Complex Numbers and Functions

Abstract

The insight into algebraic curves afforded by complex coordinates—that a complex curve is topologically a surface—has important repercussions for functions defined as integrals of algebraic functions, such as the logarithm, exponential, and elliptic functions. The complex logarithm turns out to be “many-valued,” due to the different paths of integration in the complex plane between the same endpoints. It follows that its inverse function, the exponential function, is periodic. In fact, the complex exponential function is a fusion of the real exponential function with the sine and cosine: e x+iy = e x (cos y + i sin y). The double periodicity of elliptic functions also becomes clear from the complex viewpoint. The integrals that define them are taken over paths on a torus surface, on which there are two independent closed paths. The two-dimensional nature of complex numbers imposes interesting and useful constraints on the nature of differentiable complex functions. Such functions define conformal (angle-preserving) maps between surfaces. Also, their real and imaginary parts satisfy equations, called the Cauchy–Riemann equations, that govern fluid flow. So complex functions can be used to study the motion of fluids. Finally, the Cauchy–Riemann equations imply Cauchy’s theorem. This fundamental theorem guarantees that differentiable complex functions have many good features, such as power series expansions.