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X

an(z z0 )n :

h(z) = (13.3)

;

n=;1

161

CHAPTER 13. COMPLEX INTEGRATION

162

However, it should be noted that not every function can be represented as such a sum.

Problem b: Can you think of one?

13.2 The residue theorem

It was argued at the beginning of this section that the integral of a complex function

around a closed contour in the complex plane is only nonzero when the function is not

analytic at some point in the area enclosed by the contour. In this section we will derive

what the value is of the contour integral. Let us integrate a complex function h(z) along a

contour C in the complex plane that encloses a pole of the function at the point z0 , see the

left panel of gure (13.1). Note that the integration is carried out in the counter-clockwise

direction. It is assumed that around the point z0 the function h(z) can be expanded in

a power Hseries of the form (13.3). It is our goal to determine the value of the contour

integral C h(z)dz.

000000000000000

111111111111111

C

C

111111111111111

000000000000000

111111111111111

000000000000000

000000000000000

111111111111111

111111111111111

000000000000000

C-

000000000000000

111111111111111

000000000000000

111111111111111

C+

111111111111111

000000000000000

z0

Îµ

000000000000000

111111111111111

000000000000000

111111111111111

000000000000000

111111111111111

000000000000000

111111111111111

111111111111111

000000000000000

Figure 13.1: De nition of the contours for the contour integration.

The rst step in the determination of the contour integral is to recognize that within

the shaded area in the right panel of gure (13.1) the function h(z) is analytic because we

assumed that h(z) is only non-analytic at the point z0 . By virtue of the identity (12.10)

this implies that I

h(z)dz = 0 (13.4)

C

where the path C consists of the contour C, a small circle with radius " around z0 and

the paths C + and C ; in the right panel of gure (13.1).

Problem a: Show that the integrals along C + and C ; do not give a net contribution to

the total integral: Z Z

h(z)dz + h(z)dz = 0 : (13.5)

C+ C;

Hint: note the opposite directions of integration along the paths C + and C ; .

13.2. THE RESIDUE THEOREM 163

Problem b: Use this result and expression (13.4) to show that the integral along the

original contour C is identical to the integral along the small circle C" around the

point where h(z) is not analytic:

I I

h(z)dz = h(z)dz : (13.6)

C C"

Problem c: The integration along C is in the counter-clockwise direction. Is the inte-

gration along C" in the clockwise or in the counter-clockwise direction?

Expression (13.6) is very useful because the integral along the small circle can be evaluated

by using that close to z0 the function h(z) can be written as the series (13.3). When one

does this the integration path C" needs to be parameterized. This can be achieved by

writing the points on the path C" as

z = z0 + " exp i' (13.7)

with ' running from 0 to 2 since C" is a complete circle.

Problem d: Use the expressions (13.3), (13.6) and (13.7) to derive that

I Z

1

X (n+1) 2

h(z)dz = ian " exp (i(n + 1)') d' : (13.8)

0

C n=;1

This expression is very useful because it expresses the contour integral in the coe cients

an of the expansion (13.3). It turns out that only the coe cient a;1 gives a nonzero

contribution.

Problem e: Show by direct integration that:

(

Z2 for m = 0

0

exp (im') d' = (13.9)

6

for m = 0

2

0

Problem f: Use this result to derive that only the term n = contributes to the sum

;1

in the right-hand side of (13.8) and that

I

h(z)dz = 2 ia;1 (13.10)

C

It may seem surprising that only the term n = contributes to the sum in the right

;1

hand side of equation (13.8). However, we could have anticipated this result because we

had already discovered that the contour integral does not depend on the precise choice of

the integration path. It can be seen that in the sum (13.8) each term is proportional to

"(n+1) . Since we know that the integral does not depend on the choice of the integration

path, and hence on the size of the circle C", one would expect that only terms that do not

depend on the radius " contribute. This is only the case when n + 1 = 0, hence only for

the term n = is the contribution independent of the size of the circle. It is indeed only

;1

this term that gives a nonzero contribution to the contour integral.

CHAPTER 13. COMPLEX INTEGRATION

164

The coe cient a;1 is usually called the residue and is denoted by the symbol Res h(z0 )

rather than a;1 . However, remember that there is nothing mysterious about the residue,

it is simply de ned by the de nition

Res h(z0 ) a;1 : (13.11)

With this de nition the result (13.10) can trivially be written as

I

h(z)dz = 2 iRes h(z0 ) (counter clockwise direction) : (13.12)

;

C

This may appear to be a rather uninformative rewrite of expression (13.10) but it is the

form (13.12) that you will nd in the literature. The identity (13.12) is called the residue

theorem.

Of course the residue theorem is only useful when one can determine the coe cient

a;1 in the expansion (13.3). You can nd in section (2.12) of the book of Butkov 14] an

overview of methods for computing the residue. Here we will present the two most widely

used methods. The rst method is to determine the power series expansion (13.3) of the

function explicitly.

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