lecture 4 complex variables i (see chapter 2 in...

Physics 227 Lecture 4 1 Autumn 2008

x

y

x,y

Lecture 4 Complex Variables I (See Chapter 2 in Boas.)

Although it is not immediately obvious, an extremely important and useful extension

of our usual study of real functions of real variables - all “physical” quantities are real

after all – is to consider the corresponding complex functions of a complex variable.

So where do complex numbers come from? The underlying idea is that we want to

be able to make sense of fractional powers of negative numbers. Typically we first

see this issue raised in the context of quadratic equations (you should commit this

result to memory, if you have not already done so),

2

2

0

4.

2

ax bx c

b b acx

a

(4.1)

Thus we want to understand what happens if the discriminate (the argument of the

square root) is less than zero, 2 4 0b ac . In particular, we need a definition of 1

(or by extension 1

, 1 ). The first step is to give this quantity a symbol (i.e., a

label), 1i (the symbol j is also sometimes used, typically in contexts where i is

the electric current). By definition we have the following properties

2 3 41 1, , 1.i i i i i (4.2)

Starting with two real numbers, x and y, we construct a complex number in the form

z x iy , where x is called the real part and y is called the imaginary part. Thus a

complex number is associated with two real numbers. The properties of complex

numbers are very similar to the (familiar) properties of two-dimensional vectors. The

algebra of complex numbers (addition,

subtraction and multiplication by real constants)

is identical to that of two-dimensional vectors,

but the multiplication of two complex numbers is

not identical to the multiplication of two vectors,

as we will see. As with the usual two-

dimensional vectors we can represent complex

numbers as points in a two-dimensional plane

were the real and imaginary parts play the roles

of components (as in the figure). This realization

of complex numbers is called the “rectangular


x

y

x,y

r

form”. The corresponding two-dimensional plane is called the complex plane. Such

plots of complex numbers are often called Argand diagrams.

As with usual two-dimensional vectors complex numbers can also be represented in

“cylindrical coordinates” or “polar form”. The length of the radius in this form is

called the modulus or absolute value of the complex number, modr z z . The

corresponding polar angle is called the phase of the complex number,

2 2

1

cos

sintan

cos sin .

r x y zx r

yy r

x

z r i

(4.3)

This structure is illustrated in the figure to the right.

Using the Euler formula from the last lecture we can

write the compact expression

Re coscos sin ,

Im sin

.

i

i

i

i

ee i

e

z re

(4.4)

The choice of whether to use the rectangular form or the polar form depends on the

context, i.e., we use the representation that simplifies the discussion (recall that we

are lazy and smart). One of the goals in this course is to learn to let the mathematics

do the work for us.

Note that, in order for the above expressions to make sense, we require that


x

y

z x,y

x,yz

A) is dimensionless (so that power series expansion for the exponential makes

sense) so we express the angle in radians and not degrees;

B) is not (in general) single-valued. Since 2 cos2 sin2 1ine n i n , the

angles 0 2 n , 0,1,2,3,n are all equivalent in this expression. Typically

we use just the smallest positive value (the principal angle,0 2 ) or that

angle minus 2 (but other usages can be found in literature). These different

choices for the angle will not change the value of the cosine or sine function

(since it is periodic in 2, but will be important when we take into account

fractional powers.

C) The complex number z is closely related to the complex number obtained by

reflection in the real (x) axis. This reflection changes the sign of the imaginary

part or equivalently changes to or i to –i. This new complex number is

denoted z or z and is called the complex conjugate of z,

cos sin ,

cos sin cos sin .

i

i

z x iy r i re

z x iy r i r i re

(4.5)

Again we can represent these relations

pictorially as to the right.

Next consider how we can manipulate complex numbers.

