ε-neighborhood of B.
Problem 10. Let v and w be two normalized column vectors in Cn.Does
D(v,w) := 2 arccos
(√(v∗w)(w∗v)(v∗v)(w∗w)
)provide a distance measure between v and w.
Normed Spaces 105
Problem 11. Let z1, z2 ∈ C. One can define a distance measure via
ρ(z1, z2) =|z1 − z2|√
(1 + |z1|2)(1 + |z2|2)
Let z1 = eiφ1 and z2 = eiφ2 . Find ρ(z1, z2).
Problem 12. The chordal distance between to complex numbers z1, z2
is defined as
d(z1, z2) :=|z1 − z2|√
1 + |z1|2√
1 + |z2|2.
Let z1 = eiπ/2 and z2 = e−iπ/2. Find d(z1, z2).
Problem 13. Find n× n matrices A, B such that
‖[A,B]− In‖ → min
where ‖.‖ denotes the norm and [, ] denotes the commutator.
106 Problems and Solutions
6.2 Supplementary Problems
Problem 1. Let x, y ∈ R. Is
d(x, y) =|x− y|
2 + |x− y|
a metric on R? Note that d(x, x) = 0 and d(x, y) ≥ 0.
Problem 2. Let R be the set of real numbers and x, y ∈ R. Show that
d(x, y) = |x− y|
provides a distance function. Let x = (x1, . . . , xn) ∈ Rn and y = (y1, . . . , yn) ∈Rn Show that
d(x,y) =
√√√√√ n∑j=1
(xj − yj)2
provides a distance function. Of course one always assume the positivesquare root.
Problem 3. Consider the functions f : R→ R, g : R→ R
f(x) = exp(−x2), g(x) = exp(−|x|).
Find the distance d(f, g) between f and g given by
d(f, g) =∫
R
|f(x)− g(x)|1 + |f(x)− g(x)|
dx.
Chapter 7
Complex Numbers andComplex Functions
7.1 Solved ProblemsProblem 1. Let z ∈ C. Solve the nonlinear equation
z3 = z|z|2.
Problem 2. (i) Find the complex numbers z satisfying z2 = z.(ii) Find the complex numbers z satisfying z3 = z.
Problem 3. Solve the cubic equation z3 = −1. Do the solutions forma group under multiplication? If not, what numbers have to be added toform a group. Find them by multiplication of the solutions of the cubicequation.
Problem 4. Find the square root of z = 4 exp(iπ/4).
Problem 5. Let φ ∈ [0, 2π). Consider the complex number z = 1− eiφ.Find the condition on φ such that |z| = 1.
107
108 Problems and Solutions
Problem 6. Let x, y ∈ R and z = x + iy. Find the real and imaginarypart of i/(πz).
Problem 7. Let φ ∈ [0, 2π) and θ ∈ (−π, π). Can any complex numberbe represented by
z = eiφ tan(
12θ
)?
Problem 8. Find
zdz − zdz, (zdz − zdz)⊗ (zdz − zdz).
Problem 9. Let x, c ∈ R and c 6= 0. Find the real and imaginary part ofthe function
fc(x) =x− icx+ ic
.
Problem 10. Let x, y ∈ R. Solve
x− ix+ i
y − iy + i
=xy − ixy + i
.
Problem 11. Find√i.
Problem 12. (i) Calculate
p(α, β, θ, φ) = | cos(α) cos(β) sin(θ)eiφ + sin(α) sin(β) cos(θ)|2.
(ii) Show that p(α, β, θ, φ) ≤ 1.(iii) Simplify the result from (i) for θ = π/4 and φ = 0.
Problem 13. Let z1, z2 ∈ C. Consider the distance measures
d1(z1, z2) := |z1 − z2|, d2(z1, z2) :=|z1 − z2|
1 + |z1 − z2|,
d3(z1, z2) :=|z1 − z2|√
(1 + |z1|2)(1 + |z2|2).
Here d3 is the chordal distance. Let z1 = eiφ and z2 = eiφ2 . Find d1, d2,d3.
Complex Numbers and Complex Functions 109
Problem 14. Let f(z1) and g(z2) be a pair of analytic functions of z1
and z2, respectively. We define
f(z1) g(z2) :=1
2π
∫ 2π
0
f(z1eiθ)g(z2e
−iθ)dθ.
Let
f(z1) =∞∑k=0
akzk1 , g(z2) =
∞∑k=0
bkzk2
for |zj | < R, j = 1, 2. Find f(z1) g(z2).
Problem 15. Let z = x+ iy, where x, y ∈ R. Find
∂
∂z,
∂
∂z,
∂
∂x,
∂
∂y.
Problem 16. Study the behaviour (stability) of the fixed points of thecomplex map f : C→ C, f(z) = z2.
Problem 17. Let z = x + iy (x, y ∈ R) be a nonzero complex number.We define the principal argument by z = |z| exp(iarg(z)), where arg(z) ∈(−π, π] and we define the imaginary remainder Imr(z) and the imaginaryquotient Imq(z) by
=(z) = Imr(z) + 2πImq(z)
where Imr(z) ∈ (−π, π] and Imq(z) ∈ Z. Show that
ln(ez) = <(z) + iImr(z)
and in particular
ln(ez) = z iff =(z) ∈ (−π, π].
Problem 18. Let z ∈ C and consider the analytic map
f(z) = exp(z).
Find the solutions of the equation
z = f(z).
This are the fixed points of f . We set z = x+ iy (x, y ∈ R. Then
x+ iy = exp(x+ iy) ≡ exeiy = ex(cos(y) + i sin(y)).
110 Problems and Solutions
Thus we have to solve
ex cos(y)− x = 0, ex sin(y)− y = 0.
Problem 19. Let A be an n×n matrix. Suppose f is an analytic functioninside on a closed contour Γ which encircles λ(A), where λ(A) denotes theeigenvalues of A. We define f(A) to be the n× n matrix
f(A) =1
2πi
∮Γ
f(z)(zIn −A)−1dz.
This is a matrix version of the Cauchy integral theorem. The integral isdefined on an element-by-element basis f(A) = (fjk), where
fjk =1
2πi
∮Γ
f(z)eTj (zIn −A)−1edz.
Let f(z) = z2 and
A =(
0 11 0
).
Calculate f(A).
Problem 20. Let x1, y1, x2, y2 ∈ R. Consider the two complex numbers
z1 = x1 + iy1, z2 = x2 + iy2.
In polar form we have
z1 = r1eiφ1 = r1(cos(φ1)+ i sin(φ1)), z2 = r2e
iφ2 = r2(cos(φ2)+ i sin(φ2))
where r21 = x2
1 + y21 , r2
2 = x22 + y2
2 . Now
z = z1 + z2 = (x1 + x2) + i(y1 + y2).
Find z in polar form z = reiφ.
Problem 21. Let a, b, φ be fixed real numbers. Consider the function
w(z) = z + az2 + beiφz3.
Write it as real and imginary part, with w = u+ iv and z = x+ iy.
Problem 22. Consider the z-transform
x(z) =∞∑n=0
x(n)z−n, x(n) =1
2πi
∮x(z)zn−1dz.
Complex Numbers and Complex Functions 111
Let
S(N) :=N∑n=1
x(n).
Then
S(N) =N∑n=1
x(n) =1
2πi
∮x(z)
N∑n=1
zn−1dz.
It follows that (geometric series)
S(N) =1
2πi
∮x(z)(zN − 1)
z − 1dz.
Apply this expression and the residue theorem to calculate
S(N) =N∑n=1
n3.
Problem 23. (i) Consider the complex number z = eiφ. Let n ∈ N. Findn√z.
(ii) Let z = reiφ and w = x+ iy (x, y ∈ R). Find zw.
Problem 24. Consider the complex numbers z1 = 0.4 + 0.3i, z2 = 5 + 2i.Calculate zz21 . Hint: Set z1 = r1e
iφ1 .
Problem 25. Consider the complex numbers
z1 = x1 + iy1 = 1 + 4i, z2 = x2 + iy2 = 3− 2i.
Calculate logz2(z1).
Problem 26. Let A > 0 and B ≥ 0. Consider the quadratic conformalmap in the complex w-plane of the unit disc in the complex z-plane
w(z) = Az +Bz2.
Let θ be the polar angle in the complex z-plane.(i) Show that with the notation w := u + iv, z := x + iy (with u, v, x, yreal) one has the parametric equation of the boundary
u(θ) = A cos(θ) +B cos(2θ), v(θ) = A sin(θ) +B sin(2θ).
(ii) Let C := B/A. Show that for C = 0 (i.e. B = 0) one obtains a circulardisc. Show that for C = 1/4 the curvature vanishes at θ = π. Show thatfor C = 1/2 the derivative dw/dz vanishes at the boundary, i.e. at θ = π.
112 Problems and Solutions
Problem 27. (i) Solve the equation(z + i
2
z − i2
)4
= 1.
(ii) Solve the system of equations(z1 + i
2
z1 − i2
)4
=z1 − z2 + i
z1 − z2 − i,
(z2 + i
2
z2 − i2
)4
=z2 − z1 + i
z2 − z1 − i.
These equations play a role for the Bethe ansatz for spin systems.
Problem 28. Let z be a complex number such that |z| <∞ and <(z) > 0.Consider
Z(z) =∫ ∞
0
exp(−zx− x2)dx.
Show that
Z(z) =12
∞∑k=0
(−1)kΓ( 1
2 + 12k)
k!zk.
Problem 29. Let z 6= 0. Show that the function
f(z) =ln(z)z − 1
is analytic near z = 1 and admits the Taylor expansion
f(z) =∞∑n=0
(−1)n
n+ 1(z − 1)n for |z − 1| < 1.
Problem 30. Is the function f : C→ C
f(z) = z + |z|
continuous? Find the fixed points of f .
Problem 31. Let x1, x2, y1, y2 ∈ R and z1 = x1 + iy1, z2 = x2 + iy2 withz1z1 6= 0, z2z2 6= 0. Find the conditions such that
( z1 z2 )(z2
−z1
)= 0.
Complex Numbers and Complex Functions 113
Problem 32. Let z = ρeiθ and ζ = Reiω with with R > 0 and ρ > 0.Show that
ρ2 −R2
ρ2 − 2Rρ cos(ω − θ) +R2= <
(z + ζ
z − ζ
).
7.2 Supplementary Problems
Problem 1. Show that
15
(−1 + 2i) =1√5ei(π−arctan(2)).
