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Complex Analysis

Complex Analysis

Travis Dirle

December 4, 2016

Contents

1 Complex Numbers and Functions 1

2 Power Series 3

3 Analytic Functions 7

4 Logarithms and Branches 13

5 Complex Integration 15

6 Cauchy’s Theorem and Its Consequences 17

7 Isolated Singularities 21

i

CONTENTS

ii

Chapter 1

Complex Numbers and Functions

Basic Properties:A complex number takes the form z = x + iy where x and y are real and i

is an imaginary number that satisfies i2 = −1. We call x and y the real partand the imaginary part of z, respectively, and write x = Re(z) and y = Im(z).The complex numbers can be visualized as the usual Euclidean plane by thefollowing: the complex number z = x + iy ∈ C is identified with the point(x, y) ∈ R2. Naturally, the x and y axis of R2 are called the real axis andimaginary axis.

The natural rules for adding and multiplying complex numbers: if z1 = x1 +iy1 and z2 = x2 + iy2, then

z1 + z2 = (x1 + x2) + i(y1 + y2)

and alsoz1z2 = (x1x2 − y1y2) + i(x1y2 + y1x2).

The notion of length, or absolute value of a complex number is identical tothe notion of Euclidean length in R2. We define the absolute value or modulusof a complex number z = x+ iy by

|z| = (x2 + y2)1/2

so that |z| is precisely the distance from the origin to the point (x, y). The com-plex conjugate of z = x+ iy is defined by

z = x− iy

Also,

z ± w = z̄ ± w̄, zw = z̄w̄,

z/w = z̄/w̄, |z + w| ≤ |z|+ |w|.

1

CHAPTER 1. COMPLEX NUMBERS AND FUNCTIONS

Also we have that

Re(z) =z + z

2and Im(z) =

z − z2i

We also have that

|z|2 = zz and as a consequence 1z

=z

|z|2

Any non-zero complex number z can be written in polar form

z = reiθ

where r > 0; also θ ∈ R is called the argument of z (defined uniquely up to amultiple of 2π) and is denoted by arg z, (we normalize arg by insisting that argz ∈ (−π, π] and denote this Arg z) and

eiθ = cos θ + i sin θ

We have that r = |z|, and θ is simply the angle (with positive counterclockwiseorientation) between the positive real axis and the half line starting at the originand passing through z. We also see that |ez| = eRe z and arg(ez) = Im(z).

Finally, note that if z = reiθ and w = seiφ, then

zw = rsei(θ+φ)

A connected open set in C will be called a domain.

Lemma 1.0.1. (de Moivre’s Formula) (cos θ + i sin θ)n = cos(nθ) + i sin(nθ).

Theorem 1.0.2. Suppose that z is a non-zero complex number and that n is apositive integer. then z has exactly n distinct complext nth-roots. These roots aregiven in polar form by

n√|z|[cos

(Arg z + 2kπ

n

)+ i sin

(Arg z + 2kπ

n

)]for k = 0, 1, . . . , n− 1.

2

Chapter 2

Power Series

Theorem 2.0.1. A series of complex numbers∑∞

n=1 zn converges if and only ifgiven each � > 0 there is an index N such that |

∑mk=n zk| < � holds whenever

m ≥ n ≥ N .Definition 2.0.2. If the series

∑∞n=1 |zn| converges, then the original series∑∞

n=1 zn is said to be absolutely convergent.

We then have that ∣∣∣∣∣∞∑n=1

zn

∣∣∣∣∣ ≤∞∑n=1

|zn|.

An important characteristic of an absolutely convergent series is that theterms of such a series can be permuted in any arbitrary fashion without influ-encing the convergence or the value of the sum.

Definition 2.0.3. A series that converges, but not absolutely, is called a condi-tionally convergent series.

If zn = xn + iyn, the complex series∑∞

n=1 zn converges (resp. convergesabsolutely) if and only if both real series

∑∞n=1 xn and

∑∞n=1 yn converge (resp.

converge absolutely). We have that∞∑n=1

zn =∞∑n=1

xn + i∞∑n=1

yn.

Lemma 2.0.4. A doubly infinite series of complex numbers∑∞

n=−∞ zn con-verges if and only if both the series

∑∞n=0 zn and

∑∞n=1 z−n converge, in which

event∞∑

n=−∞

zn =∞∑n=1

z−n +∞∑n=0

zn.

The notion of absolute convergence carries over to doubly infinite series,with the expected results: if

∑∞n=−∞ |zn| converges, then

∑∞n=−∞ zn converges;

moreover, the terms of the latter series can be arbitrarily rearanged without af-fecting either the convergence or the value of the sum.

3

CHAPTER 2. POWER SERIES

Definition 2.0.5. Let (fn) be a sequence of complex valued functions each ofwhich are defined in an open set U of C. Let sn be the nth partial sum:

sn = f1 + f2 + · · ·+ fn

If the sequence of functions (sn) converges pointwise in U to the limit functionf, then we write f =

∑∞n=1 fn and say that the infinite series is pointwise con-

vergent in U with sum f. If (sn) converges uniformly on a subset A of U, thenthe infinite series is uniformly convergent on A. Finally, if (sn) converges uni-formly on each compact set in U, then the infinite series is termed normallyconvergent in U. In order to certify that a series is normally convergent in U,we need only check that it is uniformly convergent on each closed disk in U.We speak of

∑∞n=1 fn as absolutely convergent in U if

∑∞n=1 |fn| is pointwise

convergent in U.

