Mathematical Methods for Physics and Engineering - Matematica.NET

APPLICATIONS OF COMPLEX VARIABLESangles) in which the phase of h(z) is independent of ρ. They make angles of π/4with the level lines through z 0 and are given byθ = − 1 2 α, θ = 1 2 (±π − α), θ = π − 1 2 α.From our previous discussion it follows that these four directions will be thelines of steepest descent (or ascent) on moving away from the saddle point. Inparticular, the two directions for which the term cos(2θ + α) in (25.62) is negativewill be the directions in which |h(z)| decreases most rapidly from its value at thesaddle point. These two directions are antiparallel, and a steepest descents pathfollowing them is a smooth locally straight line passing the saddle point. It isknown as the line of steepest descents (l.s.d.) through the saddle point. Note that‘descents’ is plural as on this line the value of |h(z)| decreases on both sides of thesaddle. This is the line which we will make the path of the contour integral ofh(z) follow. Part of a typical l.s.d. is indicated by the dashed line in figure 25.12.25.8.2 Steepest descents methodTo help understand how an integral along the line of steepest descents can behandled in a mechanical way, it is instructive to consider the case where thefunction f(z) =−βz 2 and h(z) =exp(−βz 2 ). The saddle point is situated atz = z 0 = 0, with f 0 = f(z 0 )=1andf ′′ (z 0 )=−2β, implying that A =2|β| andα = ±π +argβ, withthe± sign chosen to put α in the range 0 ≤ α