APPLICATIONS OF COMPLEX VARIABLES
25.17 Use the binomial theorem to expand, in inverse powers ofz, both the square
root in the exponent and the fourth root in the multiplier, working to O(z−^2 ).
The leading terms arey 1 (z)=Ce−z
(^2) / 4
zνandy 2 (z)=Dez
(^2) / 4
z−(ν+1). Stokes lines:
argz=0,π/ 2 ,π, 3 π/2; anti-Stokes lines: argz=(2n+1)π/4forn=0, 1 , 2 ,3.y 1
is dominant on argz=π/2or3π/2.
25.19 (a)i
√
πe−z2
, valid for allz, includingi√
πexp(β^2 ) in case (iii).
(b) The same values as in (a). The (only) saddle point, att 0 =z, is traversed in
the directionθ=+^12 πin all cases, though the path in the complext-plane varies
with each case.
(c) The same values as in (a). The level lines arev=±u. In cases (i) and (ii) the
contour turns through a right angle at the saddle point.
All threemethods giveexactanswers in this case of a quadratic exponent.
25.21 Saddle points att 1 =−zandt 2 =2zwithf′′ 1 =− 18 zandf 2 ′′=18z.
Approximation is
(π
9 zν) 1 / 2
[
cos(7νz^3 −^14 π)
1+z^2+
cos(20νz^3 −^14 π)
1+4z^2]
.
25.23 Saddle point att 0 =cos−^1 (ν/z) is traversed in the directionθ=−^14 π.Fν(z)≈
(2π/z)^1 /^2 exp [i(z−^12 νπ−^14 π)].