1) We can multiply by a real constant, which just multiplies the real and imaginary

parts separately by the constant (just like multiplication by a constant for a two-


dimensional vector), which simply changes the absolute value but not the phase of

the resulting complex number,

.icz cx icy cre (4.6)

2) We can add or subtract complex numbers just like we add or subtract two-

dimensional vectors,

1 2 1 2 1 2 .z z x x i y y (4.7)

3) We can multiply two complex numbers, needing only to be careful about the

factors of i. This process is typically simplest to consider using the polar form,

1 21 2

2

1 2 1 1 2 2 1 2 1 2 1 2 1 2

1 2 1 2 1 2 1 2

1 2 1 2

1 2 1 2 1 2 1 2

1 2 1 2 1 2 1 2

cos ,

sin .

ii i

z z x iy x iy x x i x y y x i y y

x x y y i x y y x

re r e r r e

x x y y r r

x y y x r r

(4.8)

The real part of this expression is similar (except for the minus sign) to the usual

two-dimensional scalar product of 2 vectors, while the imaginary part is analogous

to the usual two-dimensional vector product of 2 vectors (again except for the minus

sign). For example, consider 4

1 1 2 iz i e , 3

2 2 2 3 4 iz i e . We have

the product

12

1 2 2 2 3 2 2 3 4 2 5.464 1.464 .iz z i e i (4.9)

Note that multiplication of a complex number by it’s complex conjugate yields the

modulus squared,

22 2 2 ,zz x iy x iy x y r z (4.10)

which really is like the usual scalar product. Likewise the product of one complex

number by the complex conjugate of another complex number is again like a scalar


product plus i times a vector product (but now with the usual signs),

1 2

2

1 2 1 1 2 2 1 2 1 2 1 2 1 2

1 2 1 2 1 2 1 2

1 2 .i

z z x iy x iy x x i x y y x i y y

x x y y i x y y x

rr e

(4.11)

4) Finally we can also divide complex numbers, which again is typically simplest

using the polar form,

1 2

1 1 21 2 1 2 1 2 1 22

2 2 2 2

11 2 1 2 1 2 1 22

2 2

1

1cos sin .

i

z z zx x y y i y x x y

z z z r

re r r ir r

r r

(4.12)

Consider the specific example,

1

2 2

2

4 7 12

3

1 2 2 3 2 2 3 2 2 31

4 42 2 3

11 3 1 3

8

2.

4 2 2

i i

i

i i iz i

z i

i

e e

e

(4.13)

While the first version in rectangular form requires several steps, the polar version

essentially takes only a single step.

An equation involving complex numbers, like a two-dimensional vector equation,

means that there are really 2 equations, one for the real part and one for the imaginary

part,

Re LHS Re RHS ,

Im LHS Im RHS .

(4.14)


x

y

z1

z1z2

z2

Complex equations or inequalities serve to define points, lines or regions in the

complex plane. For example, the equation 3z defines a circle about the origin of

radius 3; Re 1z defines the (open) region to the right of 1x (all y); 3 4z i

specifies the single point at x = 3, y = 4.

The modulus of the difference between two complex

numbers, 2 2

1 2 1 2 1 2z z x x y y , just specifies

the distance between the 2 points in the complex plane.

Next let us consider functions of a complex variable. Take your favorite function of

the real variable x and replace x by z. In general, the function f z will be complex

valued,

,

Re , Im .

f z X z iY z

f z X z f z Y z

(4.15)

To make use of our knowledge of power series definitions of functions (from the

previous lectures) we need to understand powers of complex numbers. This is easiest

to evaluate using the polar form,

cos sin .nnn i n in nz x iy re r e r n i n (4.16)

The interested reader is encouraged to check this feature of the exponential explicitly,

e.g., 2 2 2 21 2! 1 2! 1 2 4 2!i i ie e i i i e .

Life gets more interesting for non-integer powers, e.g., 1n m (m integer),

where we find multi-valued expressions. Due to the fact that is itself multi-valued

(recall point B) above), there will always be m distinct (possibly complex) roots for 1 mz . As a specific example consider 1 4 and 1z . We have


1 0.5 0.5 1x

1

0.5

0.5

1

y

2 1 1 2x

2

1

1

2

y

0 0

2 2

1 4 1 4

4

2 2

1

1 1 .1

i i

i i

i i

i i

e e

e e iz z

e e

e e i

(4.17)

This set exhausts the 4 distinct roots (note that

1 4

6 3 2i ie e i ), although only 2 of the roots are real

numbers. The roots are always arrayed symmetrically

around a circle of radius r z

as in the figure.

A more interesting example is

1 33

1 333

1 33 5 3

8 2

8 8 2 .

8 2 2

i i

i i

i i i

e e

e e

e e e

(4.18)

These roots are illustrated in the figure at the right.

