The Wonders of Complex Analysis

Augustin-Louis Cauchy (1789–1857)

If you love mathematics and have never studied complex function theory, then you are missing something wonderful. It is one of the most beautiful branches of maths, with many amazing results. Don’t be put off by the name: complex does not mean complicated. With elementary calculus and a basic knowledge of imaginary numbers, a whole world of wonder is within your grasp.

In the early nineteenth century, Augustin-Louis Cauchy (1789–1857) constructed the foundations of what became a major new branch of mathematics, the theory of functions of a complex variable.

Complex Analysis

Complex analysis is the branch of mathematics dealing with the theory of functions of a complex variable. A function ${f(z)}$ of a complex variable ${z}$ is differentiable at a point ${z_0}$ in the complex plane if the following limit exists:

$\displaystyle f^\prime(z_0) = \lim_{ z\rightarrow z_0} \frac{f(z)-f(z_0)}{z-z_0}$

This looks like the usual definition for functions of a real variable, but it is a much stronger condition: the value of the limit must be independent of the direction in which the limit is approached; that is, it must be independent of the phase of ${z-z_0}$ .

Differentiability of a complex functions has powerful consequences. If the derivative of ${f(z)}$ exists in a region ${\Omega}$ , then the function is said to be holomorphic in ${\Omega}$ . Holomorphic functions are infinitely differentiable. This contrasts sharply from functions of a real variable, where the existence of an ${n}$ -th derivative does not imply existence of the ${(n+1)}$ -th derivative.

Every holomorphic function is analytic. That is, it can be represented in its domain by a power series (the Taylor series). Again, the contrast with real functions is sharp. For the real function

$\displaystyle f(x) = \exp(-1/x^2),\ \ \ x\ne0\,, \qquad f(0) = 0$

all derivatives vanish at the origin, so the Taylor series about ${x=0}$ (the Maclaurin series) is identically zero and does not represent the function.

If ${f(z) = u(x,y)+i v(x,y)}$ is analytic in ${\Omega}$ , where ${u(x,y)}$ and ${v(x,y)}$ are real functions, then the following equations hold:

$\displaystyle \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \,, \qquad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \,.$

These are the Cauchy-Riemann Equations. We can see from these that the real and imaginary parts of ${f(z)}$ are strongly interlinked.

Line integrals

Integrals along curves in the complex plane ${\mathbb{C}}$ are of great importance in compex analysis. If the function ${f(z)}$ of a complex variable ${z}$ is analytic in ${\Omega}$ and on its boundary ${\Gamma}$ then

$\displaystyle \oint_\Gamma f(z)\,\mathrm{d}z = 0 \,.$

This is Cauchy’s Theorem. It has many consequences. One is that, for any two points ${z_1}$ and ${z_2}$ in ${\Omega}$ , the integral

$\displaystyle \int_{z_1}^{z_2} f(z)\,\mathrm{d}z = 0$

is independent of the path (in ${\Omega}$ ) between the points.

If ${f(z)}$ is known on the boundary ${\Gamma}$ of a region ${\Omega}$ and analytic in the region, then the value of ${f(z)}$ at any point ${a}$ within the region is given using Cauchy’s integral formulas. This is somewhat amazing: the values of ${f(z)}$ and its derivatives anywhere within the region follow from a knowledge of the values of ${f(z)}$ on the boundary. These remarkable formulas are

$\displaystyle f(a) = \frac{1}{2\pi i} \oint_\Gamma \frac{f(z)}{z-a} \,\mathrm{d}z = 0$

and

$\displaystyle f^{(n)}(a) = \frac{n!}{2\pi i} \oint_\Gamma \frac{f(z)}{(z-a)^{n+1}} \,\mathrm{d}z = 0 \,.$

Laurent Series

Laurent-Annulus Suppose that ${f(z)}$ is analytic in the annular region between two concentric circles ${\mathbf{C}_1}$ and ${\mathbf{C}_2}$ and ${\gamma}$ is a simple closed curve between them (see Figure). Then ${f(z)}$ can be expanded in a doubly-infinite series

$\displaystyle f(z) = \sum_{n=-\infty}^{+\infty} a_n z^n \,, \qquad\mbox{where}\qquad a_n = \frac{1}{2\pi i}\oint_{\gamma} \frac{f(z)}{z^{n+1}}\,\mathrm{d}z \quad\mbox{for}\quad n \in \mathbb{Z} \,.$

This is Laurent’s expansion. The quantity ${a_{-1} = (1/2\pi i)\oint_{\mathbf{C}} f(z) \,\mathrm{d}z}$ is called the residue (at ${z=0}$ ) of ${f(z)}$ .

The Residue Theorem

For a simple pole at ${z=a}$ , the residue is ${\lim_{z\rightarrow a} (z-a)f(z)}$ . Suppose that ${f(z)}$ is analytic inside a simple curve ${\mathbf{C}}$ except for simple poles at points ${z_1, z_2, \dots ,z_N}$ within ${\mathbf{C}}$ and the residues at these poles are ${r_1, r_2, \dots ,r_N}$ . Then

$\displaystyle \oint_{\mathbf{C}} f(z) \,\mathrm{d}z = 2\pi i (r_1 + r_2 + \dots + r_N) \,.$

The residue theorem is valid under more general conditions than stated here. It is of great value in evaluating definite integrals such as

$\displaystyle \int_{-\infty}^{+\infty} \frac{\sin x}{x}\mathrm{d}x = \pi$

and for summing infinite series.

Conclusion

The above results are just a taste of the wonderful richness of complex analysis. There are many excellent and accessible textbooks on this subject. Drop into the library and have fun.

Sources

${\bullet}$ Wikipedia article Complex Analysis .

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