Theorem 2.0.6. (Weierstrass M-test) Suppose that each term in a function series∑∞n=1 fn is defined on a set A. If there exists a sequence (Mn) of real numbers

such that the estimate |fn(z)| ≤ Mn holds for every z in A and such that theseries

∑∞n=1Mn converges, then

∑∞n=1 fn converges absolutely and uniformly

on A.

Definition 2.0.7. Let S be a set, and f a bounded function on S. Then we definethe sup norm

‖f‖S = ‖f‖ = supz∈S|f(z)|,

Definition 2.0.8. We say that {fn} is a Cauchy sequence, if given �, there existsN such that if m,n ≥ N , then

‖fn − fm‖ < �.

Theorem 2.0.9. If a sequence {fn} of functions on S is Cauchy, then it convergesuniformly.

Recall from calculus:

Theorem 2.0.10. (The Ratio Test) Let∑an be a series with positive terms and

suppose thatlimn→∞

an+1an

= ρ

Then(i) the series converges if ρ < 1,(ii) the series diverges if ρ > 1 or ρ is infinite,(iii) inconclusive if ρ = 1.

Theorem 2.0.11. (The Root Test) Let∑an be a series with an ≥ 0 for n ≥ N ,

and suppose thatlimn→∞

n√an = ρ

4


Then(i) the series converges if ρ < 1,(ii) the series diverges if ρ > 1 or ρ is infinite,(iii) inconclusive if ρ = 1.

Theorem 2.0.12. Let∑anz

n be a power series. If it does not converge ab-solutely for all z, then there exists a number r such that the series convergesabsolutely for |z| < r and does not converge absolutely for |z| > r.

The number r is called the radius of convergence of the power series. Ifthe power series converges absolutely for all z, then we say that its radius ofconvergence is infinity. When r is 0, then the series converges absolutely onlyfor z = 0. If r is non-zero, then the power series is called a convergent powerseries.

Definition 2.0.13. Suppose that z0 ∈ C. We refer to a function series of the type∞∑n=0

an(z − z0)n = a0 + a1(z − z0) + a2(z − z0)2 + · · · ,

where a0, a1, · · · is a sequence of complex numbers, as a Taylor/power seriescentered at z0.

Definition 2.0.14. With any such Taylor series we associate an extended realnumber ρ by the rule

ρ =

(lim supn→∞

n√|an|)−1

Here we observe the conventions 1/0 = ∞ and 1/∞ = 0. The quantity ρ isknown as the radius of convergence of the series. When ρ > 0 the open disk∆(z0, ρ) is called the disk of convergence.

5


6

Chapter 3

Analytic Functions

Definition 3.0.1. Let f be a function defined in some neighborhood of a point z0.We say that f is analytic at z0 if there exists a power series

∞∑n=0

an(z − z0)n

and some r > 0 such that the series converges absolutely for |z − z0| < r, andsuch that for such z, we have

f(z) =∞∑n=0

an(z − z0)n.

Suppose f is a function on an open set U. We say that f is analytic on U if f isanalytic at every point of U.

Theorem 3.0.2. A taylor series diverges for any z satisfying |z − z0| > ρ. Ifρ > 0, the series converges absolutely and normally in the disk ∆(z0, ρ), sothe function f defined by f(z) =

∑∞n=0 an(z − z0)n is analytic in ∆(z0, ρ). The

coefficient an is then related to f through the formula

an =f (n)(z0)

n!.

If S is an arbitrary set, not necessarily open, then a function is analytic on Sif it is the restriction of an analytic function on an open set containing S.

Theorem 3.0.3. Suppose f is analytic in an open set U , that z0 ∈ U , and that∆(z0, r) ⊂ U . Then f can be represented in ∆ as a Taylor series centered at z0.This expansion is uniquely determined by f: if f(z) =

∑∞n=0 an(z − z0)n in ∆,

then the coefficient an is given by an = f (n)(z0)/n!.

Theorem 3.0.4. The power series f(z) =∑∞

n=0 an(z− z0)n defines an analyticfunction in its disc of convergence. The derivative of f is also a power series

7

CHAPTER 3. ANALYTIC FUNCTIONS

obtained by differentiating term by term the series for f, that is,

f ′(z) =∞∑n=0

nan(z − z0)n−1

Moreover, f ′ has the same radius of convergence as f.

Corollary 3.0.5. A power series is infinitely complex differentiable in its disc ofconvergence.

Definition 3.0.6. By a Laurent series centered at z0 we mean a doubly infinitefunction series of the form

∞∑n=−∞

an(z − z0)n

= · · ·+ a−2(z − z0)2

+a−1z − z0

+ a0 + a1(z − z0) + a2(z − z0)2 + · · · ,

where each an is a complex constant.

