1550251515-Classical_Complex_Analysis__Gonzalez_

654 Chapter9

9.2 ISOLATED SINGULARITIES

Definitions 9.8 If z = a i-oo is an isolated singularity of f, then in
any circular ring with center a and contained in some neighborhood of a
(Fig. 9.1) f(z) has a Laurent series representation (Section 8.18):

+oo A

f(z) = '°' Ak(z-a)k = · · + A_n +· · +--=!...__ +Ao+A1(z-a)+· · ·

L.J (z-a)n (z-a)

k=-oo

If A_n = 0 for all n > m but A_m '! 0 the point a is said to be a pole

of order m off. In this case we have

f( z ) = A-m +···+--+ A-1 A o+ A 1z-a ( ) +···

(z-a)m z-a

The sum

A-m A-1

---+···+--(z - a)m z - a

is called the principal part or the meromorphic part of f at a.

(9.2-1)

If A-n i- 0 for infinitely many values of n, then the point a is called an

isolated essential singularity of f .. The sum

00

:S A-n(Z - a)-n

n=l

is called the principal part of f at a.

Letting a= inf {k: Aki-O} we may put our definitions as follows: The

point a is a pole of f if -oo < a < 0, and it is an isolated essential

singulai;ity if a = -oo. On the other hand, the condition o: ~ 0 implies

that a is a regular point of f. ·

If a =: oo, then the Laurent expansion of J(z) in a circular ring with

center at the origin and contained in some neighborhood of oo, namely, in

Fig. 9.1