1. Bessel’s equation and Orthogonality

2. Bessel’s Equation Where the parameter γis a given number. We assume that γ is a real and non negative.

3. Bessel’s Function of the First kind Consider 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.

4. Consider Substituting into We get

5. 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

6. For s=0 For s=1 For s >1 First equation gives Indicial equation

7. Let’s first determine a solution corresponding to the root

8. Also we get for s =2,3,…………. Since It follows that and

9. But for the even terms taking s=2m