Definition 3.0.7. We assign to any such Laurent series two non-negative ex-tended real numbers ρO and ρI , its outer and inner radii of convergence, viathe formulas

ρO =

(lim supn→∞

n√|an|)−1

, ρI = lim supn→∞

n√|a−n|

When ρI < ρO, D = {z : ρI < |z − z0| < ρO} is called the ring/annulus ofconvergence of the series.

Theorem 3.0.8. The Laurent series diverges for any z satisfying |z−z0| > ρO or|z − z0| < ρI . If ρO > 0, the series

∑∞n=0 an(z − z0)n converges absolutely and

normally in the disk ∆(z0, ρO), so fO(z) =∑∞

n=0 an(z− z0)n defines a functionthat is analytic in ∆. If ρI < ∞, the series

∑∞n=1 a−1(z − z0)−n converges

absolutely and normally in the open set DI = {z : |z − z0| > ρI}, so fI(z) =∑∞n=1 a−n(z − z0)−n defines a function that is analytic in DI . If ρI < ρO, the

full Laurent series converges absolutely and normally in the set D = {z : ρI <|z − z0| < ρO}, so that the function defined by f(z) =

∑∞n=−∞ an(z − z0)n =

fI(z) + fO(z) is analytic in D. The coefficient an is then related to f through theformula

an =1

2πi

∫|z−z0|=r

f(z)dz

(z − z0)n+1

for any number r ∈ (ρI , ρO).

Theorem 3.0.9. Suppose that a function is analytic in an annulus centered at z0,then f can be represented in the annulus as a Laurent series centered at z0 withthe coefficient an as given above.

8


Definition 3.0.10. Suppose that U is a non-empty open subset of the complexplane, that f is a function whose domain contains U, and that f is (complex)differentiable at every point of U, then f is analytic/holomorphic in U. A functionwhose domain is an open set in which that function is analytic is known as ananalytic function. A function is analytic at a point z0 if f is differentiable atevery point in a neighborhood of z0.

Suppose that a function f(x + iy) = u(x, y) + iv(x, y) is differentiable atz0 = x0 + iy0. By definition this requires that z0 be interior to the domain-set off and that f ′(z0) = limz→z0(f(z) − f(z0))/(z − z0) exist. Note that there is noconstraint to the manner in which z tends to z0. We end up getting that f ′(z0) =ux(z0) + ivx(z0) = fx(z0) as well as f ′(z0) = vy(z0) − iuy(z0) = −ify(z0).From these we get:

The Cauchy-Riemann Equations:

∂u

∂x=∂v

∂yand

∂u

∂y=−∂v∂x

A necessary condition for a function f = u+ iv to be differentiable at a pointz0 is that u and v satisfy the Cauchy-Riemann equations at z0.

Theorem 3.0.11. Suppose that a function f = u + iv is defined in an opensubset U of the complex plane and that the partial derivatives ux, uy, vx, andvy exist everywhere in U. If each of these partial derivatives is continuous at apoint z0 of U and if the Cauchy-Riemann equations are satisfied at z0, then f isdifferentiable there and f ′(z0) = fx(z0) = −ify(z0).Definition 3.0.12. A real-valued function u(x, y) which is twice continuouslydifferentiable and satisfies Laplace’s equation

uxx + uyy = 0

throughout a domain D is said to be harmonic in D.

Theorem 3.0.13. If f = u + iv is analytic in a domain D, then u and v areharmonic there.

Lemma 3.0.14. Suppose that u is a real-valued function which is defined in aplane domain D and that ux(z) = uy(z) = 0 for every z ∈ D. Then u is constantin D.

Theorem 3.0.15. Suppose that a function f is analytic in a domain D and thatf ′(z) = 0 for every z ∈ D. Then f is constant on D.Theorem 3.0.16. Let f = u + iv be analytic in a domain D. If any one of thefunctions u, v, or |f | is constant in this domain, then f itself is constant in D.Theorem 3.0.17. If f, g are analytic on U, so are f + g, fg. Also f/g is analyticon the open subset of z ∈ U such that g(z) 6= 0. If g : U → V is analytic andf : V → C is analytic, then f ◦ g is analytic.

9


Theorem 3.0.18. If a function f is analytic in a domain D and if there exists apoint ζ0 of D with the property that f (n)(ζ0) = 0 for every positive integer n,then f is constant in D.

Theorem 3.0.19. Suppose that a function f is analytic and non-constant in adomain D and that z0 is a point of D for which f(z0) = 0. Then f can beuniquely represented in D in the fashion

f(z) = (z − z0)mg(z),

where m is a positive integer and g : D → C is an analytic function that obeysthe condition g(z0) 6= 0.

Corollary 3.0.20. Suppose that a function f is analytic and non-constant in adomain D and that z0 is a point of D. Then f can be uniquely represented in D inthe fashion

f(z) = f(z0) + (z − z0)mg(z)where m is a positive integer and g : D → C is an analytic function that obeysthe condition g(z0) 6= 0.

The integer m is called the multiplicity/order of f at z0. We say that f takesthe value f(z0) with order m at z0. The multiplicity of f at z0 is the smallestpositive integer m for which f (m)(z0) 6= 0.

