25 Applications of complex variables
In chapter 24, we developed the basic theory of the functions of a complex
variable,z=x+iy, studied their analyticity (differentiability) properties and
derived a number of results concerned with values of contour integrals in the
complex plane. In this current chapter we will show how some of those results
and properties can be exploited to tackle problems arising directly from physical
situations or from apparently unrelated parts of mathematics.
In the former category will be the use of the differential properties of the realand imaginary parts of a function of a complex variable to solve problems involv-
ing Laplace’s equation in two dimensions, whilst an example of the latter might
be the summation of certain types of infinite series. Other applications, such as
the Bromwich inversion formula for Laplace transforms, appear as mathematical
problems that have their origins in physical applications; the Bromwich inversion
enables us to extract the spatial or temporal response of a system to an initial
input from the representation of that response in ‘frequency space’ – or, more
correctly, imaginary frequency space.
Other topics that will be considered are the location of the (complex) zeros ofa polynomial, the approximate evaluation of certain types of contour integrals
using the methods of steepest descent and stationary phase, and the so-called
‘phase-integral’ solutions to some differential equations. For each of these a brief
introduction is given at the start of the relevant section and to repeat them here
would be pointless. We will therefore move on to our first topic of complex
potentials.
25.1 Complex potentials
Towards the end of section 24.2 of the previous chapter it was shown that the real
and the imaginary parts of an analytic function ofzare separately solutions of
Laplace’s equation in two dimensions. Analytic functions thus offer a possible way