Problem 2. Let z1 = r1eiφ1 , z2 = r2e
iφ2 with r1, r2 ≥ 0. Show that
<(z1 + z2
z1 − z2
)=
r21 + r2
2
|z1 − z2|.
Problem 3. Let z = x+ iy with x, y ∈ R. Show that
<(z2) = x2 − y2.
Problem 4. Let 0 < r < 1 and φ ∈ [0, 2π). Consider the map
f(z) = reiφ + (1− reiφ)z.
Find the fixed points. Find f(0), f(f(0)), f(f(f(0))).
Problem 5. Let x, y ∈ R. Find real and imaginary part of
z =
√x− iyx+ iy
eiφ.
Problem 6. Let z ∈ C. Show that
sin(z) = z
∞∏k=1
(1− z2
k2π2
).
114 Problems and Solutions
Problem 7. Show that the Bessel function
Jν(z) =∞∑n=0
(−1)n
n!Γ(ν + n+ 1)
(z2
)2n+ν
is an entire function of z for ν = 0, 1, . . .. Show that
Jν−1(z) + Jν+1(z) = 2ν1zJν(z).
Problem 8. (i) Show that the radius of convergence of the function
f(z1, z2) =∞∑k=0
zk1z2
is given by (z1, z2) : |z1| < 1 ∨ z2 = 0 .(ii) Show that the radius of convergence of the function
f(z1, z2) =∞∑k=0
(z1z2)k
is given by (z1, z2) : |z1| · |z2| < 1 .(iii) Show that the radius of convergence of the function
f(z1, z2) =∞∑k1=0
k1∑k2=0
(k1
k2
)zk21 zk1−k22
is given by (z1, z2) : |z1|+ |z2| < 1 .
Problem 9. (i) Show that the sum
∞∑k=0
k!zk
only converges for z = 0.(ii) Show that the sum
∞∑k=0
zk
k2
converges for |z| ≤ 1 and diverges for |z| > 1.(iii) Show that the sum
∞∑k=0
(z1z2)k
Complex Numbers and Complex Functions 115
converges for (z1, z2) : |z1| · |z2| < 1 .(iv) Show that the sum
∞∑k1=0
k1∑k2=0
(k1
k2
)zk21 zk1−k22
converges for (z1, z2) : |z1|+|z2| < 1 . The sum appears at the expansionof 1/(1− (z1 + z2)).
Problem 10. Let x ∈ R+ and f(x) =√x. Show that real function
can be extended with a power series expansion around 1 into the complexdomain
f(z) =√
1 + (z − 1) =∞∑k=0
(1/2k
)(z − 1)k
and for z = reiφ with −π/2 < φ < π/2 we have f(z) =√reiφ/2.
Problem 11. (i) Is the complex function
f(z) = z = x1 − ix2
holomorph?(ii) Is the complex function
f(z) = z2 − z2 = 4ix1x2
holomorph?
Problem 12. (i) Show that if n is a positive integer then
(r(sin(φ) + i sin(φ)))n ≡ rn(cos(nφ) + i sin(nφ)). (1)
Hint. Apply exp(iφ) ≡ cos(φ) + i sin(φ).(ii) Show that the n-th roots of unity are
ρ = cos(
2πn
)+ i sin
(2πn
), ρ2, ρ3, . . . , ρn−1, ρn ≡ 1. (1)
Problem 13. Let ε ∈ R and z ∈ C. Consider the product
f(z) =∞∏k=1
1− e−εk
1− e−ε(k+z).
116 Problems and Solutions
Find f(0). Show that
f(z) = (1− e−εz)f(z − 1).
Show that
f(m) =m∏k=1
(1− e−εk), m = 1, 2, . . .
Show that f is periodic with period 2πi/ε. Show that f has simple polesat z = −m (m = 1, 2, . . .). Show that the residues are given by
limz→−m
(z +m)f(z) =(−1)m+1 exp(− 1
2εm2 + 1
2εm)εf(m− 1)
.
Problem 14. Show that
(1 + i)1/2 = ±21/4 (cos(π/8) + i sin(π/8)) .
Problem 15. Let n = 0, 1, . . . and x ∈ R. Find the real and imaginarypart of the functions
fn(x) =1√π
(ix− 1)n
(ix+ 1)n+1.
Problem 16. Show that
1(n− k)!
=1
2πi
∮dt
et
tn−k+1
where the integration contour is a small circle around the origin in thecomplex plane.
Problem 17. We know that 0 ≤ | sin(x)| ≤ 1 and 0 ≤ | cos(x)| ≤ 1 forx ∈ R. This is no longer true for sin(z) and cos(z) with z ∈ C.(i) Let a > 0. Show that
| sin(az)| =√
sinh2(ay) + sin2(ax), | cos(az)| =√
sinh2(ay) + cos2(ax).
(ii) Find all solutions of | sin(z)| = 2 and | cos(z)| = 2.(iii) Find all solutions of cos(z) = i.
Complex Numbers and Complex Functions 117
(iv) Let n ∈ N. Show that (x, y ∈ R)
| sin(n(x+ iy))|=√
sin2(nx) cosh2(ny) + cos2(nx) sinh2(ny)
> | sinh2(ny)| → ∞
as n→∞ for any y 6= 0.
Chapter 8
Integration
8.1 Solved ProblemsProblem 1. The time-average of a continuous function f is
〈f〉 := limT→∞
12T
∫ T
−Tf(t)dt.
Find the time-average of the functions
f1(t) = cos(ωt) sin(ωt), f2(t) = cos2(ωt), f3(t) = sin2(ωt).
Problem 2. Let j, k = 1, 2, . . .. Consider the function
f(j, k) =∫ 1
0
xj−1(1− x)k−1dx, j, k = 1, 2, . . . .
Thus f(1, 1) = 1. Is
f(j − 1, k + 1) =k
j − 1f(j, k), j ≥ 2
andf(j + 1, k) = f(j, k)− f(j, k + 1) ?
Prove or disprove.
118
Integration 119
Problem 3. Let
x(τ) =
1 for 0 ≤ τ ≤ 10 otherwise
and
h(τ) =
1 for 0 ≤ τ ≤ 10 otherwise .
Find the convolution integral
y(t) =∫ ∞−∞
x(τ)h(t− τ)dτ.
Problem 4. Let T > 0 and ω = 2π/T . Let m,n ∈ N and α, β ∈ R.Calculate
I(α, β) =1T
∫ T
0
cm sin(mωt+ φm − α)cn sin(nωt+ φn − β)dt.
Problem 5. A cubic B-spline with uniform knot spacing, centered at theorign, is given by
B(x) =
16 (2− |x|)3 if 1 ≤ |x| < 2
16 ((2− |x|)3 − 4(1− |x|)3) if 0 ≤ |x| < 1
0 otherwise
Find the integral ∫ ∞−∞
B(x)dx.
Problem 6. Find the normalization constant K from the condition
1 = 4K∫ 2π
0
sin2(ω/2)dω∫ π
0
sin(θ)dθ∫ π
−πdφ.
Problem 7. Let ε ∈ R. Calculate the integral
f(ε) =∫ 2π
0
exp(εeiφ)dφ
by finding an ordinary differential equation for f together with the initialconditions. Obviously f(0) = 2π.
120 Problems and Solutions
Problem 8. Let b > a. Find the mean and variance of random variablex with uniform probabilty density function p
p(x) =
1b−a if a ≤ x ≤ b0 otherwise
Problem 9. Consider a one-dimensional lattice (chain) with lattice con-stant a. Let k be the sum over the first Brioullin zone we have
1N
∑k∈1.BZ
F (ε(k))→ a
2π
∫ π/a
−π/aF (ε(k))dk = G
whereε(k) = ε0 − 2ε1 cos(ka).
Using the identity ∫ ∞−∞
δ(E − ε(k))F (E)dE ≡ F (ε(k))
we can write
G =a
2π
∫ ∞−∞
F (E)
(∫ π/a
−π/aδ(E − ε(k))dk
)dE.
Calculate
g(E) =∫ π/a
−π/aδ(E − ε(k))dk
where g(E) is called the density of states.
Problem 10. Let
i1(t) = I2 sin2(ωt), i2(t) = I2 sin2(ωt+ φ).
Calculate
〈i1(t)i2(t)〉 := limT→∞
1T
∫ T
0
i1(t)i2(t)dt
and〈i1(t)i2(t)〉 − 〈i1(t)〉〈i2(t)〉.
Problem 11. Let α > 0. Find∫ ∞0
exp(−αx)dx =1α.
Integration 121
Problem 12. Let f(t) be a continuous function. Show that∫ x
0
dζ
∫ ζ
0
f(t)dt =∫ x
0
(x− t)f(t)dt.
Problem 13. (i) Calculate∫ ∞−∞
sech(t)tanh(t) cos(t+ t0)dt.
(ii) Calculate ∫ ∞−∞
sech2(t)tanh2(t)dt.
Problem 14. Let λ > 0. Calculate∫ ∞−∞
sin(λt)λ sinh(t)
dt.
Problem 15. Calculate∫ ∞−∞
exp(−x2 + 2ixy)dx.
Problem 16. Calculate the integral
I(λ) =∫ ∞−∞
et
(1 + et)2cos(t+ λ)dt
using the residue technique.
Problem 17. Let m be an non-negative integer. Find∫ π
0
dθ| cos2m+1(θ)|.
Problem 18. Let ε > 0. Calculate∫ ∞0
k2dk
eεk2 − 1.
122 Problems and Solutions
Problem 19. The mother Haar wavelet is given by
f(t) =
−1 for 0 ≤ t < 1/2+1 for 1/2 ≤ t < 10 otherwise
Find the Fourier transform
f(ω) =∫ ∞−∞
e−iωtf(t)dt.
Problem 20. The Poisson wavelet is given by
f(t) =(td
dt+ 1)P (t)
whereP (t) =
1π
11 + t2
.
Find the Fourier transform of f .
Problem 21. Let m ∈ Z. Calculate
am =1
2π
∫ 2π
0
eimφ2 cos(φ)dφ.
Problem 22. Let 0 < α < 1. Find the integral∫ ∞0
xα−1
1 + xdx.
Problem 23. The linear one-dimensional diffusion equation is given by
∂u
∂t= D
∂2u
∂x2, t ≥ 0, −∞ < x <∞
where u(x, t) denotes the concentration at time t and position x ∈ R. Dis the diffusion constant which is assumed to be independent of x and t.Given the initial condition c(x, 0) = f(x), x ∈ R the solution of the one-dimensional diffusion equation is given by
u(x, t) =∫ ∞−∞
G(x, t|x′, 0)f(x′)dx′
Integration 123
where
G(x, t|x′, t′) =1√
4πD(t− t′)exp
(− (x− x′)2
4D(t− t′)
).
