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 of a complex variable is differentiable at a point in the complex plane if the following limit exists:

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 .

Differentiability of a complex functions has powerful consequences. If the derivative of exists in a region , then the function is said to be *holomorphic* in . Holomorphic functions are** infinitely differentiable.** This contrasts sharply from functions of a real variable, where the existence of an -th derivative does not imply existence of the -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

all derivatives vanish at the origin, so the Taylor series about (the Maclaurin series) is identically zero and does not represent the function.

If is analytic in , where and are real functions, then the following equations hold:

These are the **Cauchy-Riemann Equations**. We can see from these that the real and imaginary parts of are strongly interlinked.

** Line integrals **

Integrals along curves in the complex plane are of great importance in compex analysis. If the function of a complex variable is analytic in and on its boundary then

This is *Cauchy’s Theorem*. It has many consequences. One is that, for any two points and in , the integral

is independent of the path (in ) between the points.

If is known on the boundary of a region and analytic in the region, then the value of at any point within the region is given using Cauchy’s integral formulas. This is **somewhat amazing:** the values of and its derivatives anywhere within the region follow from a knowledge of the values of on the boundary. These remarkable formulas are

and

**Laurent Series**

Suppose that is analytic in the annular region between two concentric circles and and is a simple closed curve between them (see Figure). Then can be expanded in a doubly-infinite series

This is Laurent’s expansion. The quantity is called the *residue* (at ) of .

**The Residue Theorem**

For a simple pole at , the residue is . Suppose that is analytic inside a simple curve except for simple poles at points within and the residues at these poles are . Then

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

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 **

Wikipedia article *Complex Analysis* .