Singularities/Residues/ Applications9.6 CHARACTERIZATION OF SOME SIMPLE
FUNCTIONS BY THE NATURE OF THEIR
SINGULARITIES663Classes of functions that are analytic in a domain except possibly on a
certain set of singular points are determined by the position and type of
those singularities. In this section we investigate the simplest classes of such
functions. We may begin by asking whether there are analytic functions
with no singularities at all in C*. The answer to this question is contained
in the following theorem.
Theorem 9.6 The functions that are regular at every point of C* are the
constant functions.
Proof Clearly, a constant function in C* is regular everywhere in the ex-tended complex plane. To prove the converse, let f be any function regular
in C*. Since f is regular at oo, lf(z)I is bounded in some neighborhood of
oo, that is, there exists Ki > 0 and R > 0 such that
IJ(z)I R
In the compact set lzl ::::; R, f is regular, hence continuous, and it follows
that for some K 2 > 0 we have
lf(z)J < K2 for lzl::::; R
Therefore,
Jf(z)I <max( Ki, K2)for all finite z. By Liouville theorem the only functions that are analytic
and bounded in C are the constant func~ions. T.hus f(z) = c for some
constant c. Since f(z) is supposed to be regular at oo [or g(z) = f(l/z)
regular at OJ we must have f ( oo) = c also. '
Corollary 9.1 A nonconstant analytic function in C must have a singular
point at oo.Examples
- A polynomial P(z) = Z::::~=O akzk with an =fa 0, n ;::: 1, has a pole of
order n at oo. - The function f(z) = sinz has an essential singularity at. oo.
Theorem 9. 7 The functions that are regular in C and have a pole at oo
are the polynomials of degree n ;::: 1.
Proof Obviously, a polynomial of degree n ;::: 1 has at oo a pole of order
n (see Section 5.16, Example 4). Next, let f be regular in C. Then f(z)