UNIT 5

Bessel function

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-

Example: Prove that-

Sol.

As we know that-

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

Hence proved.

Example: Prove that-

Sol.

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

Formula-1:

Proof:

As we know that-

On differentiating with respect to x, we obtain-

Putting r – 1 = s

Formula-2:

Proof:

We have-

Differentiating w.r.t. x, we get-

Formula-3:

Proof: We know that from formula first and second-

Now adding these two, we get-

Or

Formula-4:

Proof:

We know that-

On subtracting, we get-

Formula-5:

Proof:

We know that-

Multiply this by we get-

I.e.

Or

Formula-6:

Proof:

We know that-

Multiply by we get-

Or

Example: 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-

Example: 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

5.3. Generating function for

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

5.4. Legendre polynomials ( – Recurrence formulae &Rodrigue’s formula-

The Legendre’s equations is-

Now the solution of the given equation is the series of descending powers of x is-

Here is an arbitrary constant.

If n is a positive integer and

The above solution is

So that-

Here is called the Legendre’s function of first kind.

Note- Legendre’s equations of second kind is and can be defined as-

The general solution of Legendre’s equation is-

Here A and B are arbitrary constants.

Rodrigue’s formula-

Rodrigue’s formula can be defined as-

Legendre Polynomials-

We know that by Rodrigue formula-

If n = 0, then it becomes-

If n = 1,

If n = 2,

Now putting n =3, 4, 5……..n we get-

…………………………………..

Where N = n/2 if n is even and N = 1/2 (n-1) if n is odd.

Example: 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-

Example: 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-

Recurrence formulae for -

Formula-1:

Fromula-2:

Formula-3:

Formula-4:

Formula-5:

Formula-6:

5.5. Generating function for

Prove that is the coefficient of in the expansion of in ascending powers of z.

Proof:

Now coefficient of in

Coefficient of in

Coefficient of in

And so on.

Coefficient of in the expansion of equation (1)-

The coefficients of etc. in (1) are

Therefore-

Example: Show that-

Sol.

We know that

Equating the coefficients of both sides, we have-

Proof: is a solution of

…………………. (1)

And

is a solution of-

……………. (2)

Now multiply (1) by z and (2) by y and subtracting, we have-

Now integrate from -1 to +1, we get-

Example: Prove that-

By using Rodrigue formula for Legendre function.

On integrating by parts, we get-

Now integrating m – 2 times, we get-