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.

A curve in the complex plane is a set of points parameterized by some function \([a,b] \to \mathbb{C}\), usually denoted \(\gamma\) parameterized 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 parameterized by \(z: [a,b] \to \mathbb{C}\). The integral of \(f(z) \ dz\) along \(\gamma\) is defined as

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

\((\)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.\()\)

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

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

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

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First, parameterize 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 parameterization, 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 \(\frac{1}{z}\) is not holomorphic on the region interior to \(\gamma\), since it is not defined at \(z=0\).

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,

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

In particular, if \(\gamma\) is a closed curve, i.e. \(z(b) = z(a)\), 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.