Mathematical Methods for Physics and Engineering : A Comprehensive Guide

25.8 Approximations to integrals

Finally, we should mention that the lines in thez-plane on which the expo-

nents in the WKB solutions are purely imaginary, and the two solutions have

equal amplitudes, are usually called theanti-Stokes lines. For the general Bessel’s

equation they are the real positive and real negative axes.

25.8 Approximations to integrals

In this section we will investigate a method of finding approximations to the

values or forms of certain types of infinite integrals. The class of integrals to

be considered is that containing integrands that are, or can be, represented by

exponential functions of the general formg(z) exp[f(z) ]. The exponentsf(z)may

be complex, and so integrals of sinusoids can be handled as well as those with

more obvious exponential properties. We will be using the analyticity properties

of the functions of a complex variable to move the integration path to a part

of the complex plane where a general integrand can be approximated well by a

standard form; the standard form is then integrated explicitly.

The particular standard form to be employed is that of a Gausssian function of

a real variable, for which the integral between infinite limits is well known. This

form will be generated by expressingf(z) as a Taylor series expansion about a

pointz 0 , at which the linear term in the expansion vanishes, i.e. wheref′(z)=0.

Then, apart from a constant multiplier, the exponential function will behave like

exp[^12 f′′(z 0 )(z−z 0 )^2 ] and, by choosing an appropriate direction for the contour

to take as it passes through the point, this can be made into a normal Gaussian

function of a real variable and its integral may then be found.

25.8.1 Level lines and saddle points

Before we can discuss the method outlined above in more detail, a number of

observations about functions of a complex variable and, in particular, about the

properties of the exponential function need to be made. For a general analytic

function,

f(z)=φ(x, y)+iψ(x, y), (25.58)

of the complex variablez=x+iy, we recall that, not only do bothφandψ

satisfy Laplace’s equation, but∇φand∇ψare orthogonal. This means that the

lines on which one ofφandψis constant are exactly the lines on which the other

is changing most rapidly.

Let us apply these observations to the function

h(z)≡exp[f(z) ] = exp(φ) exp(iψ), (25.59)

recalling that the functionsφandψare themselves real. The magnitude ofh(z),

given by exp(φ), is constant on the lines of constantφ, which are known as the