Mathematical Methods for Physics and Engineering : A Comprehensive Guide

APPLICATIONS OF COMPLEX VARIABLES

forward, in that, whatever the sign ofyat any particular pointz, the curvature

always has the opposite sign. Consequently the curve always bends towards the

z-axis, crosses it, and then bends towards the axis again. Thus the curve exhibits

oscillatory behaviour. Furthermore, as−zincreases, the curvature for any given

|y|gets larger; as a consequence, the oscillations become increasingly more rapid

and their amplitude decreases.

25.6.2 Series solution of Stokes’ equation

Obtaining a series solution of Stokes’ equation presents no particular difficulty

when the methods of chapter 16 are used. The equation, written in the form

d^2 y

dz^2

−zy=0,

has no singular points except atz=∞. Every other point in thez-plane is

an ordinary point and so two linearly independent series expansions about it

(formally with indicial valuesσ= 0 andσ= 1) can be found. Those aboutz=0

take the forms

∑∞

0 anz

nand∑∞

0 bnz

n+1. The corresponding recurrence relations

are

(n+3)(n+2)an+3=an and (n+4)(n+3)bn+3=bn,

andthetwoseries(witha 0 =b 0 = 1) take the forms

y 1 (z)=1+

z^3

(3)(2)

+

z^6

(6)(5)(3)(2)

+···,

y 2 (z)=z+

z^4

(4)(3)

+

z^7

(7)(6)(4)(3)

+···.

The ratios of successive terms for the two series are thus

an+3zn+3

anzn

=

z^3

(n+3)(n+2)

and

bn+3zn+4

bnzn+1

=

z^3

(n+4)(n+3)

.

It follows from the ratio test that both series are absolutely convergent for allz.

A similar argument shows that the series for their derivatives are also absolutely

convergent for allz. Any solution of the Stokes’ equation is representable as a

superposition of the two series and so is analytic for all finitez;itisthereforean

integral function with its only singularity at infinity.

25.6.3 Contour integral solutions

We now move on to another form of solution of the Stokes’ equation (25.32), one

that takes the form of a contour integral in whichzappears as a parameter in