Theorem 3.0.21. (L’Hospital’s Rule) Let f and g be functions that are analyticand non-constant in ∆(z0, r). Assume that each of these functions has a zero atthe point z0. Then

limz→z0

f(z)

g(z)= lim

z→z0

f ′(z)

g′(z),

understood to mean that either both limits exist and are the same, or else neitherlimit exists.

Definition 3.0.22. A function that is analytic on the whole complex plane iscalled entire.

Definition 3.0.23. Let U be an open set. A subset E of U is termed a discretesubset of U if E has no limit point that belongs to U. So E must be a set ofisolated points. We speak of f as a discrete mapping of U if for each fixedcomplex number w, the set Ew = {z ∈ U : f(z) = w} is a discrete subset of U.

Theorem 3.0.24. (Discrete Mapping Theorem) If a function f is analytic andnon-constant in a domain D, then f is a discrete mapping of D.

Note that the set of zeros of a function in a domain D where that function isanalytic and non-constant, if non-empty, consists entirely of isolated points.

Theorem 3.0.25. (Principle of Analytic Continuation) If functions f and g areanalytic in a domain D and if f(z) = g(z) for all z belonging to some subset Aof D that has a limit point in D, then f(z) = g(z) for every z in D.

10


Theorem 3.0.26. If functions f and g are analytic in a domain D and if f(z)g(z) =0 for every z ∈ D, then either f(z) = 0 for every z ∈ D or g(z) = 0 for everyz ∈ D.

Some important functions:

cos z =∞∑n=0

(−1)nz2n

(2n)!, and sin z =

∞∑n=0

(−1)nz2n+1

(2n+ 1)!

cos z =eiz + e−iz

2, and sin z =

eiz − e−iz

2i

ez =∞∑n=0

zn

n!

11


12

Chapter 4

Logarithms and Branches

Definition 4.0.1. For z 6= 0 we define the logarithm of z, log z to be themultivalued function log z = log |z| + i arg z = log |z| + iArg z + 2πim form = 0,±1,±2, . . .. The values of log z are precisely the complex numbers wsuch that ew = z. We define the principle logarithm denoted Log z to be

Log z = log|z|+ iArg z, z 6= 0.In general, we have that log(z1z2) = log z1 + log z2, but this isn’t to be ex-

pected of the principle logarithm.

Definition 4.0.2. For any logarithm w of z, the complex number eλw is called theλ-power of z associated with w. The choice w = Log z gives rise to the principleλ-power of z.

zλ = eλLogz

Definition 4.0.3. If f : U → C is an analytic function and if D is a domain con-tainted in f(U), then by a branch of f−1 in D, we mean a continuous functiong : D → U that satisfies the condition f(g(z)) = z for all z ∈ D.Theorem 4.0.4. Suppose that f : U → C is an analytic function and that g is abranch of f−1 in a domain D. Let z0 be a point of D. If f ′(g(z0)) 6= 0, then g isdifferentiable at z0 and g′(z0) = 1/f ′(g(z0)). Consequently, if f ′ is free of zerosin g(D), then g is analytic in D, where its derivative satisfies g′(z) = 1/f ′(g(z)).

Definition 4.0.5. If p ≥ 2 is an integer and if D is a domain, then by a branchof the pth-root function in D, we mean an analytic function g : D → C withthe feature that (g(z))p = z for all z ∈ D.Definition 4.0.6. A branch of the logarithm function in a domain D is ananalytic function L : D → C with the property that eL(z) = z for all z ∈ D.Moreover

L′(z) =1

f ′(L(z))=

1

eL(z)=

1

z

If a branch of L of log z exists in a domain D, then D cannot contain the originfor z = eL(z) 6= 0.

13

CHAPTER 4. LOGARITHMS AND BRANCHES

Theorem 4.0.7. Suppose that a branch L of the logarithm function exists in adomain D. Then the collection of all branches of the logarithm function in Dconsists of the functions L+ 2kπi, where k is an integer.

Definition 4.0.8. If L is a branch of the logarithm function in a domain D and if λis a complex number, then the branch of the λ-power function in D associatedwith L is the function hλ : D → C defined by

hλ(z) = eλL(z)

Definition 4.0.9. If λ = 1/p for p ≥ 2, then

(h1/p(z))p = (eL(z)/p)p = eL(z) = z

making h1/p a branch of the pth-root function in D the branch of the pth-rootfunction in D associated with L.

Theorem 4.0.10. Suppose that a function f is analytic and free of zeros in adomain D. There exists a branch of log f(z) in D if and only if∫

γ

f ′(z)dz

f(z)= 0

for every closed, piecewise smooth path γ in D. If g is a branch of log f(z) in D,then the collection of all such branches consists of the functions g+ 2kπi, wherek is an integer.

Theorem 4.0.11. Suppose that U is an open set, that f : U → C is an analyticfunction, and that g is a branch of f−1 in a domain D. Then g is an analyticfunction.

