# Residue Theorem

Here we follow standard texts, such as Spiegel (1964) ^{[1]} or Levinson and Redheffer (1970). ^{[2]}

We consider a complex-valued function which is analytic everywhere in a region of the complex plane
except possibly at a point .

## Contents

## Case 1): is a simple pole

If is a simple pole, and is analytic everywhere in then we may write

By Taylor's theorem because analytic, the function has a Taylor expansion about

for every point in . Hence the Laurent expansion may

of may be formed by dividing each term of the Taylor expansion by . Hence the Laurent expansion of is

By the Cauchy integral formula

which is the coefficient of the term of the Laurent expansion of .

Thus, the integral over a closed contour in circling around the point of is given by

where is the coefficient of the order term of the Laurent expansion of about

## Case 2): is a -th order pole.

If is an -th order pole, and is analytic everywhere in then we may write

Again, by Taylor's theorem because analytic, the function has a Taylor expansion about

for every point in . Hence the Laurent expansion may

of may be formed by dividing each term of the Taylor expansion by . Hence the Laurent expansion of is

As in the previous case, the n=-1 term has the integral of the function is related to the derivative of the analytic portion of In this case

### The Residue

We call the quantity for a pole of order

the *Residue of at for a pole of order *

### Multiple poles at

Finally, if has poles at then by Cauchy's theorem and the Cauchy-Goursat theorem, we may replace the integral over the larger contour with the sum of integrals, each with a contour surrounding one and only one pole.

This the integral then in this case becomes

Finally, we note that for a case where the integration contours are clockwise, then there is a minus sign on the value of the integration result of * Two PI *i* times the sum of the residues.*