TRB PG Assistant - Legendre, Hermite and Laguerre equations - basic properties, Gamma and Beta functions Online Test (MCQ Quiz)

1. What is the differential equation known as Legendre's equation?

A) d^2y/dx^2 + x dy/dx + (n^2 - x^2)y = 0 B) (1 - x^2) d^2y/dx^2 - 2x dy/dx + n(n+1)y = 0 C) d^2y/dx^2 - x dy/dx + ny = 0 D) d^2y/dx^2 + (n+1)x dy/dx + (n^2 - x^2)y = 0

2. What are the solutions to Legendre's equation called?

A) Hermite polynomials B) Laguerre polynomials C) Legendre polynomials D) Bessel functions

3. For Legendre polynomials P_n(x), what is the value of P_n(1)?

A) 0 B) 1 C) n D) (-1)^n

4. For Legendre polynomials P_n(x), what is the value of P_n(-1)?

A) 0 B) 1 C) n D) (-1)^n

5. What is the orthogonality relation for Legendre polynomials P_n(x) over the interval [-1, 1]?

A) Integral from -1 to 1 of P_m(x)P_n(x) dx = 0 for m != n B) Integral from -1 to 1 of x^m P_n(x) dx = 0 for m != n C) Integral from -1 to 1 of P_m(x)P_n(x) dx = 2/(2n+1) delta_mn D) Integral from -1 to 1 of P_m(x)P_n(x) dx = 0 for all m, n

6. Which special function is a solution to the differential equation d^2y/dx^2 - 2x dy/dx + 2ny = 0?

A) Legendre equation B) Hermite equation C) Laguerre equation D) Bessel equation

7. What are the polynomial solutions to Hermite's differential equation called?

A) Legendre polynomials B) Hermite polynomials C) Laguerre polynomials D) Chebyshev polynomials

8. What is the generating function for Hermite polynomials H_n(x)?

A) e^(2xt - t^2) = sum(H_n(x) t^n / n!) B) e^(-t^2) = sum(H_n(x) t^n / n!) C) (1-t)^(-1/2) = sum(P_n(x) t^n) D) e^x = sum(x^n / n!)

9. What is the value of H_n(0) for even n?

A) 0 B) (-1)^(n/2) * (n!) / (n/2)! C) (-1)^(n/2) * (n!) D) 1

10. What is the value of H_n(0) for odd n?

A) 0 B) (-1)^((n-1)/2) * (n!) / ((n-1)/2)! C) (-1)^((n-1)/2) * (n!) D) 1

11. What is the orthogonality relation for Hermite polynomials H_n(x) over the interval (-infinity, infinity)?

A) Integral from -inf to inf of H_m(x)H_n(x) dx = 0 for m != n B) Integral from -inf to inf of e^(-x^2) H_m(x)H_n(x) dx = 2^n n! sqrt(pi) delta_mn C) Integral from -inf to inf of e^(-x^2) H_m(x)H_n(x) dx = 0 for m != n D) Integral from -inf to inf of H_m(x)H_n(x) dx = sqrt(pi) delta_mn

12. Which differential equation has solutions that are Laguerre polynomials?

A) x d^2y/dx^2 + (1-x) dy/dx + ny = 0 B) d^2y/dx^2 - 2x dy/dx + 2ny = 0 C) (1 - x^2) d^2y/dx^2 - 2x dy/dx + n(n+1)y = 0 D) x^2 d^2y/dx^2 + x dy/dx + (x^2 - n^2)y = 0

13. The generalized Laguerre polynomials are denoted by L_n^alpha(x). What is the value of L_n^0(x)?

A) H_n(x) B) P_n(x) C) L_n(x) D) J_n(x)

14. What is the generating function for Laguerre polynomials L_n(x)?

A) e^(-xt / (1-t)) / (1-t) = sum(L_n(x) t^n) B) e^(xt / (1-t)) / (1-t) = sum(L_n(x) t^n) C) e^(xt) = sum(L_n(x) t^n / n!) D) ln(1-t) = -sum(t^n / n)

15. What is the value of L_n(0)?

A) 0 B) 1 C) n D) (-1)^n

16. What is the orthogonality relation for Laguerre polynomials L_n(x) over the interval [0, infinity)?

A) Integral from 0 to inf of L_m(x)L_n(x) dx = 0 for m != n B) Integral from 0 to inf of e^(-x) L_m(x)L_n(x) dx = delta_mn C) Integral from 0 to inf of e^(-x) L_m(x)L_n(x) dx = (n+1) delta_mn D) Integral from 0 to inf of e^(-x) L_m(x)L_n(x) dx = 2n+1 delta_mn

17. The Gamma function, denoted by Gamma(z), is a generalization of which mathematical function?

A) Sine function B) Cosine function C) Exponential function D) Factorial function