Note that in this case only one root is a real number.

In general for the quantity 1

1n

n iz re

a) there are always n roots,

b) the roots lie evenly spaced on a circle of

radius 1 n

z .

c) the phases of the roots are given by

, 2 , 4 , , 2 1n n n n n n n n .

If our original function is just a polynomial in x, then the complex form is easy to

determine. For an initially real infinite power series (the nb are real) we can write


0 0 0

0

0

Re cos

.

Im sin

n n n in

n n n

n n n

n

n

n

n

n

n

S x b x S z b z b r e

S z X z b r n

S z Y z b r n

(4.19)

For this expression to be useful both of the series (i.e., both the real and the imaginary

parts) must converge. First consider the question of absolute convergence, which

treats both the real and imaginary parts simultaneously,

0 0 0

.n n n

n n n

n n n

S z b z b z b r

(4.20)

If S z converges then S z , X z and Y z all converge absolutely. Note that

cosr n r and sinr n r .

When the complex series does not converge absolutely, it is still possible for X and/or

Y to converge conditionally due to the alternating signs. Consider the series

1

1: 1 .

211

n

n nn

zzS z z

nnn

(4.21)

For 21 iz i e , 1z , the series S i diverges. On the other hand the real and

imaginary parts are

1 1

1 0

cos 12Re ,2

sin 12Im ,2 1

m

n m

m

n m

n

S in m

n

S in m

(4.22)


x

y

R1R2R3

z2

where we have used the facts that cos 2 1 2 0n , cos 1n

n ,

sin 2 1 2 1n

n and sin 0n . Both of these series satisfy the

alternating sign test (test 5)) and converge conditionally. In then follows that the

series S z converges absolutely everywhere inside of the unit circle in the complex

plane, 1r , and converges conditionally at certain discrete points on the unit circle,

e.g., z i . In fact, the behavior on the unit circle is even richer. At the point 1z

( 1, 0r ) the real part diverges while the imaginary part vanishes. At 1z

( 1,r ) the imaginary part again vanishes, but the real part converges

conditionally.

Clearly the concept of “interval of convergence” that was so useful for real series of

real variables is to be replaced with the concept of a “circle of convergence” for

complex series of complex variables. If a complex series of a complex variable

(0z z ) converges (absolutely) inside of a circle (about the point

0z ) in the complex

plane ( 0z z R ), we say the “radius of convergence” of the series is R. The

behavior on the circle ( 0z z R ) is typically complicated (as in the above

example), analogous to what happened at the ends of the interval of convergence, and

generally we study the behavior on the circle point-by-point (although this is not

typically required). In the region inside the circle of convergence (where the series

converges absolutely) we can apply the 4 properties of convergent power series listed

in Lecture 3 for real power series to manipulate and make use of the complex power

series.

In particular, let us consider the issue of range of convergence for the ratio of two

series. Consider two power series in z (0 0z ), 1S z and 2S z where 1S z

converges for 1z R and 2S z converges for 2z R ,

but 2 0 0S . We want to consider the ratio

1 2S z S z and so we are especially interested in the

points where 2 0S z . Let the point 2z z be the point

closest to the origin for which 2 2 0S z (but 1 2 0S z )

and define 3 2R z . The circle of convergence, i.e., the

radius of convergence, of the ratio 1 2S z S z is given

by the minimum of the three radii 1 2 3, ,R R R (see the

figure), which, in general, could be any one of the three.


As a simple example consider the ratio sin 1z z . Both the numerator and

denominator converge for 1 2z R R (the denominator is just a polynomial),

while the denominator vanishes at 2 1z z . Hence the ratio converges, i.e., the

ratio is well defined, for 3 1z R .

Now we will remind ourselves of various useful complex functions/power series

expansions. Probably the most useful is the complex exponential, which is just like

the real case,

2 3

0

1 .2! 3! !

nz

n

z z ze z

n

(4.23)

As in the real case this converges everywhere, i.e., the radius of convergence is

infinite, z R . Thus it follows from this convergent series representation that,

just as for the real case,

1 2

1 2

1

0 1 0

2 2

1 21 2

2 2

1 2 1 1 2 2

,! 1 ! !

1 12! 2!

