Legendary Polynomial & Bessel Functions MCQ Quiz - Objective Question with Answer for Legendary Polynomial & Bessel Functions - Download Free PDF

Explanation:

From the identity of Bessel functions

 = J0(x) - cos x

Option (3) is true.

Explanation:

We know that

Pn(x) Qn-2(x) - Qn(x) Pn-2(x) = , where Pn(x) and Qn(x) are Legendre’s polynomials of first and second kind respectively.

Putting n = 5 we get

P5(x) Q3(x) - Q5(x) P3(x) =  = 

Option (2) is true.

Explanation:

Putting x = 1 in the differential equation

(1 - x²)P_n''(x) - 2xP_n'(x) + n(n + 1)P_n(x) = 0

we get

0 × Pn''(1) - 2Pn'(1) + n(n + 1)Pn(1) = 0

⇒ 2Pn'(1) = n(n + 1)Pn(1) 

⇒ 2Pn'(1) = n(n + 1) (as Pn(1) = 1)

⇒ Pn'(1) = 

Option (3) is true.

Formula used:Recurrence Relations of Bessel’s Function:

This can be also written as

∫ xⁿJ_n-1(x)dx = xⁿJ_n(x)

Calculation:

We know that, from recurrence relations 

∫ xnJn-1(x)dx = xnJn(x)

Put n = 2

∫ x2J1(x)dx = x2J2(x) + C

Additional Information

Bessel’s Equation:

The differential equation, recurrence relations 

known as Bessel’s equation of order n and its particular solutions are called Bessel’s functions.

Formula used:Recurrence Relations of Bessel’s Function:

This can be also written as

∫ xⁿJ_n-1(x)dx = xⁿJ_n(x)

Calculation:

We know that, from recurrence relations 

∫ xnJn-1(x)dx = xnJn(x)

Put n = 2

∫ x2J1(x)dx = x2J2(x) + C

Additional Information

Bessel’s Equation:

The differential equation, recurrence relations 

known as Bessel’s equation of order n and its particular solutions are called Bessel’s functions.

Explanation:

From the identity of Bessel functions

 = J0(x) - cos x

Option (3) is true.

Explanation:

We know that

Pn(x) Qn-2(x) - Qn(x) Pn-2(x) = , where Pn(x) and Qn(x) are Legendre’s polynomials of first and second kind respectively.

Putting n = 5 we get

P5(x) Q3(x) - Q5(x) P3(x) =  = 

Option (2) is true.

Explanation:

Putting x = 1 in the differential equation

(1 - x²)P_n''(x) - 2xP_n'(x) + n(n + 1)P_n(x) = 0

we get

0 × Pn''(1) - 2Pn'(1) + n(n + 1)Pn(1) = 0

⇒ 2Pn'(1) = n(n + 1)Pn(1) 

⇒ 2Pn'(1) = n(n + 1) (as Pn(1) = 1)

⇒ Pn'(1) = 

Option (3) is true.

Concept Used:-

Beta function, or the Euler integral of first type, is a different type of function which is closely related to the gamma function and to binomial coefficients. The beta function for binomial coefficients m and n is given as,

Explanation:-

We have to find the value of integral .

We know that,

On adding both the equations, we get 

Now a beta function can be given as,

Put this value in equation (1),

So, the correct option is 2.