HereG(x, t|x′, t′) is called the fundamental solution of the diffusion equationobtained for the initial data δ(x− x′) at t = t′, where δ denotes the Diracdelta function.(i) Let u(x, 0) = f(x) = exp(−x2/(2σ)). Find u(x, t).(ii) Let u(x, 0) = f(x) = exp(−|x|/σ). Find u(x, t).
Problem 24. Let ω0 > 0 be a fixed frequency and t the time. Calculate
f(ω) =1√2π
∫ +∞
−∞e−|ωt|e−iωtdt.
Problem 25. Let ω0 > 0 be a fixed frequency and t the time. Calculate
f(ω) =1√2π
∫ ∞−∞
e−|ω0t|e−iωtdt.
Problem 26. The sum
limn→∞
1n
n∑k=1
exp(
2 cos(
kπ
n+ 1
))can be cast into the integral
limn→∞
1n
∫ n
0
exp(2 cos(πx))dx. (1)
Calculate this integral.
Problem 27. Let f : R → R be an analytic function. The Dirichletintegral identity is given by∫ u
0
∫ u−w2
0
f(u− w1 − w2)wµ1−11 wµ2−1
2 dw1dw2 =
Γ(µ1)Γ(µ2)Γ(µ1 + µ2)
∫ u
0
f(u− w)wµ1+µ2−1dw.
Let f(x) = e−x. Calculate the left and right-hand side of this identity.
124 Problems and Solutions
Problem 28. Let r1 > 0, r2 > 0. Find the integral
I(r1, r2) =∫ 1
−1
dx√r21 + r2
2 − 2r1r2x.
Problem 29. Show that
ea2
=∫
Rdx exp
(−x
2
2+√
2ax).
Problem 30. Let u ≥ 0 and
ρ(u) =12
exp(−√u).
Find
ρn =∫ −∞
0
unρ(u)du.
Problem 31. Calculate the integral
I =∫ ∫
D
√x2 + 4y2dydx
where D is the domain bounded by the positive x-axis, the positive y-axisand the parabola y2 = 1− x.
Problem 32. Calculate the definite integral∫ 1
0
sin(x2)dx.
Problem 33. (i) Consider the wavelet (ω0 > 0)
ψ0(t) = (eiω0t − e−ω20/2)e−t
2/2.
Show that ∫ ∞−∞
ψ0(t)dt = 0.
Hint: Use (a > 0)∫ ∞−∞
e−(ax2+bx+c)dx =√π
ae(b2−4ac)/4a.
Integration 125
(ii) We define
ψn(t) :=d
dtψn−1(t), n = 1, 2, . . .
Show that ∫ ∞−∞
tkψn(t)dt = 0, 0 ≤ k ≤ n.
Problem 34. Let b > a. Consider the integral∫ y=b
y=a
f(y)dy
where f is a continuous function in [a, b]. Apply the transformation
y(x) =12
((b− a)x+ a+ b)
so that the integration range is between −1 and +1. Then the Gaussquadrature can be applied which extends over the interval [−1,+1].
Problem 35. Let ε > 0. Find f of the equation
exp(−εt) = 1−∫ t
0
f(s)ds.
Problem 36. (i) Find the area of the set
S2 := (x1, x2) : 1 ≥ x1 ≥ x2 ≥ 0 .
(ii) Find the volume of the set
S3 := (x1, x2, x3) : 1 ≥ x1 ≥ x2 ≥ x3 ≥ 0 .
Extend the n-dimensions.
Problem 37. Let ω1, ω2 be real and positive. Find
J(ω1, ω2) =1√2π
∫ ∞−∞
exp(−12x2 + iω1x+ iω2x
2)dx.
Problem 38. Let e be the eccentricity of an ellipse, i.e.√
1− e2 = b/a.Let n ≥ 1. Calculate the integral∫ 2π
0
dx(1e − cos(x)
)n
126 Problems and Solutions
by making the substitution z = exp(ix). Apply the residue theorem.Hint. We have∫ 2π
0
dx(1e − cos(x)
)n =∫|z|=1
dz(1e −
12
(z + 1
z
))niz
=∫|z|=1
i(−1)n+12nzn−1dz(z2 − 2
ez + 1)n .
Problem 39. Let q2 > 0 and (p− q)2 > 0. Find∫ ∞0
exp(−xq2)dx,∫ ∞
0
exp(−x(p− q)2)dx.
Problem 40. Find α > 0 such that∫ +∞
−∞exp(−α|x|)dx = 1.
Afterwards calculate∫ ∞−∞
x exp(−α|x|)dx,∫ ∞−∞
x2 exp(−α|x|)dx.
Problem 41. Let c > 0. Show that
exp(
12cy2
)=
1√2π
∫ ∞−∞
dx exp(−1
2x2 +
√cyx
).
Problem 42. The Gauss invariant for two given closed loops Cα and Cβin R3 parametrized by rα(s), rβ is defined by
G(Cα, Cβ) :=1
4π
∮Cα
ds
∮Cβ
ds′drα(s)ds
× drβ(s′)ds′
rα(s)− rβ(s′)|rα(s)− rβ(s′)|3
where × denotes the vector product. Find G(Cα, Cβ) for the two curves
Cα : x21 + x2
3 = 1, Cβ : (x1 − 1)2 + x22 = 1.
Problem 43. Let x ∈ R. We define [x] as the integer part of x andx := x− [x]. Calculate the integrals∫ 7/2
0
[x]dx,∫ 7/2
0
xdx.
Integration 127
Problem 44. Let x ∈ R. Consider the integral
f(x) =∫ ∞x
eiy
ydy.
(i) Show that f(−|x|) = f(|x|)− iπ.(ii) Show that for large x
f(x) = eix(i
x+
1x2
+ · · ·).
Problem 45. Let n = 0, 1, 2, . . .. We define
an :=∫ π/2
0
cos2n(x)dx, bn :=∫ π/2
0
x2 cos2n(x)dx.
Then a0 = π/2 and b0 = π3/24. Show that using integration by parts
an = (2n− 1)(an−1 − an).
Show that for n ≥ 1 we have
an = (2n− 1)nbn−1 − 2n2bn.
Show that (n ≥ 1)
0 ≤ π2
6−
n∑k=1
1k2
= 2bnan≤ π2
4(n+ 1).
Problem 46. Calculate
I =∫ 10
0
(x− int(x))dx
where int(x) defines the largest integer less than x.
Problem 47. Let 0 < a < b < 1. Find∫ b
a
ln(1− x)x
dx.
Problem 48. Show that∫ π/2
0
cos10(x) cos8(2x) cos6(4x) cos4(6x) cos2(8x)dx =
128 Problems and Solutions
5166673π536870912
+296654976251281698120709987525225
.
Problem 49. Let z ∈ C. Let ln(1 + z) be the branch of the logarithmdefined on C \ (−∞,−1]. Calculate
In(r) = Pv
∫|z|=r
zn−1 ln(1 + z)dz, r > 0, n ∈ Z
where Pv is the principal value.
Problem 50. Let b > a and a, b be finite. Consider the integral∫ b
a
f(x)dx.
(i) Apply the transformation
x =12
(a+ b+ (b− a) tanh(y)⇔ y = tanh−1
(2x− a− bb− a
)to the integral.(ii) Apply the transformation
x =12
(a+ b+ (b− a) tanh
(π2
sinh(y)))
withdx
dy=
(b− a)π cosh(y)/4cosh2(π sinh(y)/2)
to the integral.
Problem 51. Calculate the integral∫ π
0
1 + sin(x)1 + cos(x)
dx
utilizing the transformation t = tan(x/2) with −π < x < π. From thistransformation it follows that
x = 2 arctan(t), dx =2
1 + t2dt
and x = 0→ t = 0, x = π/2→ t = 1.
Integration 129
Problem 52. Simplify the integral
I =∫ 1
0
cos(x)√x
dx
for numerical calculation.
Problem 53. Let a > 0. Let f : R → R, g : R → R be continuousfunctions with f(−x) = f(x) and g(−x) = −g(x). Show that∫ a
−af(x)g(x)dx = 0.
Problem 54. Show that∫ ∞0
e−xx2dx = 2!,∫ ∞
0
e−xx3dx = 3!.
Let n = 4, 5, . . .. Show that∫ ∞0
e−xxndx = n!.
Problem 55. Let k1 > 0, k2 > 0, k3 > 0. Find the integral∫ ∞0
sin(k1r) sin(k2r) sin(k3r)dr
r.
Problem 56. Calculate
F (s, t) =∫
Re−ixt−|x−s|dx.
Problem 57. Let x ∈ (0, 1). Calculate
P
∫ 1
0
ym
(y − x)dy
for m = 0, m = 1, m = 2, m = 3.
Problem 58. Let a, b > 0 and b > |a|. Show that∫R
dx
x2 + 2ax+ b2=
π√b2 − a2
.
130 Problems and Solutions
Problem 59. Let a, b > 0. Show that∫ ∞0
e−ax − e−bx
xdx = ln
(b
a
).
Problem 60. Let x > 0. Show that∫ ∞0
e−txdt =1x.
Problem 61. a, b ∈ R. Let
f(x) = a exp(bx).
Find the conditions on a and b such that∫ 1
0
f(x)dx = 1,∫ 1
0
xf(x)dx =12.
Problem 62. Dawson’s integral is given by
f(x) =∫ x
0
et2−x2
dt, x ≥ 0.
(i) Show that for all complex values z the function f satisfies the lineardifferential equation
df(z)dz
+ zf(z) = 1.
(ii) Let j = 1, 2, . . .. Show that
f (j+1)(z) + 2zf (j) + 2jf (j−1)(z) = 0, j = 1, 2, . . .
where f (j) indicates the jth derivative.
Problem 63. Let a, b, c ∈ R and a+ b cos(θ) + c sin(θ) 6= 0. Show that∫ +π
−π
b sin(θ)− c cos(θ)a+ b cos(θ) + c sin(θ)
dθ = 0.
Problem 64. Let a > 0. Show that∫ ∞0
cos(ax)1 + x2
dx =π
2e−a.
Integration 131
Problem 65. Show that∫ ∞0
ln(cosh(x))cosh3(x)
dx =π
4(ln(2)− 1/2).
