Mathematical Methods for Physics and Engineering : A Comprehensive Guide

25.7 WKB METHODS

The precise combination of these two solutions that is required for any particular problem
has to be determined from the problem.

When Stokes’ equation is applied more generally to functions of a complex

variable, i.e. the real variablexis replaced by the complex variablez,ithas

solutions whose type of behaviour depends upon wherezlies in the complex

plane. For the particular caseλ=−1, when Stokes’ equation takes the form

d^2 y

dz^2

=zy

and the two WKB solutions (with the inverse fourth root written explicitly) are

y 1 , 2 (z)=

A 1 , 2

z^1 /^4

exp

[

∓

2

3

z^3 /^2

]

, (25.51)

one of the solutions, Ai(z) (see section 25.6), has the property that it is real

wheneverzis real, whether positive or negative. For negative realzit has

sinusoidal behaviour, but it becomes an evanescent wave for real positivez.

Since the functionz^3 /^2 has a branch point atz= 0 and therefore has an abrupt

(complex) change in its argument there, it is clear that neither of the two functions

in (25.51), nor any fixed combination of them, can be equal to Ai(z) for all values

ofz. More explicitly, forzreal and positive, Ai(z) is proportional toy 1 (z), which

is real and has the form of a decaying exponential function, whilst forzreal and

negative, whenz^3 /^2 is purely imaginary andy 1 (z)andy 2 (z) are both oscillatory,

it is clear that Ai(z) must contain bothy 1 andy 2 with equal amplitudes.

The actual combinations ofy 1 (z)andy 2 (z) needed to coincide with these two

asymptotic forms of Ai(z) are as follows.

Forzreal and>0, c 1 y 1 (z)=

1

2

√

πz^1 /^4

exp

[

−

2

3

z^3 /^2

]

. (25.52)

Forzreal and<0, c 2 [y 1 (z)eiπ/^4 −y 2 (z)e−iπ/^4 ]

=

1

√

π(−z)^1 /^4

sin

[

2

3

(−z)^3 /^2 +

π

4

]

. (25.53)

Therefore it must be the case that the constants used to form Ai(z)from

the solutions (25.51) change aszmoves from one part of the complex plane to

another. In fact, the changes occur for particular values of the argument ofz;

these boundaries are therefore radial lines in the complex plane and are known

asStokes lines. For Stokes’ equation they occur when argzis equal to 0, 2π/3or

4 π/3.

The general occurrence of a change in the arbitrary constants used to make

up a solution, as its argument crosses certain boundaries in the complex plane, is

known as the Stokes phenomenon and is discussed further in subsection 25.7.4.