Beta and Gamma Functions - One Line Questions

1. The Gamma function has poles at non-positive integers. What are the residues at these poles? — (-1)^n / n!
2. What is the value of Γ(3/2)? — (1/2)√π
3. The Gamma function is particularly useful for defining factorials for non-integer values. For example, Γ(1/2) = √π. What is Γ(3/2)? — (1/2)√π
4. What is the value of Γ(5/2)? — (3/4)√π
5. What is the value of Γ(n) for a positive integer n? — (n-1)!
6. What is the value of Γ(1/2)? — √π
7. The Gamma function is related to the Beta function by B(m, n) = Γ(m)Γ(n) / Γ(m+n). What is the value of B(1/2, 3/2)? — √π / 2
8. What is the value of B(1, 1)? — 1
9. The Gamma function can be defined by the integral Γ(z) = ∫[0 to ∞] t^(z-1) * e^(-t) dt for complex numbers z with Re(z) > 0. What is the value of Γ(1)? — 1
10. Using the relation B(m, n) = Γ(m)Γ(n) / Γ(m+n), what is B(2, 1)? — 1
11. What is the value of Γ(1)? — 1
12. The Gamma function Γ(z) = ∫[0 to ∞] t^(z-1) * e^(-t) dt. What is the value of Γ(2)? — 1
13. What is the value of B(2, 3)? — 1/12
14. The Beta function B(m, n) can be expressed as B(m, n) = Γ(m)Γ(n) / Γ(m+n). What is B(2, 2)? — 1/12
15. The relationship B(m, n) = Γ(m)Γ(n) / Γ(m+n) is crucial for evaluating Beta functions. What is B(3, 2)? — 1/120
16. What is the value of B(3, 3)? — 1/120
17. What is the value of B(1, 2)? — 1/2
18. What is the value of B(3, 1)? — 1/6
19. The Gamma function is related to the Beta function by B(m, n) = Γ(m)Γ(n) / Γ(m+n). What is the value of B(1, 3)? — 1/6
20. The Beta function B(m, n) is defined as ∫[0 to 1] x^(m-1) * (1-x)^(n-1) dx. What is the value of B(2, 2)? — 1/6
21. What is the value of B(1/2, 1)? — 2
22. What is the value of B(2, 1/2)? — 4√π / 3
23. What is the value of Γ(4)? — 6
24. The Gamma function satisfies Γ(z+1) = zΓ(z). If Γ(3) = 2, what is Γ(4)? — 6
25. The Beta function can be expressed in terms of the Gamma function using which integral representation? — B(m, n) = ∫[0 to ∞] y^(m-1) / (1+y)^(m+n) dy
26. Which integral representation of the Beta function is particularly useful for relating it to the Gamma function? — B(m, n) = ∫[0 to ∞] y^(m-1) / (1+y)^(m+n) dy
27. Which property of the Beta function allows it to be expressed as an integral over an infinite range? — B(m, n) = ∫[0 to ∞] y^(m-1) / (1+y)^(m+n) dy
28. Which integral representation of the Beta function involves a transformation using y = x/(1-x)? — B(m, n) = ∫[0 to ∞] y^(m-1) / (1+y)^(m+n) dy
29. The Beta function, denoted by B(m, n), is defined as the integral of x^(m-1) * (1-x)^(n-1) from 0 to 1. What is its mathematical definition? — B(m, n) = ∫[0 to 1] x^(m-1) * (1-x)^(n-1) dx
30. The Beta function has a relationship with trigonometric functions. Which identity holds true? — B(m, n) = 2 * ∫[0 to π/2] (sin(θ))^(2m-1) * (cos(θ))^(2n-1) dθ
31. Which integral representation of the Beta function involves a substitution of x = sin^2(θ)? — B(m, n) = 2 * ∫[0 to π/2] (sin(θ))^(2m-1) * (cos(θ))^(2n-1) dθ
32. Which of these is a symmetric property of the Beta function? — B(m, n) = B(n, m)
33. Which of the following is NOT an identity for the Beta function? — B(m, n) = Γ(m+n) / (Γ(m)Γ(n))
34. Which of the following is a fundamental property relating the Beta function to the Gamma function? — B(m, n) = Γ(m)Γ(n) / Γ(m+n)
35. Which integral representation of the Beta function is derived from the definition of the Gamma function? — B(m, n) = Γ(m)Γ(n) / Γ(m+n)
36. What is the primary application of the Beta and Gamma functions in calculus? — Evaluation of definite integrals
37. Which property of the Beta function is demonstrated by B(m, n) = ∫[0 to 1] x^(m-1) * (1-x)^(n-1) dx? — Integral representation
38. Which of the following is NOT a valid argument for the Beta function integral definition B(m, n) = ∫[0 to 1] x^(m-1) * (1-x)^(n-1) dx for convergence? — m >= 1 and n >= 1
39. The Gamma function can be extended to complex numbers. For which values of z is Γ(z) defined by the integral ∫[0 to ∞] t^(z-1) * e^(-t) dt? — Re(z) > 0
40. Which property of the Beta function is expressed as B(m, n) = B(n, m)? — Symmetry
41. What is the relationship between the Beta function and the incomplete Beta function? — The incomplete Beta function is a generalization involving a variable upper limit of integration.
42. The Gamma function Γ(z) can be defined for complex numbers using analytic continuation. What is the value of Γ(0)? — Undefined (pole)
43. The Gamma function satisfies the Euler's reflection formula. For which values of z is it applicable? — z is not an integer
44. The Gamma function, denoted by Γ(z), is a generalization of the factorial function. For positive integers n, what is the relationship between Γ(n) and n!? — Γ(n) = (n-1)!
45. Which of the following is an alternative definition of the Gamma function for positive real numbers? — Γ(x) = ∫[0 to ∞] t^(x-1) * e^(-t) dt
46. The Gamma function satisfies the reflection formula. What is it? — Γ(z)Γ(1-z) = π / sin(πz)
47. What is the recurrence relation for the Gamma function? — Γ(z+1) = zΓ(z)
48. What is the value of B(1/2, 1/2)? — π
49. The Beta function B(m, n) is related to the Gamma function by B(m, n) = Γ(m)Γ(n) / Γ(m+n). What is B(1/2, 1/2)? — π
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