11 2

2

.

n n mz z

n n m

z z

z z

d d z z ze e

dz dz n n m

z ze e z z

z z z z z z

e

(4.24)

ASIDE: This last result that the product of 2 exponentials is an exponential with

exponent equal to the sum of the two original exponents is true as long as the two

exponents commute, 1 2 1 2 2 1, 0z z z z z z , i.e., the exponents are “c” numbers

and not matrices.

Recall from Euler’s formula that for an exponential with an imaginary argument we

have


cos sin

cos sin cos sin

cos , sin .2 2

iy

iy iy

iy iy iy iy

e y i y

e e y i y y i y

e e e ey y

i

(4.25)

These last exponential expressions for the sinusoidal functions are often useful. Note

that they lead directly to the usual double angle formulae

1 2 1 2

1 2 1 2 1 2 1 2 1 2 1 2

1 1 2 2 1 1 2 2

1 2

1 2 1 2

1sin

2

2 4 4 2

1

4

sin cos cos sin ,

i i i i

i i i i i i i i i i i i

i i i i i i i i

e ei

e e e e e e e e e e

i i i i

e e e e e e e ei

1 2 1 2

1 2 1 2 1 2 1 2 1 2 1 2

1 1 2 2 1 1 2 2

1 2

2

1 2 1 2

1cos

2

2 4 4 2

1 1

4

cos cos sin sin .

i i i i

i i i i i i i i i i i i

i i i i i i i i

e e

e e e e e e e e e e

e e e e e e e ei

(4.26)

For a complex exponent a little arithmetic with the series yields

0 0 0! ! !

cos sin .

n n m

z x iy

n n m

x iy x

x iy x iye e

n n m

e e e y i y

(4.27)


x

y

The real part of the exponent, x, tells us about the modulus of the exponential, x iy xe e , while the imaginary part of the exponent defines the phase of the

exponential, 1tan Im Rex iy x iye e y .

A related and important function is the complex logarithm, which calls out to be

studied in terms of the polar form,

ln

Re ln lnln ln 2 , 0,1,2, .

Im ln

z iz e re

z rz r i i n n

z

(4.28)

Note that the logarithm is multi-valued due to the

multi-valued phase. We will eventually represent

this by defining the logarithm to have infinitely

many “branches” in the complex plane (see Chap.

14 in Boas). The principal value (or principal

branch) is defined by 0 2 with the so-called

branch cut along the positive real axis as indicated

by the wavy line in the figure (although sometimes

it is useful to choose instead with the

branch cut along the negative real axis). Thus the logarithms of several familiar

complex numbers are

ln 1 0 0 2 0 2 ,

ln 1 0 2 ,

ln 0 2 .2

i ni ni

i ni

i i ni

(4.29)

We interpret the 2 n to be describing the points on the various sheets (each sheet is

labeled by a value of n), which are reached by passing “through” the branch cut. We

have already used this structure (without mentioning the branch cuts) in our

discussion of fractional powers and will return to it later.

Now let us return to our discussion of complex exponentials. We saw in Eq. (4.25)

that purely imaginary exponents led to the sine and cosine functions with real


arguments. What about sines and cosines of imaginary arguments, which are related

to real exponentials? We have the following definitions

2 4

3 5

2 2 2 2

cos 12 2 2! 4!

cosh ,

sin2 2 3! 5!

sinh

cosh sinh ,

1 cosh sinh cosh 1 sinh ,

cosh sinh , sinh cosh .

i iy i iy y y

i iy i iy y y

y

y y

e e e e y yiy

y

e e e e y yiy i i y

i

i y

e y y

e e y y y y

d dy y y y

dy dy

(4.30)

These are the hyperbolic functions, which, like the sinusoidal functions, are defined

by power series that converge for all finite (complex) arguments. On the other hand

the behaviors of the hyperbolic functions themselves (with real arguments) are

different from the sinusoidal functions. Instead of exhibiting periodic, bounded

behavior, the hyperbolic functions are monotonic with ranges given by

1 coshcosh cosh ,

0

sinhsinh sinh .

yy y

y

yy y

y

(4.31)

We also have the following relations


cos cos cos cos sin sin

cos cosh sin sinh ,

sin sin sin cos cos sin

sin cosh cos sinh ,

cos sin

cos cosh sinh sin sinh cosh

cos sin .

