Mathematical Methods for Physics and Engineering : A Comprehensive Guide

COMPLEX VARIABLES

y

x

A B

C D

Γ

γ

Figure 24.16 A typical cut-plane contour for use with multivalued functions

that have a single branch point located at the origin.

located at the origin is shown in figure 24.16. Here Γ is a large circle of radiusR

andγis a small one of radiusρ, both centred on the origin. Eventually we will

letR→∞andρ→0.

The success of the method is due to the fact that because the integrand is

multivalued, its values along the two linesABandCDjoiningz=ρtoz=R

arenotequal and opposite although both are related to the corresponding real

integral. Again an example provides the best explanation.

Evaluate

I=

∫∞

0

dx

(x+a)^3 x^1 /^2

,a> 0.

We consider the integrandf(z)=(z+a)−^3 z−^1 /^2 and note that|zf(z)|→0onthetwo
circles asρ→0andR→∞. Thus the two circles make no contribution to the contour
integral.
The only pole of the integrand inside the contour is atz=−a(and is of order 3).
To determine its residue we putz=−a+ξand expand (noting that (−a)^1 /^2 equals
a^1 /^2 exp(iπ/2) =ia^1 /^2 ):

1

(z+a)^3 z^1 /^2

=

1

ξ^3 ia^1 /^2 (1−ξ/a)^1 /^2

=

1

iξ^3 a^1 /^2

(

1+

1

2

ξ

a

+

3

8

ξ^2

a^2

+···

)

.

The residue is thus− 3 i/(8a^5 /^2 ).
The residue theorem (24.61) now gives
∫

AB

+

∫

Γ

+

∫

DC

+

∫

γ

=2πi

(

− 3 i

8 a^5 /^2

)

.