Mathematical Methods for Physics and Engineering : A Comprehensive Guide

APPLICATIONS OF COMPLEX VARIABLES

exp[±iφ(z) ] will be purely real. What is more, one such factor, known as the

dominantterm, will be exponentially large, whilst the other (thesubdominantterm)

will be exponentially small. AStokes lineis precisely where this happens.

We can now see how the change takes place without an observable discontinuity

occurring. Suppose thaty 1 (z) is very large andy 2 (z) is very small on a Stokes line.

Then a finite change inA 2 will have a negligible effect onY(z); in fact, Stokes

showed, for some particular cases, that the change is less than the uncertainty

iny 1 (z) arising from the approximations made in deriving it. Since the solution

with any particular asymptotic form is determined in a region bounded by two

Stokes lines to within an overall multiplicative constant and the original equation

is linear, the change inA 2 when one of the Stokes lines is crossed must be

proportional toA 1 ,i.e.A 2 changes toA 2 +SA 1 ,whereSis a constant (theStokes

constant) characteristic of the particular line but independent ofA 1 andA 2 .It

should be emphasised that, at a Stokes line, if the dominant term is not present

in a solution, then the multiplicative constant in the subdominant termcannot

change as the line is crossed.

As an example, consider the Bessel functionJ 0 (z)ofzeroorder.Itissingle-

valued, differentiable everywhere, and can be written as a series in powers ofz^2 .It

is therefore an integral even function ofz. However, its asymptotic approximations

for two regions of thez-plane, Rez>0andzreal and negative, are given by

J 0 (z)∼

1

√

2 π

1

√

z

(

eize−iπ/^4 +e−izeiπ/^4

)

, |arg(z)|<^12 π,|arg(z−^1 /^2 )|<^14 π,

J 0 (z)∼

1

√

2 π

1

√

z

(

eize^3 iπ/^4 +e−izeiπ/^4

)

, arg(z)=π,arg(z−^1 /^2 )=−^12 π.

We note in passing that neither of these expressions is naturally single-valued,

and a prescription for taking the square root has to be given. Equally, neither is

an even function ofz. For our present purpose the important point to note is

that, for both expressions, on the line argz=π/2 bothz-dependent exponents

become real. For large|z|the second term in each expression is large; this is the

dominant term, and its multiplying constanteiπ/^4 is the same in both expressions.

Contrarywise, the first term in each expression is small, and its multiplying

constant does change, frome−iπ/^4 toe^3 iπ/^4 ,asargzpasses throughπ/2 whilst

increasing from 0 toπ. It is straightforward to calculate the Stokes constant for

this Stokes line as follows:

S=

A 2 (new)−A 2 (old)

A 1

=

e^3 iπ/^4 −e−iπ/^4

eiπ/^4

=eiπ/^2 −e−iπ/^2 =2i.

If we had moved (in the negative sense) from argz= 0 to argz=−π, the relevant

Stokes line would have been argz=−π/2. There the first term in each expression

is dominant, and it would have been the constanteiπ/^4 in the second term that

would have changed. The final argument ofz−^1 /^2 would have been +π/2.