Problem 66. Let m,n = 0, 1, 2, . . .. Find the integral
fmn(t) =∫ t
0
(t− τ)mτndτ.
Problem 67. Let a, b > 0. Find the integral∫ ∞0
cos(at)− cos(bt)t
dt.
Problem 68. Let q2 > 0. Calculate the integral∫ ∞0
exp(−q2x)dx.
Problem 69. Let x > 0. Assume that f : R→ R is integrable over [0, x]for all x > 0 and limx→∞ f(x) = a. Show that
limx→∞
1x
∫ x
0
f(s)ds = a.
Utilize∣∣∣∣ 1x∫ x
0
f(s)ds− a∣∣∣∣ =
∣∣∣∣ 1x∫ x
0
(f(s)− a)ds∣∣∣∣ ≤ ∣∣∣∣ 1x
∣∣∣∣ ∫ x
0
|f(s)− a|ds.
Problem 70. Consider the function f : [0, 1]→ [0, 1]
f(x) =
1/x− int(1/x) x 6= 00 x = 0
where int(y) denotes the integer part of y. The function is integrable.(i) Let k ≥ 1 be a positive integer. Find∫ 1/k
1/(k+1)
f(x)dx.
132 Problems and Solutions
(ii) Find ∫ 1
1/k
f(x)dx.
(iii) Find ∫ 1
0
f(x)dx.
Problem 71. Show that∫ π/2
0
√sin(x)√
sin(x) +√
cos(x)dx =
π
4.
Problem 72. Calculate ∫ 3
0
(x− bxc+12
)dx
where bxc denotes the greatest integer less than or equal to x.
Hilbert Transform
Problem 73. The Hilbert transform H and its inverse is given by
g(y) =H(f(x)) =1πP
∫R
f(x)x− y
dx
f(x) =H−1(g(y)) =1πP
∫R
g(y)y − x
dy
where the Cauchy principal value is defined by
P
∫Rf(x)dx := lim
R→∞
∫ +R
−Rf(x)dx.
The Hilbert transform relates parts of the function in the same domain.Let k > 0. Find the Hilbert transform of f(x) = cos(kx).
Problem 74. Consider the Hilbert transform
H[f ] =1πP
∫ ∞−∞
f(x′)x′ − x
dx′.
Integration 133
Find H2.
Problem 75. The Hilbert transform acts on a one-dimensional functiong(s) by a convolution with the kernel 1/(πs). The singularity at s = 0 ishandled in the Cauchy principal value sense. The Fourier transform of theHilbert kernel is −i sgnσ. Thus the Hilbert transform of g is
Hg(s) =∫ ∞−∞
g(s− s′)πs′
ds′ =∫ ∞−∞
(−i sgnσ)G(σ)ei2πsσdσ
where G(σ) is the Fourier transform of g, i.e.
G(σ) =∫ ∞−∞
g(s)e−i2πσsds.
Suppose g is a function whose support is strictly less than radius R, i.e.g(s) = 0 for all |s| > R − ε for some small positive ε. Find an inversionformula.
Problem 76. The Hilbert transform of a function f ∈ L2(R) is definedas
H(f)(y) =1πPV
∫ ∞−∞
f(x)x− y
dx
where PV stands for principal value. Calculate the Hilbert transform off(x) = 1/(1 + x4).
Problem 77. The Radon transform for a function f(x, y) is given by theintegral transform
P (r, θ) = Rf(x, y) =∫ +∞
−∞f(r cos(θ)− s sin(θ), r sin(θ) + s cos(θ))ds.
The function P (r, θ) desribes the values of points on projections. Show thatthe inverse Radon transform can be given by
f(x, y) = R−1P (r, θ) = − 12πBHDP (r, θ)
where D is the partial differential operator Dg(r, θ) = ∂g/∂r with respectto r, H is the Hilbert transform operator
Hg(r, θ) = − 1π
∫g(u, θ)r − u
du
and B is the backprojection operator
Bg(r, θ) =∫ π
0
g(x cos(θ) + y sin(θ), θ)dθ.
134 Problems and Solutions
Problem 78. The Hilbert transform of a function f ∈ L2(R) is definedas
H(f)(y) =1πPV
∫ ∞−∞
f(x)x− y
dx
where PV stands for principal value. Calculate the Hilbert transform of
f(x) = exp(−x2/2).
Problem 79. Let f : R→ R
f(x) =x
sinh(x).
Then f(0) = 1. Let c > 0. Find∫ c
0
f(x)dx.
Problem 80. Let p1, p2, p3 > 0. Calculate
f(p1, p2, p3) =4π
∫ ∞0
sin(p1r) sin(p2r) sin(p3r)dr
r.
Problem 81. Find the constant K (normalization) from the condition
1 = 4K∫ 2π
0
sin2(ω/2)dω∫ π
0
sin(θ)dθ∫ π
−πdφ.
Problem 82. Consider
I =1
8π2
∫ π
−πdα
∫ π
0
sin(β)dβ∫ π
−πF (α, β, γ)dγ.
(i) Find I for F (α, β, γ) = 1.(ii) Find I for F (α, β, γ) = α+ β + γ.(iii) Find I for F (α, β, γ) = αβγ.
Problem 83. Consider the circle around (0, 0, 0) in the x1 − x2 plane
r1(t) =
x1,1(t)x1,2(t)x1,3(t)
=
cos(t)sin(t)
0
, t ∈ [0, 2π]
Integration 135
and the circle around (0, 1, 0) in the x2 − x3 plane
r2(s) =
x2,1(s)x2,2(s)x2,3(s)
=
01 + cos(s)
sin(s)
.
Then the derivatives are
dr1(t)dt
=
− sin(t)cos(t)
0
,dr2(t)dt
=
0− sin(s)cos(s)
.
Calculate (Gauss formula)
14π
∮ ∮dtds
(dr1(t)dt
× dr2(s)ds
)· r1(t)− r2(s)|r1(t)− r2(s)|3
where × denotes the vector product, · denotes the scalar product and con-tour integrations run from 0 to 2π.
Problem 84. Show that∫ ∞0
e−xx2dx = 2!,∫ ∞
0
e−xx3dx = 3!.
Problem 85. (i) Find the area of the set
S2 := (x1, x2) : 1 ≥ x1 ≥ x2 ≥ 0 .
(ii) Find the volume of the set
S3 := (x1, x2, x3) : 1 ≥ x1 ≥ x2 ≥ x3 ≥ 0 .
Extend the n-dimensions.
Problem 86. Let n be a positive integer. Find∫ π
0
cos2(nx)dx.
Problem 87. Let n ≥ 0. The Legendre polynomial of degree n is definedas
Pn(x) =1
2nn!dn
dxn(x2 − 1)
136 Problems and Solutions
with P0(x) = 1. Let n ≥ 1. Show that∫ 1
0
xkPn(x)dx = 0
for k = 0, 1, . . . , n− 1
Problem 88. Let k = 0, 1, . . .. Calculate
ck =2π
∫ π
0
arcsin(cos(θ)) cos(kθ)dθ.
Problem 89. Find α > 0 such that∫ ∞0
2αxe−2αx3/3dx = 1.
Problem 90. The complete elliptic integral of first kind K(m) can bedefined by
K(m) =∫ 1
0
((1− t2)(1−mt2))−1/2dt, |m| < 1.
The beta function can be defined by
B(a, b) =∫ 1
0
ta−1(1− t)b−1dt, <(a) > 0, <(b) > 0.
Show that 2√
2K(1/2) = B(1/4, 1/2).
Problem 91. Consider the function f : [0, 1]→ [0, 1], f(x) =√x with
I =∫ 1
0
√xdx =
23x3/2
∣∣∣∣10
=23.
Find a approximation (upper bound) of the integral in the sense of Lebesgueby partioning the ordinate interval into the intervals [0, 1/4], (1/4, 1/2],(1/2, 3/4], (3/4, 1] with a0 = 0, a1 = 1/4, a2 = 1/2, a3 = 3/4, a4 = 1.
Problem 92. Let a > 0. Calculate∫dx
(a2 − x2)3/2.
Integration 137
Set x = a sin(θ). Then dx = a cos(θ)dθ.
Double Integrals
Problem 93. Let f : R2 → R be a continuous function and f(x1, x2) =f(x2, x1) for all x1, x2 ∈ R. Let b, a ∈ R and b > a. Calculate∫ b
a
∫ b
a
f(x1, x2) sin(x1 − x2)dx1dx2.
Problem 94. Let f be a continuous function.(i) Show that the double integral∫ x
0
dξ
∫ ξ
0
f(t)dt
can be expressed by a single integral.(ii) Show that (n ≥ 2)∫ x
0
dξ1
∫ ξ1
0
dξ2 · · ·∫ ξn−1
0
f(ξn)dξn
can be expressed by a single integral.
Problem 95. Calculate the integral
I =∫ π/2
0
sin2n(θ) cos2n+1(θ)dθ
by considering the double integral∫ ∫D
(r sin(θ))2n(r cos(θ)2n+1e−r2rdrdθ
where D is the first quadrant.
Problem 96. Find the integral
I(`) =∫ 2π
0
(∫ | cos(θ)|(`/2)
0
dp
)dθ.
138 Problems and Solutions
Problem 97. Evaluate the quadruple integral
I =∫ 1
0
dx
∫ 1
0
dy
∫ 1
0
dx′∫ 1
0
dy′(1
((x− x′)2 + (y − y′)2)1/2− 1
((x− x′)2 + (y − y′)2 + 1)1/2
).
Problem 98. Let z ∈ (0, 1] and x ∈ (0, 1]. Find the double integral
f(z) = 1−∫ 1
z
dx
∫ 1
z/x
dy.
Problem 99. (i) Let r =√x2
1 + · · ·+ x2n. Show that the integral∫ ∫
r≥1
· · ·∫dx1 · · · dxn
rα
is finite for α > n and infinite for α ≤ n.(ii) Let r =
√x2
1 + · · ·+ x2n. Show that the integral∫ ∫
r≤1
· · ·∫dx1 · · · dxn
rα
is finite for α < n and infinite for α ≥ n.
Problem 100. Consider a three-dimensional probability distributionf(x1, x2, x3) such that for all j
fx(xj) =1
2√π
(1 + 2x2
j −−1 + 2x2
j√2
)e−x
2j
where fx(x) is the probability density associated with an individual vari-able. This means
fx(x1) =∫
R
∫Rf(x1, x2, x3)dx2dx3
etc. Is it possible that the probabilty density associated with the sum ofthese variables s = x1 + x2 + x3 is given by
fs(s) =1
2√π
(1 + 2s2 +
−1 + 2s2
√2
)e−s
2
Integration 139
provided that f(x1, x2, x3) is non-negative?
