Begin2.DVI

R 2 are positive constants with R 2 < R 1. The annular region of convergence is the

intersection of these two regions.

Figure 12-15. Annular region of convergence for Laurent series.

The special Laurent series where the point z 0 is the only singular point of f(z)

inside the disk |z−z 0 |< R 2 is of extreme importance in the study of complex variables.

In this special case the point z 0 is called an isolated singular point. This special series

with negative powers of z−z 0 having the form

∑∞

n=1

cn

(z−z 0 )n =···+

cm

(z−z 0 )m+···+

c 2

(z−z 0 )^2 +

c 1

(z−z 0 ) (12.194)

is called the principal part of the Laurent series and associated with the principal

part of the series is the following terminology

(i) The term c 1 is called the residue of f(z)at the isolated singular point z 0.

(ii) If there is only one term in the principal part of the Laurent series, then the

singular point z 0 is called a pole of order 1 or a simple pole.

(iii) If there are only two terms in the principal part of the Laurent series, then the

singular point z 0 is called a pole of order 2.

(iv) If there are m−terms in the principal part of the Laurent series, then the singular

point z 0 is called a pole of order m.

(v) If there is an infinite number of terms in the principal part of the Laurent series,

then the singular point z 0 is called an an essential singularity.

If you get involved with complex variable theory, the binomial expansion

(a+b)−^1 =a−^1 + (−1) a−^2 b+ (−1)(−2) a−^3 b^2 /2! + ··· converges for |b|<|a| (12.195)