Begin2.DVI

by |z|=

√

x^2 +y^2 which represents the distance of the point zfrom the origin in the

z−plane. (ii) The quantity |z−z 0 |=Rrepresents a circle of radius Rsince

|z−z 0 |=|x+iy −(x 0 +iy 0 )|=|(x−x 0 ) + i(y−y 0 )|=

√

(x−x 0 )^2 + (y−y 0 )^2 =R

or (x−x 0 )^2 + (y−y 0 )^2 =R^2

Figure 12-14. The complex z−plane.

In the theory of complex variables there is an important type of series called the

Laurent^8 series which represents a function f(z) in an expansion about a singular

point z 0 having the form of a power series having both positive and negative powers

of (z−z 0 ). The point z 0 is called the center of the Laurent series. The Laurent series

has the form

f(z) =

∑∞

n=1

cn

(z−z 0 )n

+

∑∞

n=0

αn(z−z 0 )n (12.192)

where the quantities c 1 , c 2 ,... and α 0 , α 1 ,... are constants. In expanded form the

Laurent series (12.192) becomes

f(z) = ··· + c^3

(z−z 0 )^3

+ c^2

(z−z 0 )^2

+ c^1

(z−z 0 )

+α 0 +α 1 (z−z 0 ) + α 2 (z−z 0 )^2 +··· (12.193)

The Laurent series converges in some annular region R 2 <|z−z 0 |< R 1. It can be

shown that the series

∑∞

n=0

αn(z−z 0 )nconverges for zin the circular region |z−z 0 |< R 1

and the series

∑∞

n=1

cn

(z−z 0 )n converges for the circular region |z−z^0 |> R^2 Here R^1 and

(^8) Pierre Alphonse Laurent (1813-1854) A French mathematician who studied complex analysis.