Mathematical Methods for Physics and Engineering - Matematica.NET

25Applications of complex variablesIn chapter 24, we developed the basic theory of the functions of a complexvariable, z = x + iy, studied their analyticity (differentiability) properties andderived a number of results concerned with values of contour integrals in thecomplex plane. In this current chapter we will show how some of those resultsand properties can be exploited to tackle problems arising directly from physicalsituations 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 involvingLaplace’s equation in two dimensions, whilst an example of the latter mightbe the summation of certain types of infinite series. Other applications, such asthe Bromwich inversion formula for Laplace transforms, appear as mathematicalproblems that have their origins in physical applications; the Bromwich inversionenables us to extract the spatial or temporal response of a system to an initialinput from the representation of that response in ‘frequency space’ – or, morecorrectly, 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 integralsusing the methods of steepest descent and stationary phase, and the so-called‘phase-integral’ solutions to some differential equations. For each of these a briefintroduction is given at the start of the relevant section and to repeat them herewould be pointless. We will therefore move on to our first topic of complexpotentials.25.1 Complex potentialsTowards the end of section 24.2 of the previous chapter it was shown that the realand the imaginary parts of an analytic function of z are separately solutions ofLaplace’s equation in two dimensions. Analytic functions thus offer a possible way871