Theorem 4.0.12. Let f(z) = c(z − z1)m1(z − z2)m2 · · · (z − zr)mr , wherec, z1, z2, . . . , zr are complex numbers satisfying c 6= 0 and zj 6= zk for j 6= k, andwhere m1,m2, . . . ,mr are non-zero integers. There exists a branch of log f(z)in a domain D if and only if

m1n(γ, z1) +m2n(γ, z2) + · · ·+mrn(γ, zr) = 0

is true for every closed, piecewise smooth path γ in D

14

Chapter 5

Complex Integration

Definition 5.0.1. By a path γ in the complex plane we mean a continuous func-tion of the type γ : [a, b] → C. The range of a path is called its trajectory,denoted |γ|. The initial and terminal points of the path are the points γ(a)and γ(b), respectively. When these values coincide, we call γ a closed path. Ifγ(t) 6= γ(s) for t 6= s with the possible exception that γ(a) = γ(b), we callthe path simple. A Jordan curve is a trajectory of a simple, closed path. Apath given by γ(t) = x(t) + iy(t) for a ≤ t ≤ b is termed a smooth path if itsderivative γ̇(t) = ẋ(t) + iẏ(t), with respect to the real parameter t, exists foreach t ∈ [a, b] and if the function γ̇ is continuous on [a, b].

Definition 5.0.2. For a continuous function g : [a, b]→ C with g = u+ iv, thenwe have ∫ b

a

g(t) dt =

∫ ba

u(t) dt+ i

∫ ba

v(t).

The Second Fundamental Theorem of Calculus also remains valid: if g,G :[a, b]→ C are continuous functions and Ġ(t) = g(t) for every t ∈ (a, b), then∫ b

a

g(t) dt = [G(t)]ba = G(b)−G(a).

Definition 5.0.3. Suppose γ (defined above) is a smooth path and that f is acomplex function which is defined and continuous on the trajectory of γ. Thenthe complex line integral or contour integral of f along γ is∫

γ

f(z) dz =

∫ ba

f [γ(t)]γ̇(t) dt

Definition 5.0.4. The integral of f along γ with respect to arclength is given by∫γ

f(z) |dz| =∫ ba

f [γ(t)]|γ̇(t)| dt

15

CHAPTER 5. COMPLEX INTEGRATION

Definition 5.0.5. The length `(γ) of the path γ is given by∫γ

|dz| =∫ ba

√ẋ(t)2 + ẏ(t)2 dt

Lemma 5.0.6. Suppose that f : A → C and g : A → C are continuousfunctions and that γ and β are piecewise smooth paths in A.

(i)∫γ

[f(z) + g(z)] dz =

∫γ

f(z) dz +

∫γ

g(z) dz;

(ii)∫γ

cf(z) dz = c

∫γ

f(z) dz for any constant c;

(iii) if γ + β is defined, then∫γ+β

f(z) dz =

∫γ

f(z) dz +

∫β

f(z) dz;

(iv) if β is obtainable from γ by a piecewise smooth change of parameter, then∫γ

f(z) dz =

∫β

f(z) dz;

(v)∫−γf(z) dz = −

∫γ

f(z) dz;

(vi)∣∣∣∣∫γ

f(z) dz

∣∣∣∣ ≤ ∫γ

|f(z)| |dz|

Definition 5.0.7. Suppose that U is an open set in C and that f is a functionwhose domain includes U. A function F : U → C is a primitive for f in U if Fis analytic in U and has F ′(z) = f(z) for every z in that set.

Theorem 5.0.8. Suppose that a function f is continuous in an open set U andthat F is a primitive for f in U. If γ : [a, b]→ U is a piecewise smooth path, then∫

γ

f(z) dz = [F (z)]γ(b)γ(a)

In particular, under the above hypothesis it is true that∫γ

f(z) dz = 0

for every closed, piecewise smooth path γ in U.

16

Chapter 6

Cauchy’s Theorem and ItsConsequences

Theorem 6.0.1. Suppose that a function f is continuous in a plane domain Dand that

∫γf(z) dz = 0 for every closed, piecewise smooth path γ in D. Then f

has a primitive in D.

Theorem 6.0.2. (Cauchy’s Theorem - Local Form) Suppose that ∆ is an opendisc in the complex plane and that f is a function which is analytic in ∆ (or, moregenerally, is continuous in ∆ and analytic in ∆\{z0} for some point z0 of in ∆).Then

∫γf(z) dz = 0 for every closed, piecewise smooth path γ in ∆.

Definition 6.0.3. Suppose that γ is a closed and piecewise smooth path and thatz is a point of C\|γ|. The winding number n(γ, z) or index, of γ about z isdefined to be:

n(γ, z) =1

2πi

∫γ

dζ

ζ − z.

The winding number is an integer, it records the net number of completerevolutions of a path about a point.

Lemma 6.0.4. Let γ be a closed, piecewise smooth path in the complex planeand let U = C\|γ|. Then:

(i) n(γ, z) remains constant as z varies over any component of U;(ii) n(γ, z) = 0 for any z belonging to the unbounded component of U;(iii) when γ is simple, either n(γ, z) = 1 for every z in the bounded compo-

nent of U or n(γ, z) = −1 for all such z.

For γ a simple, closed, and piecewise smooth path, we say γ is positivelyoriented if n(γ, z) = 1 for every z in the inside of the path, and negativelyoriented if n(γ, z) = −1 for all such z.

