Mathematical Methods for Physics and Engineering : A Comprehensive Guide

SPECIAL FUNCTIONS

evaluated using l’Hˆopital’s rule, or alternatively we may calculate the relevant

integral directly.

Evaluate the integral

∫b

a

Jν^2 (αx)xdx.

Ignoring the integration limits for the moment,
∫
J^2 ν(αx)xdx=

1

α^2

∫

Jν^2 (u)u du,

whereu=αx. Integrating by parts yields

I=

∫

Jν^2 (u)udu=^12 u^2 Jν^2 (u)−

∫

Jν(u)J′ν(u)u^2 du.

Now Bessel’s equation (18.73) can be rearranged as

u^2 Jν(u)=ν^2 Jν(u)−uJν′(u)−u^2 Jν′′(u),

which, on substitution into the expression forI,gives

I=^12 u^2 Jν^2 (u)−

∫

J′ν(u)[ν^2 Jν(u)−uJ′ν(u)−u^2 Jν′′(u)]du

=^12 u^2 Jν^2 (u)−^12 ν^2 J^2 ν(u)+^12 u^2 [Jν′(u)]^2 +c.

Sinceu=αx, the required integral is given by

∫b

a

Jν^2 (αx)xdx=

1

2

[(

x^2 −

ν^2

α^2

)

Jν^2 (αx)+x^2 [J′ν(αx)]^2

]b

a

, (18.89)

which gives the normalisation condition for Bessel functions of the first kind.

Since the Bessel functionsJν(x) possess the orthogonality property (18.88), we

may expand any reasonable functionf(x), i.e. one obeying the Dirichlet conditions

discussed in chapter 12, in the interval 0≤x≤bas a sum of Bessel functions of

a given (non-negative) orderν,

f(x)=

∑∞

n=0

cnJν(αnx), (18.90)

provided that theαnare chosen such thatJν(αnb) = 0. The coefficientscnare then

given by

cn=

2

b^2 Jν^2 +1(αnb)

∫b

0

f(x)Jν(αnx)xdx. (18.91)

Theintervalistakentobe0≤x≤b, as then one need only ensure that the

appropriate boundary condition is satisfied atx=b, since the boundary condition

atx= 0 is met automatically.