Mathematical Methods for Physics and Engineering : A Comprehensive Guide

24.8 Complex integrals

x φ^1 φ^2

1 x 2

− 11

y

x

w 1 w 2

w 3 w 3

−aa

s

r

w=g(z)

Figure 24.7 Transforming the upper half of thez-plane into the interior of

the region−a<r<a,s>0inthew-plane.

24.8 Complex integrals

Corresponding to integration with respect to a real variable, it is possible to

define integration with respect to a complex variable between two complex limits.

Since thez-plane is two-dimensional there is clearly greater freedom and hence

ambiguity in what is meant by a complex integral. If a complex functionf(z)is

single-valued and continuous in some regionRin the complex plane, then we can

define the complex integral off(z) between two pointsAandBalong some curve

inR; its value will depend, in general, upon the path taken betweenAandB(see

figure 24.8). However, we will find that for some paths that are different but bear

a particular relationship to each other the value of the integral doesnotdepend

upon which of the paths is adopted.

Let a particular pathCbe described by a continuous (real) parametert

(α≤t≤β) that gives successive positions onCby means of the equations

x=x(t),y=y(t), (24.32)

witht=αandt=βcorresponding to the pointsAandB, respectively. Then the

integral along pathCof a continuous functionf(z) is written
∫

C

f(z)dz (24.33)

and can be given explicitly as a sum of real integrals as follows:
∫

C

f(z)dz=

∫

C

(u+iv)(dx+idy)

=

∫

C

udx−

∫

C

vdy+i

∫

C

udy+i

∫

C

vdx

=

∫β

α

u

dx

dt

dt−

∫β

α

v

dy

dt

dt+i

∫β

α

u

dy

dt

dt+i

∫β

α

v

dx

dt

dt.

(24.34)