Theorem 6.0.5. (Cauchy’s Integral Formula - Local Form) Suppose that a func-tion f is analytic in an open disc ∆ and that γ is a closed, piecewise smooth path

17

CHAPTER 6. CAUCHY’S THEOREM AND ITS CONSEQUENCES

in ∆. Then

n(γ, z)f(z) =1

2πi

∫γ

f(ζ)dζ

ζ − zfor every z ∈ ∆\|γ|.Corollary 6.0.6. If a function f is analytic in an open set U, then it can be dif-ferentiated arbitrarily often in U and all its derivatives f ′, f ′′, . . . , f (k), . . . areanalytic there.

Theorem 6.0.7. (Morera’s Theorem) Let a function f be continuous in an openset U. Assume that

∫∂Rf(z) dz = 0 for every closed rectangle R in U whose

sides are parallel to the coordinate axes. Then f is analytic in U.

Theorem 6.0.8. Suppose that a function f is continuous in an open set U andanalytic in U\{z0} for some point z0 of U. Then f is analytic in U.Theorem 6.0.9. Suppose that a function f is analytic in an open disc ∆(z0, r)and that |f(z)| ≤ m holds throughout ∆, where m is a constant. Then for eachpositive integer k, the estimate

|f (k)(z)| ≤ k!mr(r − |z − z0|)k+1

is valid for every z ∈ ∆. In particular, |f (k)(z0)| ≤ k!mr−k.Theorem 6.0.10. (Liouville’s Theorem) The only bounded entire functions arethe constant functions on C.

Theorem 6.0.11. (Fundamental Theorem of Algebra) Any polynomial functionp(z) = a0 + a1z + · · ·+ anzn of degree n ≥ 1 has a root in C.Theorem 6.0.12. A polynomial function p(z) = a0 +a1z+ · · ·+anzn of degreen ≥ 1 has a factorization p(z) = c(z − z1)(z − z2) · · · (z − zn), in whichz1, z2, . . . , zn are the roots of p and c is a constant.

Theorem 6.0.13. (Mean Value Theorem) If f is analytic in D and z0 ∈ D, thenf(z0) is equal to the mean value of f taken around the boundary of any disccentered at z0 and contained in D. That is,

f(z0) =1

2π

∫ 2π0

f(z0 + reiθ) dθ

when D(z0, r) ⊂ D.Theorem 6.0.14. (Maximum-Modulus Principle) Let a function f be analytic ina domain D. Suppose that there exists a point z0 of D with the property that|f(z)| ≤ |f(z0)| for every z ∈ D. Then f is constant in D.Theorem 6.0.15. (Minimum-Modulus Principle) If f is a non-constant analyticfunction in a domain D, then no point z ∈ D can be a relative minimum of funless f(z) = 0.

18


Corollary 6.0.16. Let D be a bounded domain and let f : D → C be a con-tinuous function that is analytic in D. Then |f(z)| reaches its maximum at somepoint on the boundary of D.

Lemma 6.0.17. (Schwarz’s Lemma) Suppose that a function f is analytic in∆(0, 1) and that it obeys the conditions f(0) = 0 and |f(z)| ≤ 1 for everyz ∈ ∆. Then |f ′(0)| ≤ 1 and |f(z)| ≤ |z| for every z ∈ ∆. Furthermore, unlessf happens to be a function of the type f(z) = cz in ∆, where c is a constantof modulus one, it is actaully true that |f ′(0)| < 1 and that |f(z)| < |z| when0 < |z| < 1.

Theorem 6.0.18. (Hadamard’s Three-Lines Theorem) Let S be the set {z : 0 <Re z < 1}, and let f : S → C be a bounded, continuous function that is analyticin S. Suppose that |f(iy)| ≤ m0 and |f(1 + iy)| ≤ m1 for all real y, where m0and m1 are constants. Then

|f(x+ iy)| ≤ m1−x0 mx1

for all real y, whenever 0 < x < 1.

Definition 6.0.19. By a cycle we mean a finite sequence of closed, piecewisesmooth paths in C. We write σ = (γ1, γ2, . . . , γp) if γ1, γ2, . . . , γp are the pathsthat make up the cycle σ.

Suppose that σ = (γ1, γ2, . . . , γp) is a cycle in a set A and that f : A→ C isa continuous function. We have that∫

σ

f(z) dz =

∫γ1

f(z) dz +

∫γ2

f(z) dz + · · ·+∫γp

f(z) dz

In particular, for z ∈ C\|σ| we define the winding number of σ about z by

n(σ, z) =1

2πi

∫σ

dζ

ζ − z

And also,n(σ, z) = n(γ1, z) + n(γ2, z) + · · ·+ n(γp, z).

Definition 6.0.20. Let U be an open set. A cycle σ in U is said to be homologousto zero in U if n(σ, z) = 0 for every z ∈ C\U . Two cycles σ0 = (γ1, γ2, . . . , γp)and σ1 = (β1, β2, . . . , βq) in U are pronounced homologous in U if the cycle σ =(γ1, . . . , γp,−β1, . . . ,−βq) is homologous to zero in that set, or equivalently, ifn(σ0, z) = n(σ1, z) for every z ∈ C\U . Finally , two non-closed piecewisesmooth paths λ0 and λ1 are homologous in U if they share the same initial andterminal points and if the closed path γ = λ0 − λ1 is homologous to zero in U.

