Question Bank

UNIT–5

Question Bank

UNIT–5

Question-1: Define Bessel function.

Sol.The Bessel equation is-

The solution of this equations will be-

The Bessel function is denoted by and defined as-

If we put n = 0 then Bessel function becomes-

Now if n = 1, then-

The graph of these two equations will be-

General solution of Bessel equation-

Question-2: Prove that-

Sol.

As we know that-

Now put n = 1/2 in equation (1), then we get-

Hence proved.

Question-3: Prove that-

Sol.

Put n = -1/2 in equation (1) of the above question, we get-

Question-4: Prove that-

Sol.

As we know that-

On differentiating with respect to x, we obtain-

Putting r – 1 = s

Question-5: Show that-

By using recurrence relation.

Sol.

We know that-

The recurrence formula-

On differentiating, we get-

Now replace n by n -1 and n by n+1 in (1), we have-

Put the values of and from the above equations in (2), we get-

Question-6: Prove that-

Sol.

We know that- from recurrence formula

On integrating we get-

On taking n = 2 in (1), we get-

Again-

Put the value of from equation (2) and (3), we get-

By equation (1), when n = 1

Question-7: Prove that is the coefficient of in the expansion of

Proof:

As we know that-

Multiply equation (1) by (2), we get-

Now the coefficient of in the product of (3)

=

Similarly the coefficient of in the product of (3) =

So that-

That is why is known as the generating function of Bessel functions

Question-8: Express in terms of Legendre polynomials.

Sol.

By equating the coefficients of like powers of x, we get-

Put these values in equation (1), we get-

Question-9: Let be the Legendre’s polynomial of degree n, then show that for every function f(x) for which the n’th derivative is continuous-

Sol.

We know that-

On integrating by parts, we get-

Now integrate (n – 2) times by parts, we get-

Question-10: Show that-

Sol.

We know that

Equating the coefficients of both sides, we have-

Question-11: Prove that-

Sol.

By using Rodrigue formula for Legendre function.

On integrating by parts, we get-

Now integrating m – 2 times, we get-