Mathematical Methods for Physics and Engineering - Matematica.NET

25.8 APPROXIMATIONS TO INTEGRALSFinally, we should mention that the lines in the z-plane on which the exponentsin the WKB solutions are purely imaginary, and the two solutions haveequal amplitudes, are usually called the anti-Stokes lines. For the general Bessel’sequation they are the real positive and real negative axes.25.8 Approximations to integralsIn this section we will investigate a method of finding approximations to thevalues or forms of certain types of infinite integrals. The class of integrals tobe considered is that containing integrands that are, or can be, represented byexponential functions of the general form g(z) exp[ f(z) ]. The exponents f(z) maybe complex, and so integrals of sinusoids can be handled as well as those withmore obvious exponential properties. We will be using the analyticity propertiesof the functions of a complex variable to move the integration path to a partof the complex plane where a general integrand can be approximated well by astandard form; the standard form is then integrated explicitly.The particular standard form to be employed is that of a Gausssian function ofa real variable, for which the integral between infinite limits is well known. Thisform will be generated by expressing f(z) as a Taylor series expansion about apoint z 0 , at which the linear term in the expansion vanishes, i.e. where f ′ (z) =0.Then, apart from a constant multiplier, the exponential function will behave likeexp[ 1 2 f′′ (z 0 )(z − z 0 ) 2 ] and, by choosing an appropriate direction for the contourto take as it passes through the point, this can be made into a normal Gaussianfunction of a real variable and its integral may then be found.25.8.1 Level lines and saddle pointsBefore we can discuss the method outlined above in more detail, a number ofobservations about functions of a complex variable and, in particular, about theproperties of the exponential function need to be made. For a general analyticfunction,f(z) =φ(x, y)+iψ(x, y), (25.58)of the complex variable z = x + iy, we recall that, not only do both φ and ψsatisfy Laplace’s equation, but ∇φ and ∇ψ are orthogonal. This means that thelines on which one of φ and ψ is constant are exactly the lines on which the otheris changing most rapidly.Let us apply these observations to the functionh(z) ≡ exp[ f(z) ] = exp(φ) exp(iψ), (25.59)recalling that the functions φ and ψ are themselves real. The magnitude of h(z),given by exp(φ), is constant on the lines of constant φ, which are known as the905