# Contour integration

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

For a complex-valued function of a complex variable that is analytic in some region we may define the process of integration along a curve via the Riemann summation formalism

where represents an increment along the curve

See Figure 1. Such a curve is called a *contour,* though this term does not refer to a level curve,

or a curve of equal elevation as it would in cartography, but to any curve in the complex plane that does not cross itself.

Formally we may write

Several important results allow us to make sense of, and to make use of the properties of complex integration.

## Contents

## Cauchy's Theorem

If is a closed contour, and the complex valued function is analytic inside the region bounded by, and on then

This is followed by a complementary theorem by Morerra

## Morera's Theorem

If for every closed contour within a region of the complex and

then is analytic everywhere in .

## Cauchy integral theorem

If is analytic on, and inside, a region bounded by a closed contour then for a point inside C,

## Cauchy integral formulas

If is analytic on, and inside, a region bounded by a closed contour then for a point inside C, and for every integer

where is the -th derivative with respect to of .

## Residue Theorem

### Simple pole

If is analytic on, and inside, a region bounded by a closed contour then for a point
except at a point inside C, such that is a *simple pole,* that is
for analytic inside and on then

where, is called the *residue* of at

If has finite simple poles inside

### Multiple pole

If for an integer where is analytic inside and on then is said to have an -th order pole at and

Here, the where is the -th derivative of