Page 404 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 16] FUNCTIONS OF A COMPLEX VARIABLE 395

derivatives at points within C can be calculated. Thus, if a function of a complex variable has a first
derivative, it has all higher derivatives as well. This, of course, is not necessarily true for functions of
real variables.

TAYLOR’S SERIES
Let f ðzÞ be analytic inside and on a circle having its center at z ¼ a. Then for all points z in the circle
we have the Taylor series representation of f ðzÞ given by
00 000
2 3
f ðaÞ f ðaÞ
0
2! 3!
f ðzÞ¼ f ðaÞþ f ðaÞðz aÞþ ðz aÞ þ ðz aÞ þ ð8Þ
See Problem 16.21.
SINGULAR POINTS
A singular point of a function f ðzÞ is a value of z at which f ðzÞ fails to be analytic. If f ðzÞ is analytic
everywhere in some region except at an interior point z ¼ a,we call z ¼ a an isolated singularity of f ðzÞ.

1
EXAMPLE. ,then z ¼ 3isan isolated singularity of f ðzÞ.
2
If f ðzÞ¼
ðz 3Þ
sin z
EXAMPLE. has a singularity at z ¼ 0. Because lim is finite, this singularity is called a
The function f ðzÞ¼
z z!0
removable singularity.
POLES
; ðaÞ 6¼ 0, where ðzÞ is analytic everywhere in a region including z ¼ a, and if n is a
ðzÞ
If f ðzÞ¼ n
ðz aÞ
positive integer, then f ðzÞ has an isolated singularity at z ¼ a, which is called a pole of order n.If n ¼ 1,
the pole is often called a simple pole;if n ¼ 2, it is called a double pole,and so on.

LAURENT’S SERIES
If f ðzÞ has a pole of order n at z ¼ a but is analytic at every other point inside and on a circle C with
n
center at a, then ðz aÞ f ðzÞ is analytic at all points inside and on C and has a Taylor series about z ¼ a
so that
a n a nþ1 a 1 2
f ðzÞ¼ n þ n 1 þ þ þ a 0 þ a 1 ðz aÞþ a 2 ðz aÞ þ ð9Þ
ðz aÞ z a
ðz aÞ
2
This is called a Laurent series for f ðzÞ. The part a 0 þ a 1 ðz aÞþ a 2 ðz aÞ þ is called the analytic
part, while the remainder consisting of inverse powers of z a is called the principal part. More
1
X k
generally, we refer to the series a k ðz aÞ as a Laurent series, where the terms with k < 0constitute
k¼ 1
the principal part. A function which is analytic in a region bounded by two concentric circles having
center at z ¼ a can always be expanded into such a Laurent series (see Problem 16.92).
It is possible to define various types of singularities of a function f ðzÞ from its Laurent series. For
example, when the principal part of a Laurent series has a finite number of terms and a n 6¼ 0 while
a n 1 ; a n 2 ; ... are all zero, then z ¼ a is a pole of order n. If the principal part has infinitely many
terms, z ¼ a is called an essential singularity or sometimes a pole of infinite order.
1
EXAMPLE. The function e 1=z ¼ 1 þ þ 1 þ has an essential singularity at z ¼ 0.
z 2! z 2