Gamma Function Basics

Calculus Level 3

$\large \frac { \Gamma \left(\frac { 11 }{ 2 } \right) }{ \Gamma \left(\frac { 11 }{ 4 }\right)\Gamma \left(\frac { 13 }{ 4 } \right) } = \ ?$

Notation: $\Gamma (\cdot)$ denotes the gamma function .

16\sqrt { \frac { \pi }{ 2 } }

\sqrt { \frac { 20-\pi }{ 3 } }

6-\sqrt { \frac { \pi }{ 3 } }

16\sqrt { \frac { 2 }{ \pi } }

\frac { 18\pi }{ 3 }

1 solution

Michael Mendrin
Jun 17, 2018

Using the duplication formula for Gamma function, we have

$\dfrac{\Gamma(\frac{11}{2})}{\Gamma(\frac{11}{4})\Gamma(\frac{13}{4})}=$

$\dfrac{\Gamma(\frac{11}{2})}{(2^{1-\frac{11}{2}}\sqrt{\pi})\Gamma(\frac{11}{2})}=$

$16\sqrt{\frac{2}{\pi}}$