Numerical algorithms based on the theory of complex variable

Abstract

Since its introduction in the early part of the nineteenth century, the theory of complex variables has played a steadily increasing role in mathematics, and in scientific research. In some fields complex algebra is used to simplify the description of a physical system. The use of a complex impedance Z in network theory is an example of this. In other fields complex algebra seems to be a basic ingredient of the physical laws. In Wave Mechanics for example a probability density P(x,t) is related to the square modulus of a wave function &psgr;(x,t) which is itself complex, being obtained from a wave equation whose coefficients may be complex. In mathematical research itself, it is rare to find a topic which is naturally restricted to real variables, and in many topics the extension to complex variables results in a simpler theory. For example a polynomial of degree n has exactly n zeros in the field of complex numbers.