Mathematical Methods for Physics and Engineering : A Comprehensive Guide

24.8 COMPLEX INTEGRALS

(a) (b) (c)

y y y

x RRx x

R R

t t

−R −R

s=1

iR

t=0

C 3 b C 3 a

C 1 C 2

Figure 24.9 Different paths for an integral off(z)=z−^1. See the text for

details.

This very important result will be used many times later, and the following should

be carefully noted: (i) its value, (ii) that this value is independent ofR.

In the above example the contour was closed, and so it began and ended at

the same point in the Argand diagram. We can evaluate complex integrals along

open paths in a similar way.

Evaluate the complex integral off(z)=z−^1 along the following paths (see figure 24.9):

(i)the contourC 2 consisting of the semicircle|z|=Rin the half-planey≥ 0 ,

(ii)the contourC 3 made up of the two straight linesC 3 aandC 3 b.

(i) This is just as in the previous example, except that now 0≤t≤π. With this change,
we have from (24.35) or (24.36) that

∫

C 2

dz

z

=πi. (24.37)

(ii) The straight lines that make up the countourC 3 may be parameterised as follows:

C 3 a,z=(1−t)R+itR for 0≤t≤1;

C 3 b,z=−sR+i(1−s)R for 0≤s≤ 1.

With these parameterisations the required integrals may be written

∫

C 3

dz

z

=

∫ 1

0

−R+iR

R+t(−R+iR)

dt+

∫ 1

0

−R−iR

iR+s(−R−iR)

ds. (24.38)

If we could take over from real-variable theory that, for realt,

∫

(a+bt)−^1 dt=b−^1 ln(a+bt)
even ifaandbare complex, then these integrals could be evaluated immediately. However,
to do this would be presuming to some extent what we wish to show, and so the evaluation