Question

\(\Gamma(m) * \Gamma(1-m) = \frac{\pi}{sin(m\pi)}\). Check if the statement is True or False?

(a) True

(b) False

This question was posed to me by my school principal while I was bunking the class.

The above asked question is from Special Functions in section Special Functions – Gamma, Beta, Bessel and Legendre of Engineering Mathematics

Answer 1

Right option is (a) True

To elaborate: From the relation between Beta and Gamma function, we have,

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

Let \(n = 1 – m \)

\(\frac{\Gamma(m).\Gamma(1-m)}{\Gamma(1)} \)

= \(\beta(m, 1-m)\)

= \(\int_0^∞ \frac{x^{m-1}}{(1+x)} dx \)

= \( \frac{\pi}{sin(mπ)} \) ( by method of residues).