Mathematical Methods for Physics and Engineering : A Comprehensive Guide

APPLICATIONS OF COMPLEX VARIABLES

25.19 The functionh(z)ofthecomplexvariablezis defined by the integral

h(z)=

∫i∞

−i∞

exp(t^2 − 2 zt)dt.

(a) Make a change of integration variable,t=iu, and evaluateh(z)usinga
standard integral. Is your answer valid for all finitez?
(b) Evaluate the integral using the method of steepest descents, considering in
particular the cases (i)zis real and positive, (ii)zis real and negative and
(iii)zis purely imaginary and equal toiβ,whereβis real. In each case sketch
the corresponding contour in the complext-plane.
(c) Evaluate the integral for the same three cases as specified in part (b) using
the method of stationary phases. To determine an appropriate contour that
passes through a saddle pointt=t 0 ,writet=t 0 +(u+iv) and apply the
criterion for determining a level line. Sketch the relevant contour in each
case, indicating what freedom there is to distort it.
Comment on the accuracy of the results obtained using the approximate methods
adopted in (b) and (c).
25.20 Use the method of steepest descents to show that an approximate value for the
integral

F(z)=

∫∞

−∞

exp[iz(^15 t^5 +t)]dt,

wherezis real and positive, is

(

2 π

z

) 1 / 2

exp(−βz)cos(βz−^18 π),

whereβ=4/(5

√

2).

25.21 The stationary phase approximation to an integral of the form

F(ν)=

∫b

a

g(t)eiνf(t)dt, |ν| 1 ,

wheref(t) is a real function oftandg(t) is a slowly varying function (when

compared with the argument of the exponential), can be written as

F(ν)∼

(

2 π

|ν|

) 1 / (^2) ∑N
n=1
g(tn)
√
An
exp

{

i

[

νf(tn)+

π

4

sgn

(

νf′′(tn)

)]}

,

where thetnare theNstationary points off(t) that lie ina<t 1 <t 2 <···<

tN<b,An=|f′′(tn)|, and sgn(x) is the sign ofx.

Use this result to find an approximation, valid for large positive values ofν,

to the integral

F(ν, z)=

∫∞

−∞

1

1+t^2

cos[ (2t^3 − 3 zt^2 − 12 z^2 t)ν]dt,

wherezis a real positive parameter.
25.22 The Bessel functionJν(z)isgivenfor|argz|<^12 πby the integral around a
contourCof the function

g(z)=

1

2 πi

t−(ν+1)exp

[

z

2

(

t−

1

t

)]

.

The contour starts and ends along the negative realt-axis and encircles the origin

in the positive sense. It can be considered to be made up of two contours. One

of them,C 2 ,startsatt=−∞, runs through the third quadrant to the point