18. What is the integral definition of the Gamma function for Re(z) > 0?

A) Integral from 0 to infinity of x^(z-1) e^(-x) dx B) Integral from 0 to infinity of x^z e^(-x) dx C) Integral from 0 to 1 of x^(z-1) e^(-x) dx D) Integral from -infinity to infinity of x^(z-1) e^(-x) dx

19. What is the fundamental recurrence relation for the Gamma function?

A) Gamma(z+1) = z Gamma(z) B) Gamma(z+1) = (z+1) Gamma(z) C) Gamma(z) = z Gamma(z-1) D) Gamma(z+1) = Gamma(z) / z

20. What is the value of Gamma(n) for a positive integer n?

A) n! B) (n-1)! C) (n+1)! D) n

21. What is the value of Gamma(1/2)?

A) sqrt(pi) B) pi C) sqrt(pi)/2 D) 1

22. What is the Beta function, denoted by B(x, y)?

A) Integral from 0 to 1 of t^(x-1) (1-t)^(y-1) dt B) Integral from 0 to infinity of t^(x-1) (1-t)^(y-1) dt C) Integral from -1 to 1 of t^(x-1) (1-t)^(y-1) dt D) Integral from 0 to 1 of t^x (1-t)^y dt

23. What is the relationship between the Beta function and the Gamma function?

A) B(x, y) = Gamma(x)Gamma(y) / Gamma(x+y) B) B(x, y) = Gamma(x+y) / (Gamma(x)Gamma(y)) C) B(x, y) = Gamma(x)Gamma(y) D) B(x, y) = Gamma(x+y)

24. What is the value of B(1, 1)?

A) 1 B) 2 C) 1/2 D) 0

25. What is the value of B(x, y) if x and y are positive integers?

A) (x-1)!(y-1)! / (x+y-1)! B) x!y! / (x+y)! C) (x-1)!(y-1)! D) (x+y-1)! / ((x-1)!(y-1)!)

26. Which of the following is a property of the Beta function?

A) B(x, y) = B(y, x) B) B(x, y) = -B(y, x) C) B(x, y) = B(x+1, y+1) D) B(x, y) = B(x-1, y-1)

27. Legendre polynomials P_n(x) satisfy the recurrence relation: (2n+1)xP_n(x) = (n+1)P_{n+1}(x) + nP_{n-1}(x). What is the correct form of this relation?

A) (2n+1)xP_n(x) = (n+1)P_{n+1}(x) + nP_{n-1}(x) B) (2n+1)xP_n(x) = nP_{n+1}(x) + (n+1)P_{n-1}(x) C) (2n-1)xP_n(x) = nP_{n+1}(x) + (n-1)P_{n-1}(x) D) (2n+1)xP_n(x) = P_{n+1}(x) + P_{n-1}(x)

28. Which of the following is a Rodrigues' formula for Legendre polynomials P_n(x)?

A) P_n(x) = 1/(2^n n!) d^n/dx^n (x^2 - 1)^n B) P_n(x) = 1/(2^n) d^n/dx^n (x^2 - 1)^n C) P_n(x) = 1/(2^n n!) d^n/dx^n (x^2 + 1)^n D) P_n(x) = d^n/dx^n (x^2 - 1)^n

29. The Hermite polynomials H_n(x) satisfy the recurrence relation: H_{n+1}(x) = 2xH_n(x) - 2nH_{n-1}(x). What is the correct form?

A) H_{n+1}(x) = 2xH_n(x) - 2nH_{n-1}(x) B) H_{n+1}(x) = 2xH_n(x) + 2nH_{n-1}(x) C) H_{n+1}(x) = xH_n(x) - nH_{n-1}(x) D) H_{n+1}(x) = 2xH_n(x) - H_{n-1}(x)

30. What is the Rodrigues' formula for Hermite polynomials H_n(x)?

A) H_n(x) = (-1)^n e^(x^2) d^n/dx^n (e^(-x^2)) B) H_n(x) = e^(x^2) d^n/dx^n (e^(-x^2)) C) H_n(x) = (-1)^n d^n/dx^n (e^(-x^2)) D) H_n(x) = (-1)^n e^(-x^2) d^n/dx^n (e^(x^2))

31. The Laguerre polynomials L_n(x) satisfy the recurrence relation: (n+1)L_{n+1}(x) = (2n+1-x)L_n(x) - nL_{n-1}(x). What is the correct form?

