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6. Week 6: The Fundamental Theorems of Complex Analysis

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Last week, we discussed some techniques for computing path integrals, either using parameterised curves (as in vector calculus) or finding a holomorphic antiderivative. This week, we’ll be introducing a series of integral identities that allow us to relate the integral of a holomorphic function \(f(z)\) over path \(\gamma \) to some “special values” of \(f\) inside that path \(\gamma \).

The first of these is Cauchy’s theorem:

Theorem 1. Cauchy's Theorem

Let \(D\subseteq \mathbb{C}\) be a domain, and let \(f: D\to \mathbb{C}\) be a holomorphic function. Then, \[\oint _{\partial D} f(z) dz = 0.\]

This path integral is oriented, and \(\partial D\) may have more than one “piece”. By convention, we traverse \(\partial D\) in the direction where \(D\) lies “to the left” of \(\partial D\). This means that the boundary of the unit disc is oriented counterclockwise, for instance.

Cauchy has more theorems:

Theorem 2. Cauchy's Integral Formulas

Let \(D\subseteq \mathbb{C}\) a domain, \(f: D\to \mathbb{C}\) holomorphic, and let \(z_0\in D\). Then, for all \(n \geq 0\), \[f ^{(n)} \left( z_0 \right) = \frac{n!}{2 \pi i} \oint _{\gamma } \frac{f(z)}{ \left( z-z_0 \right) ^{n+1}} dz\] whenever \(\gamma \) is the boundary of a disc centred at \(z_0\) contained in \(D\).

These are profoundly important formulae in simply computing integrals over closed paths, particularly when the integrand is not quite holomorphic and has perhaps some problem points on the inside.

Notably, Cauchy’s integral formulae only apply for very specific closed paths surrounding the point \(z_0\). However, the integrand is holomorphic on the domain \(\mathbb{C}\setminus \left\lbrace z_0 \right\rbrace\) — one can therefore use Cauchy’s theorem (i.e. the “path independence” of the integral) to justify replacing \(\gamma \) with almost any closed loop bounding \(z_0\).

Example 3.

Using Cauchy’s integral formula, compute \[\oint _{\gamma _R } \frac{e ^{iz}}{z^2+1} dz, \] where \(\gamma _R\) is the boundary of \(S_R = \left\lbrace z\in \mathbb{C} : \left\lvert z \right\rvert < R, \Im z > 0 \right\rbrace\) for \(R > 1\).

First, we define the function \(f(z) = \frac{e ^{iz}}{z+i}\) so that \[ \frac{e ^{iz}}{z^2+1} = \frac{e ^{iz}}{(z-i)(z+i)} = \frac{f(z)}{z-i}.\] Let \(D\) be a very small disc centred at the point \(i\). By Cauchy’s formula, since \(f\) is holomorphic away from \(z=-i\) and hence is holomorphic on \(D\), we have \[\oint _{D} \frac{e ^{iz}}{z^2+1} dz = \oint _{D} \frac{f(z)}{z-i} dz = f(i) \cdot 2\pi i.\] However, \(\frac{f(z)}{z-i}\) is holomorphic on \(S_R\setminus D\), as this set never contains \(\pm i\). Thus, by Cauchy’s theorem, \[\oint _{\partial \left( S_R\setminus D \right)} \frac{f(z)}{z-i} dz = \oint _{\gamma _R} \frac{f(z)}{z-i} dz - \oint _{\partial D} \frac{f(z)}{z-i} dz = 0.\] Hence, we have \[\oint _{\gamma _R} \frac{e ^{iz}}{z^2+1} dz = 2\pi i \cdot f(i) = \frac{\pi }{e}\] for all \(R\).

Exercise 4.

Using the preceeding example, determine \[\int _{-\infty}^{\infty} \frac{\cos x}{x^2+1}dx.\]

Hint
Express \(\cos z\) as the real part of \(e ^{iz}\). With \(\gamma _R\) as in the preceeding example, take a limit as \(R\to\infty\). The part of \(\gamma _R\) that lies on the real axis will approach the above integral. Use the \(ML\) bound to show that the integral over the semicircular arc tends to zero.

The above example therefore demonstrates a general framework for computing contour integrals over closed paths: we use Cauchy’s theorem to transform the path into just a circle, and then we use Cauchy’s integral formula to actually determine the integral.

The above exercise illustrates a framework for using Cauchy’s theorem and Cauchy’s integral formula for computing ordinary integrals we might encounter in applications. By “joining together” two ends of an infinite integral, perhaps using some naughtry tricks along the way, we can relate a totally real integral to some special values of the integrand in the complex plane!

Exercise 5.

Using a similar argument as above, compute \[\int _{-\infty}^{\infty} \frac{\cos x}{x^4+2x^2+1} dx.\]

Exercise 6.

Using a similar argument as above, compute \[\int _{-\infty}^{\infty} \frac{\sin^2x}{x^2+1}dx.\]

Exercise 7.

Using a similar argument as above, compute \[\int _{-\infty}^{\infty} \frac{1}{\left(x^2-\sqrt 2 x + 1\right)^3}dx.\]

Already, we’re capable of tackling some nasty-looking integrals that one may reasonably expect to encounter in applications. In the coming weeks, we’ll be developing some other theoretical tools similar to Cauchy’s theorem and Cauchy’s integral formula that allow us to determine the values of contour integrals with multiple “poles” in the integrand. These will all come together beautifully and supply us with the tools of “contour integration”, a general framework that allows us to take (usually real) integrals and relate them to integrals up in the complex plane.