Theorem 6.0.21. (Cauchy’s Theorem - Global) Let σ be a cycle in an open setU. Then

∫σf(z) dz = 0 for every function f that is analytic in U if and only if σ

is homologous to zero in U.

19


Corollary 6.0.22. If a function f is analytic in an open set U and if σ0 and σ1are cycles in U that are homologous in this set, then

∫σ0f(z) dz =

∫σ1f(z) dz.

Corollary 6.0.23. If a function f is analytic in an open set U and λ0 and λ1 arenon-closed piecewise smooth paths in U that are homologous in this set, then∫λ0f(z) dz =

∫λ1f(z) dz.

Theorem 6.0.24. (Cauchy’s Integral Formula - Global) Suppose that a functionf is analytic in an open set U and that σ is a cycle in U which is homologous tozero in this set. Then

n(σ, z)f(z) =1

2πi

∫σ

f(ζ)dζ

ζ − zfor every z ∈ U\|σ|.Theorem 6.0.25. Suppose that a function f is analtyic in an open set U, that k isa non-negative integer, and that σ is a cycle in U which is homologous to zero inthis set. Then

n(σ, z)f (k)(z) =k!

2πi

∫σ

f(ζ)dζ

(ζ − z)k+1

for every z ∈ U\|σ|.Theorem 6.0.26. (Goursat’s Theorem) Let γ be a Jordan contour, and let Dbe the inside of |γ| with f : D → C continous and also analytic in D. Then∫γf(z) dz = 0 and

f (k)(z) =k!

2πi

∫γ

f(ζ)dζ

(ζ − z)k+1

for every z ∈ D and every non-negative integer k.Theorem 6.0.27. Let z be a point of the complex plane. If α and β are closed,piecewise smooth paths in C\{z} that are freely homotopic in C\{z}, thenn(α, z) = n(β, z).

Corollary 6.0.28. Let U be an open set in the complex plane, and let α and β beclosed, piecewise smooth paths in U. If α and β are freely homotopic in U, thenthey are homologous in this set.

Definition 6.0.29. A closed path γ in a set A is said to be contractible / nullhomotopic in A if γ is freely homotopic in that set to a constant path.

Theorem 6.0.30. Let U be an open set and let γ be a closed, piecewise smoothpath in U. If γ is contractible in U, then γ is homologous to zero in this set.

Definition 6.0.31. A domain D is simply connected under the condition thatevery closed and piecewise smooth path in D, hence every cycle in D, is homol-ogous to zero in that domain.

Theorem 6.0.32. Let D be a domain. Then D is simply connected if and only ifevery function that is analytic in D possesses a primitive in this domain.

20

Chapter 7

Isolated Singularities

Definition 7.0.1. We say that a function f has an isolated singularity at a pointz0 provided there exists an r > 0 with the property that f is analytic in thepunctured disc ∆∗(z0, r), yet not analytic in the full open disc ∆(z0, r). Thissituation can come about for one of two reasons: either z0 does not belong tothe domain-set of f, or, z0 is a member of the domain-set but is a point at whichf is discontinuous. We call f analytic modulo isolated singularities in an openset U if there is a discrete subset E of U, the singular set of f in U, with thefeature that f is analytic in the open set U\E, but has a singularity at each pointof E.

Recall that if f is analytic in ∆∗(z0, r) and continuous in ∆(z0, r) then f isactaully analytic in ∆(z0, r).

Definition 7.0.2. Assume f has an isolated singularity at z0. Let ∆∗(z0, r) be thepunctured disc in which f is analytic. We know that f can be represented in ∆∗ asa sum of a Laurent series centered at z0. The singularity falls into one of threecategories depending on the character of the Laurent expansion. We say f hasa removable singularity at z0 if an = 0 for every negative index n; to have apole at z0 if an 6= 0 holds for at least one, but for at most finitely many negativevalues of n; and to have an essential singularity at z0 if an 6= 0 is true for aninfinite number of negative integers n.

An isolated singularity of a function f at a point z0 is removable if and only iff(z0) can be defined, as to render f differentiable at z0.

Theorem 7.0.3. (Riemann Extension Theorem) Let a function f have an isolatedsingularity at a point z0. The singularity is removable if and only if f is boundedin some punctured disc centered at z0.

Theorem 7.0.4. Let a function f have an isolated singularity at a point z0. Thesingularity is removable if and only if limz→z0 |f(z)| exists.

21

CHAPTER 7. ISOLATED SINGULARITIES

We say that f has a pole of order m at z0 if we can write f as

f(z) =a−m

(z − z0)m+ · · ·+ a−1

z − z0+∞∑n=0

an(z − z0)n,

with a−m 6= 0. Multiplying both sides by (z − z0)m we obtain for all z ∈ ∆∗

(z − z0)mf(z) = a−m + a−m+1(z − z0) + · · · =∞∑n=0

an−m(z − z0)n.

