Page 420 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 16] FUNCTIONS OF A COMPLEX VARIABLE 411

(d)cot z; z ¼ 5 ,a pole of order 1. Then:

cos z z 5 1
Residue is lim ðz 5 Þ ¼ lim lim cos z ¼ lim ð 1Þ
z!5 sin z z!5 sin z z!5 z!5 cos z
¼ð 1Þð 1Þ¼ 1
where we have used L’Hospital’s rule, which can be shown applicable for functions of a complex
variable.

16.25. If f ðzÞ is analytic within and on a simple closed curve C except at a number of poles a; b; c; ...
interior to C, prove that
þ
f ðzÞ dz ¼ 2 i fsum of residues of f ðzÞ at poles a; b; c; etc.g
C
Refer to Fig. 16-8.
By reasoning similar to that of Problem 16.12 (i.e., by con-
structing cross cuts from C to C 1 ; C 2 ; C 3 ; etc.), we have
þ þ þ
f ðzÞ dz ¼ f ðzÞ dz þ f ðzÞ dz þ
C C 1 C 2
For pole a, Fig. 16-8
a m a 1
f ðzÞ¼ m þ þ þ a 0 þ a 1 ðz aÞþ
ðz aÞ ðz aÞ
þ
hence, as in Problem 16.23, f ðzÞ dz ¼ 2 ia 1 :
C 1

b n b 1
Similarly for pole b; f ðzÞ¼ n þ þ þ b 0 þ b 1 ðz bÞþ
ðz bÞ ðz bÞ
þ
so that f ðzÞ dz ¼ 2 ib 1
C 2
Continuing in this manner, we see that
þ
f ðzÞ dz ¼ 2 iða 1 þ b 1 þ Þ ¼ 2 i (sum of residues)
C

e dz
þ z
16.26. Evaluate where C is given by (a) jzj¼ 3=2; ðbÞjzj¼ 10.
2
C ðz 1Þðz þ 3Þ
z
e e
Residue at simple pole z ¼ 1is lim ðz 1Þ 2 ¼
z!1 16
ðz 1Þðz þ 3Þ
Residue at double pole z ¼ 3is
z
d e z ðz 1Þe e z 5e 3
lim 2 ¼ lim
z! 3 dz ðz þ 3Þ 2 z! 3 2 ¼ 16
ðz 1Þðz þ 3Þ ðz 1Þ
(a)Since jzj¼ 3=2encloses only the pole z ¼ 1,
e ie

the required integral ¼ 2 i ¼
16 8