Mathematical Methods for Physics and Engineering : A Comprehensive Guide

APPLICATIONS OF COMPLEX VARIABLES

which there is no obvious solution for a generaln(x). However, on the assumption

thatd^2 φ/dx^2 is small, an iterative solution can be found.

As a first approximationφ′′is ignored, and the solution

dφ

dx

≈±k 0 n(x)

is obtained. From this, differentiation gives an approximate value for

d^2 φ

dx^2

≈±k 0

dn

dx

,

which can be substituted into equation (25.47) to give, as a second approximation

fordφ/dx, the expression

dφ

dx

≈±

[

k^20 n^2 (x)±ik 0

dn

dx

] 1 / 2

=±k 0 n

(

1 ±

i

2 k 0 n^2

dn

dx

+···

)

≈±k 0 n+

i

2 n

dn

dx

.

This can now be integrated to give an approximate expression forφ(x) as follows:

φ(x)=±k 0

∫x

x 0

n(u)du+

i

2

ln[n(x)], (25.48)

where the constant of integration has been formally incorporated into the lower

limitx 0 of the integral. Now, noting that exp(i^12 ilnn)=n−^1 /^2 , substitution of

(25.48) into equation (25.46) gives

y±(x)=

A

n^1 /^2

exp

[

±ik 0

∫x

x 0

n(u)du

]

(25.49)

as two independent WKB solutions of the original equation (25.42). This result is

essentially the same as that in (25.45) except that the amplitude has been divided

by

√

n(x), i.e. by [f(x)]^1 /^4 .Sincef(x) may be complex, this may introduce an

additionalx-dependent phase into the solution as well as the more obvious change

in amplitude.

Find two independent WKB solutions of Stokes’ equation in the form

d^2 y

dx^2

+λxy=0,withλreal and> 0.

The form of the equation is the same as that in (25.42) withf(x)=x,andtherefore
n(x)=x^1 /^2. The WKB solutions can be read off immediately using (25.49), so long as
we remember that althoughf(x) is real, it has four fourth roots and that therefore the
constant appearing in a solution can be complex. Two independent WKB solutions are

y±(x)=

A±

|x|^1 /^4

exp

[

±i

√

λ

∫x

√

udu

]

=

A±

|x|^1 /^4

exp

[

±i

2

√

λ

3

x^3 /^2

]

.

(25.50)