A) (n+1)L_{n+1}(x) = (2n+1-x)L_n(x) - nL_{n-1}(x) B) (n+1)L_{n+1}(x) = (2n+1+x)L_n(x) - nL_{n-1}(x) C) nL_{n+1}(x) = (2n+1-x)L_n(x) - (n+1)L_{n-1}(x) D) (n+1)L_{n+1}(x) = (n+1-x)L_n(x) - nL_{n-1}(x)

32. What is the Rodrigues' formula for Laguerre polynomials L_n(x)?

A) L_n(x) = 1/n! e^x d^n/dx^n (x^n e^(-x)) B) L_n(x) = e^x d^n/dx^n (x^n e^(-x)) C) L_n(x) = 1/n! d^n/dx^n (x^n e^(-x)) D) L_n(x) = (-1)^n/n! e^x d^n/dx^n (x^n e^(-x))

33. What is the relationship between the Gamma function and the factorial for non-negative integers?

A) Gamma(n+1) = n! B) Gamma(n) = n! C) Gamma(n+1) = (n+1)! D) Gamma(n) = (n-1)!

34. Which integral representation is valid for the Beta function B(x, y) for x > 0 and y > 0?

A) Integral from 0 to 1 of t^(x-1) (1-t)^(y-1) dt B) Integral from 0 to infinity of t^(x-1) / (1+t)^(x+y) dt C) Integral from 0 to infinity of x^(t-1) e^(-t) dt D) Integral from 0 to 1 of x^t dt

35. What is the value of Gamma(3)?

A) 2 B) 6 C) 1 D) 24

36. What is the value of B(2, 3)?

A) 1/12 B) 1/24 C) 1/6 D) 1/10

37. The differential equation d^2y/dx^2 + (1-x) dy/dx + ny = 0 is related to which special function?

A) Legendre B) Hermite C) Laguerre D) Bessel

38. What are the first few Legendre polynomials?

A) P0(x)=1, P1(x)=x, P2(x)=(3x^2-1)/2 B) P0(x)=1, P1(x)=1, P2(x)=x^2 C) P0(x)=1, P1(x)=x, P2(x)=x^2 D) P0(x)=0, P1(x)=x, P2(x)=3x^2

39. What are the first few Hermite polynomials?

A) H0(x)=1, H1(x)=2x, H2(x)=4x^2-2 B) H0(x)=1, H1(x)=x, H2(x)=x^2 C) H0(x)=1, H1(x)=2x, H2(x)=4x^2 D) H0(x)=0, H1(x)=2x, H2(x)=4x^2-2

40. What are the first few Laguerre polynomials?

A) L0(x)=1, L1(x)=1-x, L2(x)=(x^2-4x+2)/2 B) L0(x)=1, L1(x)=x, L2(x)=x^2 C) L0(x)=1, L1(x)=1-x, L2(x)=x^2-4x+2 D) L0(x)=0, L1(x)=1-x, L2(x)=(x^2-4x+2)/2

41. Which property of the Gamma function is crucial for relating it to factorials?

A) Gamma(z+1) = z Gamma(z) B) Gamma(z)Gamma(1-z) = pi / sin(pi z) C) Gamma(z) = Integral from 0 to inf of x^(z-1) e^(-x) dx D) Gamma(1/2) = sqrt(pi)

42. The integral form of the Beta function B(x, y) is symmetric in x and y. What does this imply?

A) B(x, y) = B(y, x) B) B(x, y) = -B(y, x) C) B(x, y) = 1/B(y, x) D) B(x, y) = B(x+y, x+y)

43. What is the differential equation for the associated Legendre equation?

A) (1-x^2)d^2y/dx^2 - 2x dy/dx + [n(n+1) - m^2/(1-x^2)]y = 0 B) (1-x^2)d^2y/dx^2 - 2x dy/dx + [n(n+1) + m^2/(1-x^2)]y = 0 C) x d^2y/dx^2 + (1-x) dy/dx + ny = 0 D) d^2y/dx^2 - 2x dy/dx + 2ny = 0

44. What is the relation between the Gamma function and the Beta function that is most commonly used?

A) B(x, y) = Gamma(x)Gamma(y) / Gamma(x+y) B) B(x, y) = Gamma(x+y) / (Gamma(x)Gamma(y)) C) B(x, y) = Gamma(x) + Gamma(y) D) B(x, y) = Gamma(x) - Gamma(y)

45. The differential equation d^2y/dx^2 - 2x dy/dx + 2ny = 0 is known as Hermite's equation. What is the value of n for the polynomial solution H_0(x)?

A) 0 B) 1 C) 2 D) n

46. What is the value of Gamma(n+1) for a positive integer n?

A) n! B) (n+1)! C) n D) 1

47. Which of the following is NOT a property of Legendre polynomials P_n(x)?

A) P_n(1) = 1 B) P_n(-1) = (-1)^n C) Integral from -1 to 1 of P_n(x) dx = 0 for n > 0 D) P_n(0) = 0 for all n

48. What is the limit of Gamma(z) as z approaches infinity along the real axis?

A) Infinity B) 0 C) 1 D) Euler-Mascheroni constant

Legendre, Hermite and Laguerre equations - basic properties, Gamma and Beta functions - Online Test

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