Mathematical Methods for Physics and Engineering - Matematica.NET

APPLICATIONS OF COMPLEX VARIABLES25.17 Use the binomial theorem to expand, in inverse powers of z, both the squareroot in the exponent and the fourth root in the multiplier, working to O(z −2 ).The leading terms are y 1 (z) =Ce −z2 /4 z ν and y 2 (z) =De z2 /4 z −(ν+1) . Stokes lines:arg z =0,π/2,π,3π/2; anti-Stokes lines: arg z =(2n +1)π/4 forn =0, 1, 2, 3. y 1is dominant on arg z = π/2 or3π/2.25.19 (a) i √ πe −z2 , valid for all z, including i √ π exp(β 2 ) in case (iii).(b) The same values as in (a). The (only) saddle point, at t 0 = z, is traversed inthe direction θ =+ 1 π in all cases, though the path in the complex t-plane varies2with each case.(c) The same values as in (a). The level lines are v = ±u. In cases (i) and (ii) thecontour turns through a right angle at the saddle point.All three methods give exact answers in this case of a quadratic exponent.25.21 Saddle points at t 1 = −z and t 2 =2z with f 1 ′′ = −18z and f′′ 2 =18z.Approximation is[( π) 1/2 cos(7νz 3 − 1 π) 4+ cos(20νz3 − 1 π)]4.9zν 1+z 2 1+4z 225.23 Saddle point at t 0 =cos −1 (ν/z) is traversed in the direction θ = − 1 4 π. F ν(z) ≈(2π/z) 1/2 exp [ i(z − 1 2 νπ − 1 4 π)]. 926