Mathematical Methods for Physics and Engineering - Matematica.NET

APPLICATIONS OF COMPLEX VARIABLESFinally, putting the various values into the formula yields( ) 1/2 2πF(x) ∼ + g(i)exp[f(i)]exp[ 1 i(π − α)]2A( ) 1/2 (2π x 1/2=+2x 3/2 2π exp − 2 )3 x3/2 exp[ 1 i(π − π)]2(1=2 √ πx exp − 2 )1/4 3 x3/2 .This is the leading term in the asymptotic expansion of F(x),which,asshowninequation(25.39), is a particular contour integral solution of Stokes’ equation. The fact that it tendsto zero in a monotonic way as x → +∞ allows it to be identified with the Airy function,Ai(x).We may ask why the saddle point at t = −i was not used. The answer to this is asfollows. Of course, any path that starts and ends in the right sectors will suffice, butif another saddle point exists close to the one used, then the Taylor expansion actuallyemployed is likely to be less effective than if there were no other saddle points or if therewere only distant ones.An investigation of the same form as that used at t =+i shows that the saddle at t = −iis higher by a factor of exp( 4 3 x3/2 ) and that its l.s.d. is orientated parallel to the imaginaryt-axis. Thus a path that went through it would need to go via a region of largish negativeimaginary t, over the saddle at t = −i, and then, when it reached the col at t =+i, bendsharply and follow part of the same l.s.d. as considered earlier. Thus the contribution fromthe t = −i saddle would be incomplete and roughly half of that from the t =+i saddlewould still have to be included. The more serious error would come from the first of these,as, clearly, the part of the path that lies in the plane Re t = 0 is not symmetric and is farfrom Guassian-like on the side nearer the origin. The Gaussian-path approximation usedwill therefore not be a good one, and, what is more, the resulting error will be magnified bya factor exp( 4 3 x3/2 ) compared with the best estimate. So, both on the grounds of simplicityand because the effect of the other (neglected) saddle point is likely to be less severe, wechoose to use the one at t =+i. ◭25.8.3 Stationary phase methodIn the previous subsection we showed how to use the saddle points of anexponential function of a complex variable to evaluate approximately a contourintegral of that function. This was done by following the lines of steepest descentthat passed through the saddle point; these are lines on which the phase of theexponential is constant but its amplitude is varying at the maximum possiblerate for that function. We now introduce an alternative method, one that entirelyreverses the roles of amplitude and phase. To see how such an alternative approachmight work, it is useful to study how the integral of an exponential function ofa complex variablecan be represented as the sum of infinitesimal vectors in thecomplex plane.We start by studying the familiar integralI 0 =∫ ∞−∞exp(−z 2 ) dz, (25.67)912