The name “complex analysis” is unfortunate. It’s called complex because it studies functions of complex numbers, which are also unfortunately named.

Complex numbers often simplify calculations, and functions of a convex variable have amazing properties that can greatly simplify analysis. (They can also make pretty pictures, such as the phase plot below of a function with three singularities.)

When a function of a real variable is differentiable, that tells us a few useful things about the function. But when a function of a complex variable is differentiable, that tells an enormous amount. Because of the extra freedom of motion in two dimensions, requiring that the limit in the definition of a derivative exist is a very restrictive condition. And yet many of the functions that come up most in application satisfy this restriction: the gamma function, Bessel functions, most well-known probability distributions, etc.

Requiring that a complex function be differentiable rules out a lot of ill-behaved functions and leaves well-behaved functions that we care about in application. A function is called **analytic** where it is differentiable. If it’s differentiable everywhere in the complex plane, it’s called **entire**, which is even more restrictive and tells us even more. Usually we’re interested in functions that are analytic but with some restrictions. These functions are sometimes called **holomorphic** or **meromorphic**. Knowing that a function is analytic, even in a small neighborhood, tells us that it is infinitely differentiable, it has a power series, it cannot have local maxima or minima, it satisfies a mean value theorem, it is uniquely determined by its values on any set of points with a limit point, etc.

“The shortest path between two truths in the real domain passes through the complex domain.” — Jacques Hadamard

Because complex functions have such incredible properties, they are useful even for problems that at first don’t seem to involve complex numbers. Far from wanting to *avoid* complex functions because they are “complex,” applied mathematicians look for ways to *introduce* complex functions because they simplify analysis.

The best known example of this is **contour integration**. The integral of a complex function along a closed path doesn’t depend on the path itself but on certain values (“residues”) associated with places inside the path where the function has a singularity. This means that it is often easier to integrate a real function of a real variable by converting it into a problem involving a contour integral in the complex plane.

Another example of where people simplify a problem by introducing complex functions is generating functions. In calculus, you might have a homework problem to find the power series coefficients of a a function. Generating functions turn this on its head, taking a sequence of numbers and making them the power series coefficients of a function. (Electrical engineers do the same thing in signal processing, but they talk about **z-transforms** rather than generating functions, though they’re essentially the same thing.) Generating functions make discrete problems easier by turning them into continuous problems.

Another common application of complex analysis is **conformal mapping**, using the magical properties of analytic functions to map a region of one shape into a region of another shape in a way that has lots of nice mathematical properties. You might, for example, want to transform a mechanical problem on a complicated domain into an equivalent problem on a circular disk.

If you’d like help using the power of complex analysis to simplify your work, please give me a call or send me an email.