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-
