Simple pole

A simple pole is a pole of order one. Its residue is the nonzero value of the holomorphic extension of

(z - a) f (z)

at the pole.

Table of contents
- Regular part at a simple pole Simple pole

Regular part at a simple pole ( $\frac{N ( z )}{( z - z _{0} ) D ( z )} = \frac{N ( z _{0} ) / D ( z _{0} )}{z - z _{0}} + \frac{N ^{'} D - N D ^{'}}{D ^{2}} (z_{0}) + O (z - z_{0})$ )

For holomorphic

N, D

with

N (z_{0}) D (z_{0}) \neq = 0

, write the quotient as

h (z) / (z - z_{0})

where

h = N / D

. Its Taylor series immediately gives the displayed Laurent series. The residue is

h (z_{0})

and the constant regular term is

h^{'} (z_{0})

; the latter requires differentiating both numerator and remaining denominator. Higher regular coefficients follow from the higher Taylor coefficients of

h

.

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