Mathematical Methods for Physics and Engineering - Matematica.NET

APPLICATIONS OF COMPLEX VARIABLESstationary, the magnitude of any factor, g(z), multiplying the exponential function,exp[ f(z)]∼ exp[ Ae iα (z − z 0 ) 2 ], is at least comparable to its magnitude elsewhere,then this result can be used to obtain an approximation to the value of theintegral of h(z) =g(z) exp[ f(z) ]. This is the basis of the method of stationaryphase.Returning to the behaviour of a function exp[ f(z) ] at one of its saddle points,we can now see how the considerations of the previous paragraphs can be appliedthere. We already know, from equation (25.62) and the discussion immediatelyfollowing it, that in the equationh(z) ≈ g(z 0 ) exp(f 0 )exp{ 1 2 Aρ2 [cos(2θ + α)+i sin(2θ + α)]}(25.71)the second exponent is purely imaginary on a level line, and equal to zero atthe saddle point itself. What is more, since ∇ψ = 0 at the saddle, the phase isstationary there; on one level line it is a maximum and on the other it is aminimum. As there are two level lines through a saddle point, a path on whichthe amplitude of the integrand is constant could go straight on at the saddlepoint or it could turn through a right angle. For the moment we assume that itruns continuously through the saddle.On the level line for which the phase at the saddle point is a minimum, we canwrite the phase of h(z) as approximatelyarg g(z 0 )+Imf 0 + v 2 ,where v is real, iv 2 = 1 2 Aeiα (z − z 0 ) 2 and, as previously, Ae iα = f ′′ (z 0 ). Thene iπ/4 dv = ±√A2 eiα/2 dz, (25.72)leading to an approximation to the integral of∫∫ ∞ Ah(z) dz ≈±g(z 0 ) exp(f 0 ) exp(iv )√ 2 2 exp[ i( 1 4 π − 1 2 α)]dv−∞= ± g(z 0 ) exp(f 0 ) √ π exp(iπ/4)√A2 exp[ i( 1 4 π − 1 2 α)]√2π= ±A g(z 0) exp(f 0 ) exp[ 1 2i(π − α)]. (25.73)Result (25.68) was used to obtain the second line above. The ± ambiguity is againresolved by the direction θ of the contour; it is positive if −3π/4