iz

y

z x iy x iy x iy

x y i x y

z x iy x iy x iy

x y i x y

e z i z

x y y i x y y

e x i x

(4.32)

We can now combine our new knowledge of complex exponentials and logarithms to

explore the behavior of complex powers and roots, being careful to include the issue

of multi-valued phases. Consider two complex numbers, 1

1 1 1 1

iz x iy re and

2

2 2 2 2

iz x iy r e and evaluate

3 3 3 32 2 1ln

1 3 3

3 2 1 2 1 2 1 1

3 2 1 2 1 2 1 1

cos sin ,

Re ln ln 2 ,

Im ln ln 2 .

z x iy xz z ze z e e e e y i y

x z z x r y n

y z z y r x n

(4.33)

Thus such an expression will in general be multi-valued. Let’s consider some

specific examples. First consider 1 1 2 21, 1, 1 4, 0x y x y

1 311 ln 2 2ln 1 4 444

3

16

916

18

1716

2516

1 ln 28

1

16 2

2 ,

i ini

i

i

i

i

xi e e

ny

e

e

e

e

(4.34)

i.e., just the 4 roots we expected for a complex number to the (real) power ¼. For a

more interesting result consider 1 1 2 21, 1, 0, 2x y x y


2 ln 2 22 34

3

ln 2 8 4 4 82

421ln 2

, , ,1, , , .

i i ini

i

x ni e

y

e e e e e e

(4.35)

So in this case there are an infinite number of distinct values of this expression. An

infinite number of values is the typical result whenever 2 2Im 0z y (or if

2 2Re z x is an irrational number). Finally consider an example of the full case

1 1 2 21, 1, 1 3, 1x y x y

1 1 31 ln 1 ln 2 23 3 43

3

12 6 6 12

ln 2 2114 8 2 4 1012 26 34

210 43

1 ln 2 26 4

12ln 2

12 2 3

1 3 , , ,1, , ,

2 3 1 , , , , , ,

, , ,3 1

i i i i ini

ii

i

x ni e e

ny

n m e e e e

e e e n m e e e e e

e ee n m

2 8 14

.

, , ,e e e

(4.36)

Note that the multiple values of 3y yield only 3 distinct phases while there are an

infinite number of overall magnitudes, which are different for different phases!

To finish this particular discussion we raise some final warnings about the care

needed when dealing with complex numbers. Consider the product involving 3

complex numbers, 32

1 1

zzz z , which you might expect could be rewritten as

2 3 1 2 332ln?

1 1 1 .z z z z zzzz z z e

(4.37)

If we just use our definitions and keep all multi-valued variables, we find the

logarithms of the two sides of the equation have the forms

2 3 1 1

2 1 1 1 3 1 1 2

ln RHS ln 2 ,

ln LHS ln 2 ln 2 .

z z r i in

z r i in z r i in

(4.38)


Thus the left-hand-side has more values than the right-hand-side, i.e., the cases when

1 2n n n . The same warning applies to expressions like 33 3 ?

1 2 1 2

zz zz z z z and

3 2 32 ?

1 1

z z zzz z . As a specific example consider

1 1

1 2 1 2

21? 2

2 2ln 2 2

2 2 2 2ln 2 2 .i

ii ini

ii in ni i i

ii i n in in ni i i

i i e

i e e e

i e e e e

(4.39)

The expression on the RHS of the first line agrees with the last expression in the last

line only for 2 0n . We must use care for such expressions. Typically in physics

applications the situation will help define how to proceed, i.e., which of the possible

values contribute.

Actually this issue is not unfamiliar. Consider what happens when we invert an

exponential by taking a logarithm. In the complex variable world we must use

ln 2ze z in , which leads to the questions above, i.e., new values appear that

were not present when we lived just on the real axis. Similarly for other inversions

we have

1 1

1

1 2 2

2

1 2 2

cos2

ln 1 2 ,

iz ize ez z z

z i z z n

(4.40)

and

1 1

1

1 2 2

2

1 2 2

sin2

ln 1 2

iz ize ez z z

i

z i iz z n

(4.41)

where the principal value is the logarithm alone (familiar from real analysis).

lecture 4 complex variables i (see chapter 2 in...

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