Question

What is the value of \(\Gamma(5.5)\)?

(a) \(\displaystyle\frac{11*9*7*5*3*1*\sqrt{\pi}}{32} \)

(b) \(\displaystyle\frac{9*7*5*3*1*\sqrt{\pi}}{32} \)

(c) \(\displaystyle\frac{9*7*5*3*1*\sqrt{\pi}}{64} \)

(d) \(\displaystyle\frac{11*9*7*5*3*1*\sqrt{\pi}}{64}\)

The question was asked by my school principal while I was bunking the class.

I'm obligated to ask this question of Special Functions in division Special Functions – Gamma, Beta, Bessel and Legendre of Engineering Mathematics

Answer 1

Correct answer is (b) \(\displaystyle\frac{9*7*5*3*1*\sqrt{\pi}}{32} \)

The explanation is: \(\Gamma\left(\frac{11}{2}\right) = \frac{9}{2} * \Gamma\left(\frac{9}{2}\right) = \frac{9}{2} * \frac{7}{2} * \Gamma\left(\frac{7}{2}\right) = \frac{9}{2} * \frac{7}{2} * \frac{5}{2} * \Gamma\left(\frac{5}{2}\right)\)

= \(\frac{9}{2} * \frac{7}{2} * \frac{5}{2} * \frac{3}{2} * \Gamma\left(\frac{3}{2}\right)\)

= \(\frac{9}{2} * \frac{7}{2} * \frac{5}{2} * \frac{3}{2} * \frac{1}{2} * \Gamma\left(\frac{1}{2}\right)\)

= \(\displaystyle\frac{9*7*5*3*1*\sqrt{\pi}}{32}\).