The last series is a Taylor series that converges at every point in ∆. If we letg(z) =

∑∞n=0 an−m(z − z0)n we see that it is analytic in ∆ and that g(z0) =

a−m 6= 0 and that

f(z) =g(z)

(z − z0)mfor every z ∈ ∆∗

Theorem 7.0.5. Let m be a positive integer. A function f that is analytic in apunctured disc ∆∗(z0, r) has a pole of order m at z0 if and only if f can berepresented in ∆∗ in the fashion

f(z) =g(z)

(z − z0)m,

where g is a function that is analytic in ∆(z0, r) and obeys the condition g(z0) 6=0.

Definition 7.0.6. The residue at z0 of the function f (denoted Res(z0, f)), is thecoefficient a−1 in its Laurent expansion. Or it is the coefficient of (z− z0)m−1 inthe Taylor expansion of g about z0. Thus we have that

Res(z0, f) =1

(m− 1)!limz→z0

dm−1

dzm−1[(z − z0)mf(z)].

If an analytic function f has a zero of order m at z0, then 1/f has a pole oforder m at z0. Also, if a function f has a pole of order m at z0, then 1/f has azero of order m there, in the sense that 1/f has a removable singularity at z0 anthat upon its removal, 1/f acquires a zero of order m at that point.

Theorem 7.0.7. Let a function f have an isolated singularity at a point z0. Thesingularity is a pole if and only if limz→z0 |f(z)| =∞. Moreover, the singularityis a pole of order m if and only if m is the unique positive exponent for whichlimz→z0 |z − z0|m|f(z)| is a positive real number.

Theorem 7.0.8. If neither of two functions f and g has worse than a pole at apoint z0, then none of the functions f ′, f + g, fg, and, unless g vanishes identi-cally in some punctured disc centered at z0, f/g has worse than a pole at z0.

22


Definition 7.0.9. Let U be an open set. A function f is called meromorphic inU provided f has at no point of U worse than a pole.

Theorem 7.0.10. Let a function f have an isolated singularity at a point z0. Thesingularity is essential if and only if limz→z0 |f(z)| fails to exist either in thestrict sense or as an infinite limit.

Theorem 7.0.11. (Casorati-Weierstrass Theorem) If a function f is analytic ina punctured disc ∆∗(z0, r) and has an essential singularity at its center, thenf(∆∗) is dense in the complex plane, i.e., the set C\f(∆∗) has no interior points.

Theorem 7.0.12. (Picard’s Theorem) If a function f is analytic in a punctureddisc ∆∗(z0, r) and has an essential singularity at its center, then the set C\f(∆∗)contains at most one point.

Theorem 7.0.13. (Residue Theorem) Suppose that a function f is analytic mod-ulo isolated singularities in an open set U, that E 6= ∅ is the singular set of f inU, and that σ is a cycle in U\E which is homologous to zero in U. Then∫

σ

f(z) dz = 2πi∑z∈E

n(σ, z)Res(z, f).

Theorem 7.0.14. (Argument Principle) Assume that a function f is meromorphicin an open set U. Let γ be a Jordan contour in U such that the Jordan curve |γ|does not pass through any zero or pole of f and such that the inside D of |γ| iscontained in U. Then

1

2πi

∫γ

f ′(z)dz

f(z)= Z − P,

where Z and P indicate the number of zeros and the number of poles, respec-tively, that f has in D, multiplicity being taken into account.

Theorem 7.0.15. (Rouché’s Theorem) If D is the domain inside the trajectoryof a Jordan contour, if f and g are functions that are analytic in some open setwhich contains D, and if the inequality

|f(z)− g(z)| < |f(z)|+ |g(z)|

holds at every point z in ∂D, then f and g have the same number of zeros in D,provided that zero-counts are made with due regard for multiplicity. A restate-ment of the theorem says that if

|g(z)| < |f(z)| for all z ∈ ∂D

then f and f + g have the same number of roots in D

. The classical Rouché’s Theorem says that if

|f(z)− g(z)| < |f(z)| for all z ∈ ∂D

then f and g have the same number of roots in D.

23


Theorem 7.0.16. Let R be a rational function of x and y whose domain includesthe circle K(0, 1). Then∫ 2π

0

R(cos θ, sin θ) dθ = 2π

p∑k=1

Res(zk, f),

where f(z) = z−1R[(z + z−1)/2, (z − z−1)/2i] and z1, z1, . . . , zp are the polesof f in the disc ∆(0, 1).

Theorem 7.0.17. If f(z) = (a0 + a1z + · · · + anzn)/(b0 + b1z + · · · + bmzm)is a rational function in which m ≥ n+ 2 and in which the denominator has noreal roots, then for c ≥ 0∫ ∞

−∞f(x)eicx dx = 2πi

p∑k=1

Res[zk, f(z)eicz],

where z1, z2, . . . , zp are the poles of f in the half-plane H = {z : Im z > 0}.Furthermore, if all the coefficients of f are real numbers, then∫ ∞

−∞f(x) cos(cx) dx = Re

{2πi

p∑k=1

Res(zk, f(z)eicz)

}and ∫ ∞

−∞f(x) sin(cx) dx = Im

{2πi

p∑k=1

Res(zk, f(z)eicz)

}If c > 0, then the conclusions are valid even if m = n+ 1.

24

Complex Numbers and FunctionsPower SeriesAnalytic FunctionsLogarithms and BranchesComplex IntegrationCauchy's Theorem and Its ConsequencesIsolated Singularities

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