Problem 101. Consider a one-dimensional chain of length N with openend boundary conditions. The counting is from left to right starting at0. The canonical partition function Z(β) (β > 0) is given by the multipleintegral
ZN (β) =∫ 1
−1
ds0
∫ 1
−1
ds1 · · ·∫ 1
−1
dsN−1eβ|s0−s1|eβ|s1−s2| · · · eβ|sN−2−sN−1|.
Show that there is a coordinate transformation which decouples the sites.Find Z2(β) and Z3(β).
Problem 102. Let f : Rn → Rn be an analytic function. Consider themap
xj = f(xj−1) = · · · = f(x0).
To study the evolution of phase-space distributions, we can introduce theevolution operator U(x′,x, j) such that any initial phase-space distributionρ(x, 0) evolves into
ρ(x′; j) =∫
Ω
U(x′,x; j)ρ(x, 0)dx
where Ω is the phase space area. Find U(x′,x; j).
Problem 103. Let
S := (x, y) ∈ R2 : x, y ≥ 0, 0 ≤ x21 + x2
2 ≤ 1 .
Let m,n be nonnegative integers. Find the integral∫S
x2m+1y2n+1dxdy.
Problem 104. Calculate ∫ ∫A
dxdy
whereA = (x, y) : x, y ≥ 0, x+ y ≤ 1, x ≥ 1/3 .
Problem 105. Calculate the integral
I =∫ ∞
1
dx
∫ ∞1
dy
√x2 − 1
√y2 − 1
(x+ y)6
140 Problems and Solutions
using the substitution x = cosh(α), y = cosh(β).
Problem 106. Find the integral
I =∫ ∞
1
dx
∫ ∞1
dy
√x2 − 1
√y2 − 1
(x+ y)6.
Hint. Use the substitutions x = cos(α), y = cosh(β) and show that theintegral can be written as
I =∫ ∞
0
dα
∫ ∞0
dβsinh2(α) sinh2(β)
(cosh(α) + cosh(β))6.
Problem 107. Calculate the double integral
12π2
∫ π
0
∫ π
0
dαdα′ ln(2− cos(α)− cos(α′))
utilizing the identity
cos(α) + cos(α′) ≡ 2 cos(α+ α′
2
)cos(α− α′
2
)the transformation x = (α+ α′)/2, y = (α− α′)/2 and the integral∫ π
0
ln(1 + sin(x))dx = −π ln(2) +G
where G is the Catalan constant.
Problem 108. Let a > 0 and R ≥ 0. Find
Ia(R) =∫ R
0
∫ R
0
dxdy
(x− y)2 + a2≡∫ R
0
∫ R
0
dxdy
x2 + y2 − 2xy + a2.
Problem 109. Calculate the integral∫ 1
0
∫ 1
0
∫ 1
0
∫ 1
0
∫ 1
0
∫ 1
0
dx1dx2dx3dx4dx5dx6
1 + x1 + x2 + x3 + x4 + x5 + x6.
Problem 110. Let n1, n2 ∈ Z. Consider a function f : R2 → R with aperiod 2π for both x1 and x2
f(x1, x2) =∞∑
n1,n2=−∞cn1,n2e
i(n1x1+n2x2)
Integration 141
and ∫ 2π
0
∫ 2π
0
f(x1, x2)dx1dx2 = 0.
Then
cn1,n2 =1
4π2
∫ 2π
0
∫ 2π
0
f(x1, x2)e−i(n1x1+n2x2)dx1dx2
(i) Letf(x1, x2) = sin(x1) + cos(x2).
Find cn1,n2 .(ii) Let
f(x1, x2) = sin(x1) cos(x2).
Find cn1,n2 .
Problem 111. Find a non-negative analytic function f : R2 → R suchthat ∫
R
∫Rf(x, y)dxdy = 1
and ∫R
∫Rx2dxdy =
12,
∫R
∫Ry2dxdy =
12.
Problem 112. Consider the compact set
S := (x, y) : y ≥ x2, y ≤√x, x, y ≥ 0 .
Thus S ⊂ [0, 1]× [0, 1]. Find
I(S) =∫S
dµ =∫S
dxdy.
Problem 113. Let a > 0. Consider the compact set (lemniscate)
S := (x, y) : (x2 + y2)2 ≤ 2a2xy .
FindI(S) =
∫S
dµ =∫S
dxdy.
Introduce polar coordinates x(r, φ) = r cos(φ), y(r, φ) = r sin(φ). Thus wehave
x2 + y2 = r2, xy = r2 cos(φ) sin(φ) ≡ 12
sin(2φ).
142 Problems and Solutions
Problem 114. Calculate the integrals∫ 1
0
dx2
(∫ 1−x2
0
dx1
)dx2,
∫ 1
0
dx3
(∫ 1−x3
0
dx2
(∫ 1−x3−x2
0
dx1
)).
Extend to n-dimensions. Give an interpretation of the result.
Problem 115. (i) Is the subset of R2
S2 = (x1, x2) : x1 ≥ 0, x2 ≥ 0, x1 + x2 ≤ 1
convex?(ii) Calculate
A =∫S2
dx1dx2.
(iii) The area of a triangle in the plane R2 with vertices at (x1, y1), (x2, y2),(x3, y3) is given by
A = ±12
det
x1 y1 1x2 y2 1x3 y3 1
where the sign is chosen so that the area is nonnegative. Let (x1, y1) =(0, 0), (x2, y2) = (1, 0), (x3, y3) = (0, 1). Find A. Compare to (ii).(iv) Is the subset of R3
S3 = (x1, x2, x3) : x1 ≥ 0, x2 ≥ 0, x3 ≥ 0
convex?(v) Calculate
V =∫S3
dx1dx2dx3.
Problem 116. Consider
I =1
8π2
∫ π
−πdα
∫ π
0
sin(β)dβ∫ π
−πF (α, β, γ)dγ.
Find I for F (α, β, γ) = 1, F (α, γ, β) = α+ β + γ, F (α, β, γ) = αβγ.
Problem 117. Let γ > 0. A random variable X is said to be Lorentzianwith parameters (α, γ) if its probability density is given by
fX(x) =1π
γ
(x− α)2 + γ2, x ∈ R.
Integration 143
Let X, Y be two independent Lorentzian random variables with parameters(α, γ) and (β, δ), respectively. Let ε ≥ 0. Show that(a) εX is distributed Lorentzian (εα, εγ)(b) ε+X is distributed Lorentzian (ε+ α, γ)(c) −X is distributed Lorentzian (−α, γ)(d) X + Y is distributed Lorentzian (α+ β, γ + δ)
(e) X−1 is distributed Lorentzian(
αα2+γ2 ,
γα2+γ2
)Problem 118. A random variable X is said to be Lorentzian if its prob-ability density pX is a the form
pX(x) =1π
γ
(x− α)2 + γ2
where γ > 0. We say that X is (α, γ) to indicate that the random variableis Lorentzian with the probability density given above. Let X, Y be twoindependent random variables with (α, γ) and (β, δ). Let ε be a nonnegativereal number. Show that
εX is (εα, εγ)
ε+X is (ε+ α, γ)
−X is (−α, γ)
X + Y is (α+ β, γ + δ)
X−1 is(
α
α2 + γ ∗ 2,
γ
α2 + γ2
).
Problem 119. Consider the set
M = (x1, x2, x3) : x21 + x2
2 ≤ 1 and x21 + x2
3 ≤ 1
which is subset of R3. Obviously M is compact and measureable
I(M) =∫M
dµ =∫
R3χMdµ =
∫R3χMdx1dx2dx3
where χM is the indicator function. Find I(M).
Problem 120. Given a smooth Hamilton function
H(p,q) =n∑j=1
p2j
2+ U(q)
144 Problems and Solutions
with n degrees of freedom (p = (p1, . . . , pn), q = (q1, . . . , qn). Let V (E) bethe classical phase space volume at energy E of a smooth Hamilton functionis given by
V (E) =∫
R2nΘ(E −H(p,q))dnpdnq
where Θ is the step function. Assume that U(εq) = εmU(q).(i) Consider the transformation
p = E1/2p′, q = E1/nq′
with the inverse transformation
p′ = E−1/2p, q′ = E−1/nq.
Find dnp′dnq′ and H(p′,q′).(ii) Calculate V (E) with the assumption that E > 0. Find the asymptoticbehaviour.
Problem 121. Let a > 0. Find the area of the surface in R3 given bythe intersection of a hyperbolic paraboloid x3 = x1x2/a and the cylinderx2
1 + x22 = a2. We have the parameter representation
x1(r, φ) = r cos(φ), x2(r, φ) = r sin(φ), x3(r, φ) =x1x2
a=
1ar2 cos(φ) sin(φ).
Note that sin(φ) cos(φ) ≡ 12 sin(2φ).
8.2 Programming Problems
Problem 1. Calculate the integral∫ 1
0
| cos(2πx)|dx
using the random numnber generator described in problem 7, chapter 10,page 250, Problems and Solutions in Scientific Computing. Compare to theexact result by solving the integral.
Integration 145
8.3 Supplementary Problems
Problem 1. Let α > 0. Show that∫ ∞−∞
sech(αt)dt =2α.
Problem 2. Consider the function φ : R→ R
φ(x) :=
1 for x ∈ [0, 1]0 otherwise
Findψ(x) := φ(2x)− φ(2x− 1).
Draw the function. Calculate ∫ +∞
−∞ψ(x)dx.
Problem 3. Show that
e−x2/2 =
12π
∫ ∞−∞
exp(−1
2y2 + ixy
)dy.
Problem 4. Let ` > 0 and r0 > 0. Find the integral∫ r0
0
r3√1 + r2/`2
dr.
Problem 5. Let 0 ≤ r < 1. Consider the Hilbert space L2[0, 2π] andf(θ) ∈ L2[0, 2π]. Show that
12π
∫ 2π
0
f(θ)dθ +1π
∫ 2π
0
∞∑j=1
rjf(θ) cos(j(φ− θ))dθ
=1
2π
∫ 2π
0
f(θ)1− r2
1− 2r cos(φ− θ) + r2dθ.
