Bessel's equation describes functions that arise in various physical problems, such as vibrating membranes and radar. The equation can be solved using an extended power series method to derive Bessel functions of the first kind, which are characterized by their orthogonality properties and represent solutions as a sum of integer powers of x.

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 kindConsider the Bessel’s EquationComparing withWe getWhich 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 changesThen by equating the coefficient s of x to the power (r+s), to zero, we get

6. For s=0For s=1For s >1First 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