Mathematical Methods for Physics and Engineering - Matematica.NET

25.8 APPROXIMATIONS TO INTEGRALSFrom the observations contained in the two previous paragraphs, we deducethat a path that follows the lines of steepest descent (or ascent) can never forma closed loop. On such a path, φ, and hence |h(z)|, must continue to decrease(increase) until the path meets a singularity of f(z). It also follows that if a levelline of h(z) forms a closed loop in the complex plane, then the loop must enclosea singularity of f(z). This may (if φ →∞) or may not (if φ →−∞) produce asingularity in h(z).We now turn to the study of the behaviour of h(z) at a saddle point and howthis enables us to find an approximation to the integral of h(z) alongacontourthat can be deformed to pass through the saddle point. At a saddle point z 0 ,atwhich f ′ (z 0 ) = 0, both ∇φ and ∇ψ are zero, and consequently the magnitude andphase of h(z) are both stationary. The Taylor expansion of f(z) at such a pointtakes the formf(z) =f(z 0 )+0+ 1 2! f′′ (z 0 )(z − z 0 ) 2 +O(z − z 0 ) 3 . (25.60)We assume that f ′′ (z 0 ) ≠ 0 and write it explicitly as f ′′ (z 0 ) ≡ Ae iα , thus definingthe real quantities A and α. Ifithappensthatf ′′ (z 0 ) = 0, then two or moresaddle points coalesce and the Taylor expansion must be continued until the firstnon-vanishing term is reached; we will not consider this case further, though thegeneral method of proceeding will be apparent from what follows. If we alsoabbreviate the (in general) complex quantity f(z 0 )tof 0 , then (25.60) takes theformf(z) =f 0 + 1 2 Aeiα (z − z 0 ) 2 +O(z − z 0 ) 3 . (25.61)To study the implications of this approximation for h(z), we write z − z 0 asρe iθ with ρ and θ both real. Then|h(z)| = | exp(f 0 )| exp[ 1 2 Aρ2 cos(2θ + α)+O(ρ 3 )]. (25.62)This shows that there are four values of θ for which |h(z)| is independent of ρ (tosecond order). These therefore correspond to two crossing level lines given by(θ = 1 2 ±12 π − α) (and θ = 1 2 ±32 π − α) . (25.63)The two level lines cross at right angles to each other. It should be noted thatthe continuations of the two level lines away from the saddle are not straightin general. At the saddle they have to satisfy (25.63), but away from it the linesmust take whatever directions are needed to make ∇φ = 0. In figure 25.12 one ofthe level lines (|h| = 1) has a continuation (y = 0) that is straight; the other doesnot and bends away from its initial direction x =1.So far as the phase of h(z) is concerned, we havearg[ h(z)]=arg(f 0 )+ 1 2 Aρ2 sin(2θ + α)+O(ρ 3 ),which shows that there are four other directions (two lines crossing at right907