Problem 6. Show that∫ ∞0
e−kx sin(ky) sin(kγ)dk
k=
14
ln(x2 + (y − γ)2
x2 + (y + γ)2
).
146 Problems and Solutions
Note thatf(x) =
∫ ∞0
e−kxdk
k⇒ df
dx= − 1
x.
Problem 7. Show that∫ ∞0
x2
(x2 + a2)3=
x(x2 − a2)8a2(x2 + a2)3
+1
8a2arctan(x/a).
Problem 8. Show that
3π
∫ π/6
0
ln(2 cos(x))dx = 0.338314... .
Problem 9. The content (n-dimensional volume) bounded by a hyper-sphere of radius r is known to be
Vn =2rnπn/2
nΓ(n/2)
where Γ is the gamma function. Let r = 1. Show that
limn→∞
Vn = 0.
Problem 10. Let k = 0, 1, 2, . . . and
yk :=∫ 1
0
xk
1 + x+ x2dx.
Show thatyk+2 + yk+1 + yk =
1k + 1
.
Note that
y0 =∫ 1
0
11 + x+ x2
dx =2√3
arctan(
2x+ 1√3
)∣∣∣∣x=1
x=0
y1 =∫ 1
0
x
1 + x+ x2dx =
12
ln(1 + x+ x2)∣∣∣∣10
− 12
∫ 1
0
11 + x+ x2
dx =12
ln(1 + x+ x2)∣∣∣∣10
− 12y0.
Problem 11. Let c ∈ C with |c| <∞ and <(c) > 0. Consider the integral
I(c) =∫ ∞
0
exp(−cx− x2)dx.
Integration 147
Show that
I(c) =12
∞∑k=0
(−1)kΓ(1/2 + k/2)
k!ck.
Problem 12. Find α > 0 such that∫ ∞0
2αx2e−2αx3/3dx = 1.
Problem 13. Let x = (x1 x2 · · · xn), y = (y1 y2 · · · yn) and x · y =x1y1 + · · ·+ xnyn. Show that
exp(ay2) =1
πn/2
∫Rn
exp(−x2 + 2a1/2x · y)dx1 . . . dxn.
Problem 14. Let b > a. Show that∫ b
a
√y − ab− y
(1− γ
y
)dy
y − x= π
(1− γ
x
√a
b
).
Problem 15. Let c > 0 and k ∈ R. Show that∫ ∞0
exp(−cs2) cos(2ks)ds =12
√π√c
exp(−k2/c).
This integral plays a role in optics.
Problem 16. Let x > 0. Show that
2∫ ∞
0
e−s2ds =
∫ ∞x2
eu√u
Problem 17. Let x ≥ 0. Consider the function f defined by
f(x) =∫ x
0
ln(1 + y)y
dy.
Show that for small X we can write
f(x) = x− x2
4+x3
9− x4
16+ · · ·
148 Problems and Solutions
Show that
f(x) =π2
6+
12
ln2(x)− f(
1x
).
Give a value of f(1).
Problem 18. Calculating∫cos3(x) sin3(x)dx
student Alice tells you the result is
14
sin4(x)− 16
sin6(x) + C1
and student Bob tells that
16
cos6(x)− 14
cos4(x) + C2
is the result. C1, C2 are the constants of integration. Discuss.
Problem 19. Let f : Rn+1 → R be an analytic function, gj : R → R(j = 1, . . . , n) are analytic functions and c1, . . . , cn are constants. Consider
I(ε) =∫ g1(ε)
c1
dx1
∫ g2(ε)
c2
dx2 · · ·∫ gn(ε)
cn
f(x1, x2, . . . , xn, ε).
Show that
dI(ε)dε
=∫ g1(ε)
c1
dx1
∫ g2(ε)
c2
dx2 · · ·∫ gn(ε)
cn
∂f(x1, x2, . . . , xn, ε)∂ε
+dg1
dε
∫ g2(ε)
c2
dx2
∫c3
g3(ε) · · ·∫ gn(ε)
cn
f(x1, x2, . . . , xn, ε)
+dg2
dε
∫ g1(ε)
c1
dx1
∫ g3(ε)
c3
dx3 · · ·∫ gn(ε)
cn
f(x1, x2, . . . , xn, ε)
+ · · ·
+dgndε
∫ g1(ε)
c1
dx1
∫ g2(ε)
c2
dx2 · · ·∫ gn−1(ε)
cn−1
dxn−1f(x1, x2, . . . , xn, ε).
Problem 20. Show that∫ ∞0
tdt
e2πt − 1=
14π2
∫ ∞0
τdτ
eτ − 1=
14π2
π2
6=
124.
Integration 149
Problem 21. Let k ∈ N. Show that∫ 2π
0
cosk(θ)dθ = 2(1 + (−1)k)π(k − 1)!!k!!
.
Problem 22. Let k ∈ N. Show that∫ 2π
0
cosk(θ)dθ = 2(1 + (−1)k)π(k − 1)!!k!!
.
Problem 23. Show that∫ 1
0
√1−√xdx =
815.
Hint. Set x = sin4(y). Then dx = 4 sin3(y) cos(y)dy.
Problem 24. Let a > 0. Show that∫x2dx
(a2 − x2)3/2=
x√a2 − x2
− arcsin(x/a) + C
Problem 25. Show that
2∫ ∞
0
sin(2px) sin(qx)x
dx = ln|2p+ q||2p− q|
.
Problem 26. For a λ/2 antenna we obtain the expression
E(r, θ) = −ωI0 sin θ4πε0c2r
∫ λ/4
−λ/4cos(k`) sin(ω(t− c−1(r − ` cos θ)))d`.
Calculate E(r, θ).
Problem 27. Let <(a) > 0 and n = 0, 1, . . .. Show that∫ ∞−∞
xne−ax2+pxdx=
∂n
∂pn
∫ ∞−∞
e−ax2+pxdx
=∂n
∂pn
√π
aep
2/(4a).
150 Problems and Solutions
Problem 28. Let a 6= 0. Show that∫ ∞−∞
sin(ax)x
dx = πsgn(a).
Problem 29. Let c > 1. Show that∫ c
1
sin(x2)dx =12
∫ c2
1
sin(τ)√τdτ.
Problem 30. Show that∫ 1
0
x(1 + x2)1/2dx =13
(23/2 − 1)
setting u = 1 + x2 and hence du = 2xdx.
Problem 31. Let (Bτ , τ ≥ 0) be the linear Brownian motion startingfrom 0. Show that
Γ+ :=∫ 1
0
ds 1(Bs>0)
follows the arcsine distribution, i.e.
P (Γ+ ∈ dτ) =1π
dτ√τ(1− τ)
.
Problem 32. Let a > 0. Show that∫R
sin2(ax/2)x2
dx =12πa.
Problem 33. Let k = 1, 2, . . .. Study the integrals
Lk(bcc) =1π3
∫ π
0
∫ π
0
∫ π
0
(cos(x1) cos(x2) cos(x3))kdx1dx2dx3
Lk(sc) =1π3
13k
∫ π
0
∫ π
0
∫ π
0
(cos(x1) + cos(x2) + cos(x3))kdx1dx2dx3
Lk(fcc) =1π3
13k
∫ π
0
∫ π
0
∫ π
0
(cos(x1) cos(x2)+cos(x2) cos(x3)+cos(x3) cos(x1))dx1dx2dx3.
Integration 151
They play a role in solid state physics for the body-centred cubic lattice,simple cubic lattice, face-center cubic lattice.
Problem 34. Draw the functions
f1(x) = cos(2πx)f2(x) = cos(2π(cos(2πx)))f3(x) = cos(2π(cos(2π(cos(2πx))))).
Extend it to fn(x). Find the integral∫ 2π
0
fn(x)dx.
Problem 35. Show that (Fresnel’s integral)∫ ∞0
cos(x2)dx =∫ ∞
0
sin(x2)dx =√π
2√
2.
Problem 36. Show that
1π
∫ ∞−∞
sech(α− β)eiωβdβ = sech(12πω)eiωα.
Problem 37. Let σ > 0. Show that
14σ2
∫ ∞0
x5/2 exp(− x2
8σ2
)dx =
(3210
)1/4
Γ(3/4)σ3/2.
Problem 38. Let α > 0. Show that∫ ∞0
xe−αx2J0(βx)dx =
12αe−β
2/(4α).
Problem 39. Show that ∫ ∞0
dx
cosh3(x)=π
4.
Chapter 9
Functional Equations
A functional equation is any equation that specifies a function in implicitform. An example is the Gamma function Γ(z)
Γ(z + 1) = zΓ(z)
with Γ(1) = Γ(2) = 1. The trigonometric function sin(x) and cos(x) satisfya system of functional equations. So functional equations are equationswhere the unkonws are functions. In most cases it is assumed that thefunctions are continouos.
9.1 Solved ProblemsProblem 1. To solve a number of nonlinear functional equation it ishelpful to have the solution of the linear functional equation
f(x+ y) = f(x) + f(y) (1)
where we assume that the function f is continouos.
Problem 2. Find the solution of the functional equations
f(x+ y) = f(x)f(y) (1)
where we assume that f is continuous.
152
Functional Equations 153
Problem 3. Find the solution of the functional equation
f(xy) = f(x) + f(y) (1)
where we assume that the function f is continuous.
Problem 4. Find the solution of the functional equation
f(xy) = f(x)f(y). (1)
We assume that the function f is continuous.
Problem 5. Find the solution of the Jensen equation
f
(x+ y
2
)=f(x) + f(y)
2(1)
where we assume that f is continuous.
Problem 6. Show that the functional equation
f(x+ y) =f(x) + f(y)1− f(x)f(y)
admits the solutions f(x) = tan(cx), where c is a constant.
Problem 7. Show that the functional equation
f(x+ y) =f(x) + f(y)
1 + f(x)f(y)C2
(1)
admits the solutionf(x) = C tanh(cx). (2)
Problem 8. Show that the functional equation
f(x+ y) =f(x)f(y)f(x) + f(y)
admits the solutionf(x) =
c
x.
Problem 9. Show that the functional equation
f(x+ y) =f(x) + f(y)− 1
2f(x) + 2f(y)− 2f(x)f(y)− 1
154 Problems and Solutions
admits the solutionf(x) =
11 + tan(cx)
.
Problem 10. Show that the functional equation
f(x+ y) = f(x)f(y) +√f(x)2
√f(y)2 − 1
admits the solutionf(x) = cosh(cx).
Problem 11. Show that the functional equation
f(x+ y + axy) = f(x)f(y)
admits the solutionf(x) = (1 + ax)c.
