Bessel’s equation and Orthogonality

Bessel’s Equation

Where the parameter γ is a given number. We assume that γ is a real and non negative.

Bessel’s Function of the First kindConsider the Bessel’s

Equation

Comparing with

We get

Which can not be expressed as a non negative powers of x, so they are not analytic but a(x)and b(x) are analytic as they can be expressed as a non negative powers of x.

So, Extended Power series method is applicable.

Consider

Substituting into

We get

To make uniform power changing the index m to s with appropriate changes

Then by equating the coefficient s of x to the power (r+s), to zero, we get

For s=0

For s=1

For s >1

First equation gives Indicial equation

Let’s first determine a solution corresponding to the root

Also we get for s =2,3,………….

Since and It follows that

But for the even terms taking s=2m