# Contour Integration

**Contour integration** is a method of evaluating integrals of functions along oriented curves in the complex plane. It is an extension of the usual integral of a function along an interval in the real number line. Contour integrals may be evaluated using direct calculations, the Cauchy integral formula, or the residue theorem.

## Definitions

A *curve* in the complex plane is a set of points parametrised by some function \([a,b] \to \mathbb{C}\), usually denoted \(\gamma\) paramatrised by \(z\). \(\gamma\) is *differentiable* if \(z'(t)\) exists for all \(t \in [a,b]\).

Let \(f: U \to \mathbb{C}\) and \(\gamma\) be a differentiable curve parametrized by \(z: [a,b] \to \mathbb{C}\). The integral of \(f(z) \ dz\) along \(\gamma\) is defined as

\[\int_\gamma f(z) \ dz = \int_a^b f(z(t))z'(t) \ dt.\]

(If \(f: U \to \mathbb{C}\) is a holomorphic function, then one may think of it as a function from a subset of \(\mathbb{R}^2\) to the whole (\mathbb{R}^2). In other words, \(f\) implicitly defines a vector field on \(\mathbb{R}^2\). Accordingly, one naturally integrates holomorphic functions in the same way as one integrates vector fields, using line integrals.)

## Basic examples

Some contour integrals may be computed by hand. For example:

Let \(\gamma\) denote the circle centered at \(O\) with radius \(1\). Then compute

\[\int_{\gamma} \frac{1}{z} \, dz.\]

First, parametrize the circle by \(\gamma(t) = e^{2\pi i t}\), so that as \(t\) ranges over \([0,1]\), the entire circle is traced out once. Note that \(\gamma(0) = \gamma(1)\); a curve with this property is called

closed.With this parametrization, one computes

\[\int_{\gamma} \frac{1}{z} \, dz = \int_{0}^{1} \frac{1}{\gamma(t)} \gamma'(t) \, dt = \int_{0}^{1} e^{-2\pi i t} \cdot 2\pi i e^{2\pi i t} \, dt = \int_{0}^{1} 2\pi i \, dt = 2\pi i.\ _\square \]

Note that in the above example, the function \(1/z\) is not holomorphic on the region interior to \(\gamma\), since it is not defined at \(z=0\).

## Cauchy-Goursat theorem

Take \(f: U \to \mathbb{C}\) to be holomorphic. The *primitive* of \(f\) is a function \(F: U \to \mathbb{C}\) such that \(F'(z) = f(z)\) for all \(z \in U\).

Suppose \(\gamma\) is a curve parametrised by \(z: [a,b] \to \mathbb{C}\) and \(f\) has a primitive. Then, since complex differentiation follows the chain rule,

\[\begin{align} \int_\gamma f(z) \ dz &= \int_a^b f(z(t)) z'(t) \ dt \\ &= \int_a^b F'(z(t)) z'(t) \ dt \\ &= \int_a^b \frac{d}{dt} F(z(t)) \ dt \\ &= F(z(b))-F(z(a)). \end{align}\]

In particular, if \(\gamma\) is a closed curve, ie \(z(b) = z(a)\), then \(\displaystyle \int_\gamma f(z) \ dz = 0\). There is a result that all holomorphic functions have primitives. Thus, the only nontrivial contour integrals along closed curves are those which enclose poles of the function being integrated.

**Cite as:**Contour Integration.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/contour-integration/