Problem 12. Show that the functional equation
f(x+ y)− f(x− y) = 4√f(x)f(y)
admits the solutionf(x) = cx2.
Problem 13. Solve the functional equation
f(x+ y) + f(x− y) = 2f(x)f(y)
assuming that f is a continuous function.
Problem 14. Show that the trigonometric functions f(x) = cos(x) andg(x) = sin(x) satisfy the system of functional equations
g(x+ y) = g(x)f(y) + f(x)g(y)f(x+ y) = f(x)f(y)− g(x)g(y)g(x− y) = g(x)f(y)− g(y)f(x)f(x− y) = f(x)f(y) + g(x)g(y).
Problem 15. Show that the Jacobi elliptic functions satisfy the systemof functional equations
f(x± y) =f(x)g(y)h(y)± f(y)g(x)h(x)
1− k2f(x)2f(y)2
Functional Equations 155
g(x± y) =g(x)g(y)∓ f(x)f(y)h(x)h(y)
1− k2f(x)2f(y)2
h(x± y) =h(x)h(y)∓ k2f(x)f(y)g(x)g(y)
1− k2f(x)2f(y)2.
Problem 16. Let a be a positive integer with a ≥ 2. Let 1 ≤ x ≤ a.Consider the equation
g(x− 1)− 2g(x) + g(x+ 1) = −λg(x).
Show that
gj(x) = sin(jπx/(a+ 1)), λj(x) = 2(1− cos(jπ/(a+ 1)))
satisfy this equation.
Problem 17. Let a, c, ε be positive constants. Solve the functional equa-tion
g((θ + c)(mod 1)) = ag(θ) + ε sin(2πθ).
156 Problems and Solutions
9.2 Supplementary Problems
Problem 1. Show that the functional equation
f(x+ y) =f(x) + f(y) + 2f(x)f(y)
1− f(x)f(y)
admits the solutionf(x) =
cx
1− cx.
Problem 2. Consider the analytic function f : R→ R
f(x) = arctan(x).
Show that
f(x) + f(y) = f
(x+ y
1− xy
).
What happens at xy = 1?
Problem 3. Consider the functional equation
g(x) = αg(g(x/α))
with g(0) = 0 and g′(x = 0) = 1. Show that
g(x) =x
1− cx
is a solution with c an arbitarary constant.
Problem 4. We know that
tan(α+ β) ≡ tan(α) + tan(β)1− tan(α) tan(β)
.
Show thattan(α+ β + γ) =
tan(α) + tan(β) + tan(γ)− tan(α) tan(β) tan(γ)1− tan(α) tan(β)− tan(α) tan(γ)− tan(β) tan(γ)
.
Chapter 10
Inequalities
10.1 Solved ProblemsProblem 1. Let a, b ∈ R. Show that
2ab ≤ a2 + b2.
Problem 2. Let a, b ∈ R. Show that
|a+ b| ≤ |a|+ |b|.
Problem 3. Let a, b ∈ R+. Show that
12
(a+ b) ≥√ab.
Problem 4. Let x ≥ 0 and 0 < p < 1. Show that
1p
(1− xp) ≥ 1− x.
Problem 5. Let x ∈ (0, 1). Show that
x(1− x) < x.
157
158 Problems and Solutions
Problem 6. Let n ∈ N.(i) Show that n < 2n.(ii) Show that n2 < 4n.(iii) Show that if n ≥ 4 then 2n < n!.
Problem 7. Let x, y be two nonnegative real numbers. Show that
xy ≤(x+ y
2
)2
≡ 14
(x2 + y2 + 2xy).
Problem 8. Let a, b, c, d be nonnegative real numbers. Show that
(abcd)1/4 ≤ 14
(a+ b+ c+ d).
Problem 9. Let xj (j = 1, . . . , n) be nonegative real numbers. Showthat
x1x2 · · ·xn ≤(x1 + x2 + · · ·+ xn
n
)n.
Problem 10. Let a, b, c, d be nonnegative real numbers. Show that
a4 + b4 + c4 + d4 ≥ 4abcd.
Problem 11. Let x ∈ R and n ∈ N. Show that (Bernoulli inequality)
(1 + x)n ≥ 1 + nx.
Problem 12. Let x ≥ 0. Show that
1− e−x ≥ x
1 + x.
Problem 13. Let
en := 1 +11!
+12!
+ · · ·+ 1n!
where n = 1, 2, . . .. Let m > n. Show that
|em − en| = em − en ≤2
(n+ 1)!.
Inequalities 159
Problem 14. Does the inequality
1 + 2 cos(θ)− cos(2θ) ≤ 2
hold for all θ ∈ [0, 2π)?
Problem 15. Let x > 0. Show that
1 +x
2− x2
8≤√
1 + x ≤ 1 +x
2.
Problem 16. Let x > 0. Show that
1− 1x≤ ln(x) ≤ x− 1
with equality iff x = 1.
Problem 17. Show that if a twice differentiable function f : R→ R hasa second order derivative which is non-negative (positive) everywhere, thenthe function is convex (strictly convex).
Problem 18. Let a1, a2, . . . , an be positive numbers and b1, b2, . . . , bn benonnegative numbers such that
n∑j=1
bj > 0.
Show that (log-sum inequality)
n∑j=1
(aj log
ajbj
)≥
n∑j=1
aj
log
(∑nj=1 aj
)(∑n
j=1 bj
)with the conventions based on continuity arguments
0 · log 0 = 0, 0 · logp
0=∞, p > 0.
Show that equality holds if and only if aj/bj = constant for all j =1, 2, . . . , n.
Problem 19. Let xj (j = 1, 2, . . . , n) be positive real numbers. Showthat
(x1x2 · · ·xn)1/n ≤∑nk=1 xkn
.
160 Problems and Solutions
This is the arithmetic-geometric mean inequality.
Problem 20. Let n be a positive integer and a, b ≥ c/2 > 0. Show that
|a−n − b−n| ≤ 4nc−n−1|a− b|.
Problem 21. Let X ⊂ R be an interval. A function ψ : X → R isconvex if for all x1, x2 ∈ X and numbers α1, α2 ≥ 0 with α1 + α2 = 1,
ψ(α1x1 + α2x2) ≤ α1ψ(x1) + α2ψ(x2). (1)
This means that every chord of the graph of ψ lies above the graph. Letψ : X → R be convex, let x1, x2, . . . , xn ∈ X, and let α1, α2, . . . , αn ≥ 0satisfy
∑nj=1 αj = 1. Show that (Jensen’s inequality)
ψ
(n∑i=1
αixi
)≤
n∑i=1
αiψ(xi). (2)
Problem 22. Consider the differentiable function f : [0,∞)→ R
f(x) =x
1 + x.
Let a, b ∈ R+. Show that
f(|a+ b|) ≤ f(|a|+ |b|). (1)
Problem 23. Let a, b ∈ R. Show that
|a+ b|1 + |a+ b|
≤ |a|1 + |a|
+|b|
1 + |b|.
Problem 24. Show that f : R→ R, f(x) = |x| is convex.
Problem 25. Let n ∈ N and n ≥ 2. Show that
en ln(n+ 1) < en+1 ln(n).
Problem 26. Show that the function f : (0,∞)→ R
f(x) = x ln(x)
Inequalities 161
is convex.
Problem 27. Let x, y ∈ R. Show that
x+ y
1 + |x+ y|≤ |x|
1 + |x|+|y|
1 + |y|.
Problem 28. Let A, B be n× n matrices over C. Show that
‖AB‖ ≤ ‖A‖ · ‖B‖.
Problem 29. Let A, B be n× n matrices over C. Show that
‖A+B‖ ≤ ‖A‖+ ‖B‖.
Problem 30. Let H be a Hilbert space. Let ψ, φ ∈ H. Assume thatψ 6= 0, φ 6= 0. Show that
|〈φ, ψ〉| ≤√〈φ, φ〉
√〈ψ,ψ〉 (1)
where 〈 , 〉 denotes the scalar product in the Hilbert space.
Problem 31. Let n ≥ 2. Let x1, x2, . . . , xn be given positive real numberwith
x1 < x2 < · · · < xn.
Let λ1, . . . , λn ≥ 0 and∑nj=1 λj = 1. Show that n∑
j=1
λjxj
n∑j=1
λjx−1j
≤ A2G−2
whereA =
12
(x1 + xn), G = (x1xn)1/2.
Problem 32. Let x,y ∈ Rn (column vectors) and ε > 0. Show that
2xTy ≤ εxTx +1εyTy.
Problem 33. Let a, b ∈ R and ε > 0. Show that
2ab ≤ εa2 + ε−1b2.
162 Problems and Solutions
Problem 34. Show that the analytic function f : R2 → R
f(α, β) = sin(α) sin(β) cos(α− β)
is bounded between −1/8 and 1.
Problem 35. Let n be an integer and n ≥ 2. Show that
en ln(n+ 1) < en+1 ln(n).
Problem 36. Let a, b, c be positive numbers. Assume that
x2
a2+y2
b2+z2
c2= 1.
Show thatx+ y + z ≤
√a2 + b2 + c2.
Use the Lagrange multiplier method.
Problem 37. Let a, b, c, d be positive numbers and s := a + b + c + d.Show that
s
s− a+
s
s− b+
s
s− c+
s
s− d≥ 16
3.
Problem 38. Let x ≥ 0 and 0 < p < 1. Show that
1p
(1− xp) ≥ 1− x.
Problem 39. Let b be a real number such that the elements of the infinitesequence (ak)∞k=1 satisfy
ak+m ≤ ak + am + b
for all k,m = 1, 2, . . .. Show that
a := limk→∞
akk
exists and ak ≥ ka− b for all k.
Problem 40. Let n ∈ N and n ≥ 2. Show that
1√1
+1√2
+ · · ·+ 1√n>√n.
Inequalities 163
Problem 41. Let a, b ≥ 0. Show that
a+ b
1 + a+ b≤ a+ b+ ab
1 + a+ b+ ab≤ a
1 + a+
b
1 + b.
Problem 42. Show that
1π2
∫ π/2
0
∫ π/2
0
sin2(x− y)(x− y)2
dxdy <14.
Problem 43. Let x1, x2, x3 ∈ R. Show that
x21 + x2
2 + x23 ≥ x1x2 + x2x3 + x3x1.
Why could it be useful to consider the function f : R3 → R
f(x1, x2, x3) = x21 + x2
2 + x23 − x1x2 − x2x3 − x3x1 ?
