Question Number 141535 by mohammad17 last updated on 20/May/21
$${how}\:{can}\:{solve}\:{this}\:{with}\:{by}\:{steps}\:{please} \\ $$$$\frac{\Gamma\left(\frac{\mathrm{3}}{\mathrm{2}}\right)\Gamma\left(\frac{\mathrm{7}}{\mathrm{2}}\right)}{\mathrm{2}\Gamma\left(\mathrm{5}\right)}\:{i}\:{want}\:{solution}\:{step}\:{by}\:{step} \\ $$$${because}\:{i}\:{understand}\:{this}? \\ $$$$ \\ $$
Commented by Dwaipayan Shikari last updated on 20/May/21
$$\Gamma\left(\frac{\mathrm{3}}{\mathrm{2}}\right)=\Gamma\left(\frac{\mathrm{1}}{\mathrm{2}}+\mathrm{1}\right)=\frac{\mathrm{1}}{\mathrm{2}}\Gamma\left(\frac{\mathrm{1}}{\mathrm{2}}\right)=\frac{\sqrt{\pi}}{\mathrm{2}} \\ $$$$\Gamma\left(\frac{\mathrm{7}}{\mathrm{2}}\right)=\Gamma\left(\frac{\mathrm{5}}{\mathrm{2}}+\mathrm{1}\right)=\frac{\mathrm{5}}{\mathrm{2}}\Gamma\left(\frac{\mathrm{3}}{\mathrm{2}}+\mathrm{1}\right)=\frac{\mathrm{5}}{\mathrm{2}}.\frac{\mathrm{3}}{\mathrm{2}}\Gamma\left(\frac{\mathrm{1}}{\mathrm{2}}+\mathrm{1}\right)=\frac{\mathrm{5}.\mathrm{3}.\mathrm{1}}{\mathrm{2}^{\mathrm{3}} }\Gamma\left(\frac{\mathrm{1}}{\mathrm{2}}\right)=\frac{\mathrm{15}\sqrt{\pi}}{\mathrm{8}} \\ $$$$\frac{\Gamma\left(\frac{\mathrm{3}}{\mathrm{2}}\right)\Gamma\left(\frac{\mathrm{7}}{\mathrm{2}}\right)}{\mathrm{2}\Gamma\left(\mathrm{5}\right)}=\frac{\mathrm{15}\pi}{\mathrm{32}.\mathrm{4}!}=\frac{\mathrm{3}.\mathrm{5}\pi}{\mathrm{3}.\mathrm{256}}=\frac{\mathrm{5}\pi}{\mathrm{256}} \\ $$