Mathematics of Physics and Engineering

222 Singularities of Complex Functions

of / at zo = 2; the Laurent series at the point ZQ = 2 must converge when
0 < \z — 21 < R for some R, and that is the series you compute in Exercise
4.4.4.
Finally, let us emphasize once again that a Laurent series does not have
to be around a singular point. We go back to the equality (1 — z)_1 =
Sfc>o •^2 ~1_fc, which is true for \z\ > 1. Even though we have infinitely many
negative powers of z, zero is not an essential singularity of f(z) = 1/(1 — z),
because, for f(z) — 1/(1 — z), the point ZQ = 0 is not a singularity at all!
EXERCISE 4.4.5.G Find the expansion of the function from (4-4-8) in the
domain {z : 1 < \z + 1| < 3}. Hint: use geometric series; your answer should
be Xlfcl-oo ak(z + 1) for suitable numbers ak-
EXERCISE 4.4.6^ We say that the point z = oo is an isolated singular point
of the function f = f(z) if and only if the point z = 0 is an isolated singular
point of the function h(z) = f(l/z). Show that z = oo is (a) a removable
singularity of a rational function P(z)/Q(z) if the degree of P is less than
or equal to the degree of Q; (b) A pole of order n for a polynomial of degree
n; (c) an essential singularity for f(z) = sin 2.

4.4.2 Residue Integration

Let ZQ be an isolated singular point of the function / = f(z) and let

oo

f{z)= Y, ck(z-z 0 )k (4.4.10)

fc= —oo

be the corresponding Laurent series expansion of / at ZQ. The coefficient

c_i in this expansion is called the residue of the function / = f(z) at the

point zo and is denoted by Kesf(z). As we saw in the previous section,

Z = ZQ

there could be several different expansions of / in powers of (z — ZQ); the

expansion we use in (4.4.10) is around the point z§ and must converge when

0 < \z — ZQ\ < R for some R.

The Latin word residuus means "left behind," and we will see next that

c_i is the only coefficient in the expansion (4.4.10) that contributes to the

integral of / along a closed curve around ZQ.

Recall that

I dz |2*i, n=l,

Jc(z 0 ) (* - zo)n [0, n = 0,-1,±2,±3,...,