Mathematical Methods for Physics and Engineering : A Comprehensive Guide

COMPLEX VARIABLES

A

B

C 1

C 2

C 3

x

y

Figure 24.8 Some alternative paths for the integral of a functionf(z) between

AandB.

The question of when such an integral exists will not be pursued, except to state

that a sufficient condition is thatdx/dtanddy/dtare continuous.

Evaluate the complex integral off(z)=z−^1 along the circle|z|=R, starting and finishing

atz=R.

The pathC 1 is parameterised as follows (figure 24.9(a)):

z(t)=Rcost+iRsint, 0 ≤t≤ 2 π,

whilstf(z)isgivenby

f(z)=

1

x+iy

=

x−iy

x^2 +y^2

.

Thus the real and imaginary parts off(z)are

u=

x

x^2 +y^2

=

Rcost

R^2

and v=

−y

x^2 +y^2

=−

Rsint

R^2

.

Hence, using expression (24.34),
∫

C 1

1

z

dz=

∫ 2 π

0

cost

R

(−Rsint)dt−

∫ 2 π

0

(

−sint

R

)

Rcostdt

+i

∫ 2 π

0

cost

R

Rcostdt+i

∫ 2 π

0

(

−sint

R

)

(−Rsint)dt (24.35)

=0+0+iπ+iπ=2πi.

With a bit of experience, the reader may be able to evaluate integrals like

the LHS of (24.35) directly without having to write them as four separate real

integrals. In the present case,
∫

C 1

dz

z

=

∫ 2 π

0

−Rsint+iRcost

Rcost+iRsint

dt=

∫ 2 π

0

idt=2πi. (24.36)