Page 418 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 16] FUNCTIONS OF A COMPLEX VARIABLE 409

sin z
; z ¼ : Let z ¼ u: Then z ¼ þ u and
z
ðcÞ
!
sin z sinðu þ Þ sin u 1 u 3 u 5
z ¼ u ¼ u ¼ u u 3! þ 5!
u 2 u 4 ðz Þ 2 ðz Þ 4
3! 5! 3! 5!
¼ 1 þ þ ¼ 1 þ þ
z ¼ is a removable singularity.
The series converges for all values of z.
z
; z ¼ 1: Let z þ 1 ¼ u. Then
ðdÞ
ðz þ 1Þðz þ 2Þ
z u 1 u 1 2 3 4
¼ ¼ ð1 u þ u u þ u Þ
u
ðz þ 1Þðz þ 2Þ uðu þ 1Þ
1 2 3
¼ þ 2 2u þ 2u 2u þ
u
1 2
¼ þ 2 2ðz þ 1Þþ 2ðz þ 1Þ
z þ 1
z ¼ 1isa pole of order 1,or simple pole.
The series converges for values of z such that 0 < jz þ 1j < 1.
1
; z ¼ 0; 2:
3
ðeÞ
zðz þ 2Þ
Case 1, z ¼ 0. Using the binomial theorem,
z
1 1 1 ð 3Þð 4Þ z 2 ð 3Þð 4Þð 5Þ z 3

3 ¼ 3 ¼ 8z 1 þð 3Þ 2 þ 2! 2 þ 3! 2 þ
zðz þ 2Þ 8zð1 þ z=2Þ
1 3 3 5 2
¼ þ z z þ
8z 16 16 32
z ¼ 0is a pole of order 1,or simple pole.
The series converges for 0 < jzj < 2.
Case 2, z ¼ 2. Let z þ 2 ¼ u. Then
u 2 3 4
1 1 1 1 u u u
3 ¼ ðu 2Þu 3 ¼ 3 ¼ 2u 3 1 þ þ 2 þ 2 þ 2 þ
2
zðz þ 2Þ 2u ð1 u=2Þ
1 1 1 1 1
2u 4u 8u 16 32
¼ 3 2 u
1 1 1 1 1
3 2 16 32
¼ ðz þ 2Þ
8ðz þ 2Þ
2ðz þ 2Þ 4ðz þ 2Þ
z ¼ 2isa pole of order 3.
The series converges for 0 < jz þ 2j < 2.
RESIDUES AND THE RESIDUE THEOREM
16.23. Suppose f ðzÞ is analytic everywhere inside and on a simple closed curve C except at z ¼ a which is
a pole of order n. Then
a n a nþ1 2
f ðzÞ¼ n þ n 1 þ þ a 0 þ a 1 ðz aÞþ a 2 ðz aÞ þ
ðz aÞ
ðz aÞ
where a n 6¼ 0. Prove that