Question

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) \(\frac{\pi}{2}\)

The question was posed to me by my school teacher while I was bunking the class.

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

Answer 1

The correct answer is (a) \(\sqrt{\pi}\)

Best explanation: \(\Gamma\left(\frac{1}{2}\right) = \int_{0}^{\infty} x^{\frac{-1}{2}} e^{-x}dx\)

= \(\int_{0}^{\infty} e^{-y^2} dy\)

= \(\int_{0}^{\infty} e^{-x^2} dx\)

= \(\Gamma\left(\frac{1}{2}\right)^2 = \int_{0}^{\frac{\pi}{2}} \int_{0}^{\infty} e^{-r^2} rdrd\theta\)

= \(4 * \frac{\pi}{2} * \frac{1}{2}\)

= \(\pi\)

= \(\Gamma\left(\frac{1}{2}\right) = \sqrt{\pi}\).