Special Functions - Gamma Questions And Answers - Sanfoundry

This set of Ordinary Differential Equations Multiple Choice Questions & Answers (MCQs) focuses on “Special Functions -1 (Gamma)”.

1. Which of the following is true? a) Γ(n+1) = nΓ(n) for any real number b) Γ(n) = nΓ(n+1) for any real number c) Γ(n+1) = nΓ(n) for n>1 d) Γ(n) = nΓ(n+1) for n>1 View AnswerAnswer: c Explanation: Γ(n+1) = n! = n. (n-1)! = n.Γ(n). Hence Γ(n+1) = nΓ(n) for n>1.

2. Γ(n+1) = n! can be used when ____________ a) n is any integer b) n is a positive integer c) n is a negative integer d) n is any real number View AnswerAnswer: b Explanation: \( \int_{0}^{\infty} x^{n} e^{-x}dx \) = \( \mid e^{-x} x^{n} \mid_{0}^{\infty} + n \int_{0}^{\infty} x^{n-1} e^{-x}dx \) = \( n \Gamma(n) \).

3. Which of the following is not a definition of Gamma function? a) \(\Gamma(n) = n!\) b) \(\Gamma(n) = \int_{0}^{\infty} x^{n-1} e^{-x}dx\) c) \(\Gamma(n+1) = n\Gamma(n)\) d) \(\Gamma(n) = \int_{0}^{1} log \left({1 \atop y}\right)^{n-1}\) View AnswerAnswer: a Explanation: Each and every option represents the definition of Gamma function except Γ(n) = n! as Γ(n+1) = n! if n is a positive number. advertisement

4. Gamma function is said to be as Euler’s integral of second kind. a) True b) False View AnswerAnswer: a Explanation: Euler’s integral of first kind is nothing but the Beta function and Euler’s integral of second kind is nothing but Gamma function. These integrals were considered by L.Euler.

5. What is the value of \(\Gamma\left(\frac{1}{2}\right)\)? a) \(\sqrt{\pi}\) b) \(\left(\frac{\sqrt{\pi}}{\sqrt{2}}\right)\) c) \(\left(\frac{\sqrt{\pi}}{2}\right)\) d) \(\frac{\pi}{2}\) View AnswerAnswer: a Explanation: \(\Gamma\left(\frac{1}{2}\right) = \int_{0}^{\infty} x^{\frac{-1}{2}} e^{-x}dx\) = \(\int_{0}^{\infty} e^{-y^2} dy\) = \(\int_{0}^{\infty} e^{-x^2} dx\) = \(\Gamma\left(\frac{1}{2}\right)^2 = \int_{0}^{\frac{\pi}{2}} \int_{0}^{\infty} e^{-r^2} rdrd\theta\) = \(4 * \frac{\pi}{2} * \frac{1}{2}\) = \(\pi\) = \(\Gamma\left(\frac{1}{2}\right) = \sqrt{\pi}\). 📌 Limited Seats! Register Now - Free AI-ML Certification (February 2026)

6. Is the given statement true or false? \(\displaystyle\beta(m, n) = \frac{\Gamma(m).\Gamma(n)}{\Gamma(m+n)}\) a) True b) False View AnswerAnswer: a Explanation: We know, \(\Gamma(n) \int_{0}^{\infty} x^{n-1} e^{-x}dx.\) So, the product of two factorials is:

\(\Gamma(m).\Gamma(n) = \int_{0}^{\infty} x^{m-1} e^{-x}dx \int_{0}^{\infty} y^{n-1} e^{-y}dy\).

= \( \int_{0}^{\infty} \int_{0}^{\infty} x^{m-1} y^{n-1} e^{-x} e^{-y} dxdy \) advertisement

Now, we do a change of variables where x= uv and y= u(1-v) which implies u varies from 0 to ∞ and v varies from 0 to 1. Jacobian of this gives –u.

\( \Gamma(m).\Gamma(n) = \int_{0}^{1} \int_{0}^{\infty} e^{-u} u^{m-1} v^{m-1} u^{n-1} (1-v)^{n-1} ududv\).

= \( \Gamma(m + n).\beta(m, n) \)

Therefore, \(\displaystyle\beta(m, n) = \frac{\Gamma(m).\Gamma(n)}{\Gamma(m+n)}\).