Question

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

I had been asked this question in unit test.

I need to ask this question from Special Functions in section Special Functions – Gamma, Beta, Bessel and Legendre of Engineering Mathematics

Answer 1

The correct answer is (b) n is a positive integer

For explanation I would say: \( \int_{0}^{\infty} x^{n} e^{-x}dx \)

= \( \mid e^{-x} x^{n} \mid_{0}^{\infty} + n \int_{0}^{\infty} x^{n-1} e^{-x}dx \)

= \( n \Gamma(n) \).