Page 417 - Schaum's Outline of Theory and Problems of Advanced Calculus

408 FUNCTIONS OF A COMPLEX VARIABLE [CHAP. 16

2 3
h h
0 00 000
2! 3!
f ða þ hÞ¼ f ðaÞþ hf ðaÞþ f ðaÞþ f ðaÞþ
By Cauchy’s integral formula (Problem 16.15), we have
1 þ f ðzÞ dz
f ða þ hÞ¼ ð1Þ
2 i C z a h
By division
1 1
z a h ¼ ðz aÞ½1 h=ðz aފ
( )
1 h h 2 h n h nþ1
2
¼ 1 þ þ þ þ n þ n ð2Þ
ðz aÞ ðz aÞ ðz aÞ ðz aÞ ðz a hÞ
ðz aÞ
Substituting (2)in(1) and using Cauchy’s integral formulas, we have
1 þ f ðzÞ dz h þ f ðzÞ dz h n þ f ðzÞ dz
þ R n
2 i C z a 2 i C ðz aÞ 2 i C ðz aÞ
f ða þ hÞ¼ þ 2 þ þ nþ1
h 2 h n
0 00 f ðnÞ
2! n!
¼ f ðaÞþ hf ðaÞþ f ðaÞþ þ ðaÞþ R n
h nþ1 þ f ðzÞ dz
where R n ¼
2 i nþ1
C ðz aÞ ðz a hÞ

Now when z is on C, f ðzÞ @ M and jz aj¼ R,sothat by (4), Page 394, we have, since 2 R is
z a h
the length of C
nþ1 M
jR n j @ jhj 2 R
2 R nþ1
As n !1; jR n j! 0. Then R n ! 0and the required result follows.
If f ðzÞ is analytic in an annular region r 1 @ jz aj @ r 2 ,wecan generalize the Taylor series to a
Laurent series (see Problem 16.92). In some cases, as shown in Problem 16.22, the Laurent series can be
obtained by use of known Taylor series.

16.22. Find Laurent series about the indicated singularity for each of the following functions. Name the
singularity in each case and give the region of convergence of each series.
e z
; z ¼ 1: Let z 1 ¼ u: Then z ¼ 1 þ u and
2
ðaÞ
ðz 1Þ
( )
e z e 1þu e u e u 2 u 3 u 4
2 ¼ u 2 ¼ e u 2 ¼ u 2 1 þ u þ 2! þ 3! þ 4! þ
ðz 1Þ
e e e eðz 1Þ eðz 1Þ 2
2 z 1 2! 3! 4!
¼ þ þ þ þ þ
ðz 1Þ
z ¼ 1isa pole of order 2,or double pole.
The series converges for all values of z 6¼ 1.
1
z cos ; z ¼ 0:
ðbÞ
z

1 1 1 1 1 1 1
z cos ¼ z 1 þ þ ¼ z þ þ
z 2! z 2 4! z 4 6! z 6 2! z 4! z 3 6! z 5
z ¼ 0isan essential singularity.
The series converges for all values of z 6¼ 0.