Essential Mathematical Methods for the Physical Sciences: 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 real and imaginary parts of a function of a complex variable to solve problems involving 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 of a 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.