Mathematical Methods for Physics and Engineering : A Comprehensive Guide

COMPLEX VARIABLES

must be made in terms of entirely real integrals. For example, the first is given by
∫ 1

0

−R+iR

R(1−t)+itR

dt=

∫ 1

0

(−1+i)(1−t−it)

(1−t)^2 +t^2

dt

=

∫ 1

0

2 t− 1

1 − 2 t+2t^2

dt+i

∫ 1

0

1

1 − 2 t+2t^2

dt

=

1

2

[

ln(1− 2 t+2t^2 )

] 1

0

+

i

2

[

2tan−^1

(

t−^12

1

2

)] 1

0

=0+

i

2

[π

2

−

(

−

π

2

)]

=

πi

2

.

The second integral on the RHS of (24.38) can also be shown to have the valueπi/2. Thus
∫

C 3

dz

z

=πi.

Considering the results of the preceding two examples, which have common

integrands and limits, some interesting observations are possible. Firstly, the two

integrals fromz=Rtoz=−R, alongC 2 andC 3 , respectively, have the same

value, even though the paths taken are different. It also follows that if we took a

closed pathC 4 , given byC 2 fromRto−RandC 3 traversed backwards from−R

toR, then the integral roundC 4 ofz−^1 would be zero (both parts contributing

equal and opposite amounts). This is to be compared with result (24.36), in which

closed pathC 1 , beginning and ending at the same place asC 4 , yields a value 2πi.

It is not true, however, that the integrals along the pathsC 2 andC 3 are equal

for any functionf(z), or, indeed, that their values are independent ofRin general.

Evaluate the complex integral off(z)=Rezalong the pathsC 1 ,C 2 andC 3 shown in

figure 24.9.

(i) If we takef(z)=Rezand the contourC 1 then

∫

C 1

Rezdz=

∫ 2 π

0

Rcost(−Rsint+iRcost)dt=iπR^2.

(ii) UsingC 2 as the contour,

∫

C 2

Rezdz=

∫π

0

Rcost(−Rsint+iRcost)dt=^12 iπR^2.

(iii) Finally the integral alongC 3 =C 3 a+C 3 bis given by

∫

C 3

Rezdz=

∫ 1

0

(1−t)R(−R+iR)dt+

∫ 1

0

(−sR)(−R−iR)ds

=^12 R^2 (−1+i)+^12 R^2 (1 +i)=iR^2 .

The results of this section demonstrate that the value of an integral between the

same two points may depend upon the path that is taken between them but, at

the same time, suggest that, under some circumstances, the value is independent

of the path. The general situation is summarised in the result of the next section,