Problem 44. Let x1, x2, . . . , xn be positive numbers. Assume that thenumbers satisfy
n∑j=1
xj = 1,n∑j=1
x2j = b2.
Show that
maxxj : 1 ≤ j ≤ n ≤ 1n
(1 +√n− 1
√nb2 − 1).
Problem 45. Consider the function f : R→ R, f(x) = x2. Show that
|f(x2)− f(x1)| ≤ 4|x2 − x1|
for all x1 and x2 in [−2, 2]. Hint. Apply
x21 − x2
2 ≡ (x1 + x2)(x1 − x2).
Problem 46. Let n ∈ N. Show that
n
2< 1 +
12
+13
+ · · ·+ 12n − 1
≤ n.
164 Problems and Solutions
Problem 47. Let
0 ≤ a1 ≤ a2 ≤ · · · ≤ an, 0 ≤ b1 ≤ b2 ≤ · · · ≤ bn
and r ≥ 1. Show that (Chebyshev inequality) 1n
n∑j=1
arj
1/r 1n
n∑j=1
brj
1/r
≤
1n
n∑j=1
(ajbj)r
1/r
.
Problem 48. Let a, x, y ∈ R. Show that
|√a2 + x2 −
√a2 + y2| ≤ |x− y|.
Problem 49. Let b > a > 0. Show that(1− a
b
)< ln
(b
a
)<b
a− 1.
Problem 50. Show that∣∣∣∣∫ 1
0
cos(nx)x+ 1
∣∣∣∣ ≤ ln(2)
for n ∈ N.
Problem 51. Let n ≥ 2. Show that
2n <(
2nn
)< 22n.
Problem 52. Consider the two manifolds
x21 + x2
2 = 1, y21 + y2
2 = 1.
Show that|x1y1 + x2y2| ≤ 1.
Hint. Set
x1(t) = cos(t), x2(t) = sin(t), y1(t) = cos(τ), y2(t) = sin(τ).
Inequalities 165
Problem 53. Let n ∈ N and x > 0. Show that
xn+1 +1
xn+1> xn +
1xn.
Problem 54. Find all x ∈ R and x > 0 such that
|√x−√
2| < 1.
Problem 55. Let j, k be positive intgers. Find all pairs (j, k) such thatthe following four inequalities are satisfied
j + k < 10, j + k ≥ 6,j
k> 1,
j
k< 2.
Problem 56. Let n be a positive integer. Show that
1√1
+1√2
+ · · ·+ 1√n≥√n.
Problem 57. Let z = x+ iy with x, y ∈ R. Show that
1√2
(|x|+ |y|) ≤ |z| ≤ |x|+ |y|.
166 Problems and Solutions
Matrices
Problem 58. Let A, B be n× n matrices over C. Show that
|tr(AB∗)|2 ≤ tr(AA∗)tr(BB∗).
Problem 59. Let A and B be two n × n matrices over R. It can beshown that
treA+B ≤ tr(eAeB) ≤ 12
tr(e2A + e2B)
treA+B ≤ tr(eAeB) ≤ (trepA)1/p(treqB)1/q
where p > 1, q > 1 with1p
+1q
= 1.
Is
(trepA)1/p(treqB)1/q ≤ 12
tr(e2A + e2B) ?
Prove or disprove.
Problem 60. Let A, B be hermitian matrices. Then
tr(eA+B) ≤ tr(eAeB).
Assume that
A =(
0 00 1
), B =
(0 11 0
).
Calculate the left and right-hand side of the inequality. Does equality hold?
Problem 61. Let A, B be positive definite matrices. Show that
tr(AB)2p+1≤ tr(A2B2)2p , p a non-negative integer
Problem 62. Let A be an n× n matrix with ‖A‖ < 1.(i) Show that (In +A)−1 exists.(ii) Show that
‖(In +A)−1‖ ≤ 11− ‖A‖
.
Inequalities 167
Problem 63. Let A be an n×n matrix over R. Assume that ajj ≥ 1 forall j and
n∑j 6=k
a2jk < 1.
Show that A is inververtible.
Problem 64. Let A be an n × n matrix over C. Let In be the n × nidentity matrix. Assume that
n∑k=1
|ajk| < 1
for each j. Show that In −A is invertible.
Problem 65. Let A be an n× n positive definite matrix over R. Let Bbe an n× n positive semidefinite matrix over R. Show that
det(A+B) ≥ det(A).
Problem 66. Let A be an n × n matrix over R. Show that there existsnonnull vectors x1, x2 in Rn such that
xT1 Ax1
xT1 x1≤ xTAx
xTx≤ xT2 Ax2
xT2 x2
for every nonnull vector x in Rn.
Problem 67. Let A be an n × n skew-symmetric matrix over R. Showthat
det(In +A) ≥ 1
with equality holding if and only if A = 0.
Problem 68. Let A be an n×n matrix over R. Assume that ajj ≥ 1 forj = 1, 2, . . . , n and
n∑j 6=k
a2jk < 1.
Show that A is invertible.
Problem 69. Let A be an n× n matrix over C. Assume that
|ajj | >n∑k 6=j
|ajk|
168 Problems and Solutions
for all j = 1, 2, . . . , n. Show that A is invertible.
Problem 70. Let A, B be n× n positive definite matries. Show that
tr(A ln(A))− tr(A ln(B)) ≥ tr(A−B).
Problem 71. Let v be a normalized (column) vector in Cn and let A bean n× n hermitian matrix. Is
v∗eAv ≥ ev∗Av
for all normalized v? Prove or disprove.
Inequalities 169
10.2 Supplementary Problems
Problem 1. Let n ∈ N and n ≥ 2. Show that
1√n− 1
> 2√n− 2
√n− 1 >
1√n.
Use this reult to show that
1 +1√2
+1√3
+ · · ·
diverges.
Problem 2. Let b > a. Show that
b− a1 + b2
< arctan(b)− arctan(a) <b− a1 + a2
.
Note that arctan(−x) = − arctan(x).
Problem 3. Let n be is positive integer. Isn+1√
(n+ 1)! > n√n!
for all n?
Problem 4. Let a, b, c ∈ C with |a| = |b| = |c| = 1. Suppose that=(a) ≥ 0, =(b) ≥ 0, =(c) ≤ 0. Show that∣∣∣∣−1 + ab+ bc+ ca
4
∣∣∣∣ ≤ 1√2.
Problem 5. Let n be a positive integer with n ≥ 2. Show that
12
+14
+18
+ · · ·+ 12n−1
< 1.
Problem 6. Let n ≥ 2 and x1, x2, . . . , xn be positive numbers. Assumethat the numbers satisfy
n∑j=1
xj = 1,n∑j=1
x2j = b2.
Show that
maxxj : 1 ≤ j ≤ n ≤ 1n
(1 +√n− 1
√nb2 − 1
).
170 Problems and Solutions
Problem 7. Show that
1 +14
+19
+ · · ·+ 1n2≤ 2− 1
n.
Problem 8. Let x ≥ 0. Show that 1 + x ≤ 3x.
Problem 9. Let A be an n× n matrix. Is
|det(In +A)| ≤ exp(‖A‖)?
Does it depend on the chosen norm?
Problem 10. Let zj (j = 1, . . . , p) be fixed complex numbers. Is
p∏j=1
|z − zj | >p∏j=1
(|zj | − |z|) ?
Problem 11. Let x ∈ (0, 1). Show that
x > ln(1 + x) > x− x2
2.
Problem 12. Let x, y ∈ R. Show that
|x+ y|1 + |x+ y|
≤ |x|1 + |x|
+|y|
1 + |y|.
Problem 13. Let x1 > 0, x2 > 0, α1 > 0, α2 > 0 and α1 +α2 = 1. Showthat
xα11 xα2
2 ≤ α1x1 + α2x2.
Apply ln(x).
Inequalities 171
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Index
z-transform, 110
Airy functions, 49algebraic, 88Alternating function, 47Angular distance, 51Arc length, 42Associated Legendre polynomials, 76
Bargmann space, 85Bell polynomials, 86Bernoulli inequality, 158Bernoulli numbers, 1, 85Bernoulli polynomials, 84Bernstein basis polynomials, 77Bernstein polynomials, 77Bessel function, 114Beta function, 136Bethe ansatz, 3Braid-like relation, 76Brillouin function, 87Buchberger algorithm, 89
Cantor function, 4Cantor series, 15Cantor series approximation, 5Cauchy integral theorem, 110Cauchy principal value, 132Cauchy sequence, 103Cesaro sum, 4Chebyshev inequality, 164Chebyshev polynomials, 73Chordal distance, 105Convex, 160Convex function, 56
Convex set, 56convolution integral, 119Coulomb potential, 47Critcal points, 74Cubic B-spline, 119Cubic equation, 94
Dawson’s integral, 130Diffeomorphism, 28Diffusion equation, 122Dirichlet integral identity, 123Dirichlet kernel, 23Dominant tidal potential, 75
Elementary symmetric functions, 75Euler dilogarithm function, 50Euler summation formula, 1
Fejer kernel, 8Fermat number, 21Fibonacci numbers, 15, 37Fixed points, 109fixed points, 74Fresnel’s integral, 151
Gamma function, 22, 53, 68Gauss sum, 12
Haar wavelet, 122Hankel matrix, 82, 83Harmonic series, 1Hermite polynomial, 74Hermite polynomials, 73Hilbert transform, 132–134Hurwitz zeta-function, 68
177
178 Index
Invariant, 77
Jensen equation, 153
Kernel, 44Kustaanheimo-Stiefel transformation,
39
Laguerre polynomials, 73Legendre map, 36Legendre polynomials, 75Lemniscate, 141log-sum inequality, 159Logarithmic derivative, 42Lorenz model, 79Lotka Volterra model, 80
Newton’s method, 27, 29
Pade approximant, 64Partition, 30Poisson summation formula, 16Poisson wavelet, 122Polar coordinates, 141Polar form, 110Principal argument, 109Principal value, 133, 134Prolate spheroidal coordinates, 47Pyramide, 10
Quartic equation, 87Quintic equation, 84
Radon transform, 133Recursion relation, 75recursion relation, 89Residue theorem, 111Resultant, 82Rudin-Shapiro substitution, 36
Schwarzian derivative, 60Sonine polynomials, 87Spherical cosine theorem, 75Star product, 32Sum rule, 74
Symmetric group, 80
Toroidal coordinates, 47Trace, 78
Vector product, 35
Wronksi determinant, 45