Sequences, Series, and Special Functions 565~ cos(2k + 1)() = 1; 1 I 1; e1
L..J 2k 1 2 n cot 2
k=O +(o < 1e1 < 7r)
*7. Suppose that the series f(z) = z=:=o anzn has radius of convergence
R = 1. Show that if an 2:: 0 for all n, then z = 1 is a singular point
of f(z) (Pringsheim theorem).
8. Let f(z) = z::=o anzn with R = 1. If an is real for all n and Sn =
ao + ai + · · · + an ---+ oo (or if Sn ---+ -oo ), then z = 1 is a singular
point of f(z).8.8 Zeros of Continuous Functions
Definitions 8.2 Let f: D ---+ C be a continuous complex function (in
particular, of class 1J or 1-t) defined in the open set D. A point z 0 E D is
said to be a zero (also, a root) off iff f(zo) = 0. More generally, zo E D
is said to be an a-point iff f(zo) = a. Clearly, the a-points off are the
zeros of F(z) = f(z) - a, and conversely.
A zero z 0 of f is called a zero of order m (or of multiplicity m ), where
m ;::: 1 is a positive integer, iff there is a continuous function (respectively,
of class 1J or 1-t) g: D ---+ C such thatf(z) = (z - zo)mg(z)with g(zo) #-0. If such function exists, then
andf(z)g(z)-
- (z - zo)m
for z #-zo
g(zo) = lim f(z) #- 0
z-+zo (z - zo)m(8.8-1)Zeros of order one are also called simple zeros, those of order two double ze-
ros, those of order three triple zeros, and so on. The integer m is sometimes
called the order or local degree off at zo.Example f(z) = (z - 1)^2 (z + i) has a double zero at 1 and a simple
zero at i.The multiplicity of an a-point of f is defined to be the same as the
multiplicity of this point as a zero of F(z) = f(z) - a.
A zero of order m of 1/ f is called a pole of order m off. A function
that is analytic in a region except possibly for poles is called meromorphic
in that region.