Mathematical Methods for Physicists: A concise introduction - Site Map

EVALUATION OF REAL DEFINITE INTEGRALS
and the given integrand becomes a rational function of z, say, f …z†. As ranges
from 0 to 2, the variable z ranges once around the unit circle jzj ˆ1 in the
counterclockwise sense. The given integral takes the form
I
f …z† dz
iz ;
C
the integration being taken in the counterclockwise sense around the unit circle.
Example 6.30
Evaluate
Z 2
0
d
3 2cos ‡ sin :
Solution:
Let z ˆ e i , then dz ˆ ie i d, ord ˆ dz=iz, and
then
Z 2
0
sin ˆ z z1
2i
I
d
3 2 cos ‡ sin ˆ
C
; cos ˆ z ‡ z1
;
2
2dz
…1 2i†z 2 ‡ 6iz 1 2i ;
where C is the circle of unit radius with its center at the origin (Fig. 6.18).
We need to ®nd the poles of
1
…1 2i†z 2 ‡ 6iz 1 2i :
z ˆ 6i
q
…6i† 2 4…1 2i†…1 2i†
2…1 2i†
ˆ 2 i; …2 i†=5;
Figure 6.18.
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