Question Number 33896 by math khazana by abdo last updated on 26/Apr/18
$${let}\:\Gamma\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:\:{t}^{{x}−\mathrm{1}} \:{e}^{−{t}} {dt} \\ $$ $$\left.\mathrm{1}\right)\:{find}\:\Gamma\left({x}+\mathrm{1}\right)\:{interms}\:{of}\:\Gamma\left({x}\right)\:\:\:{with}\:{x}>\mathrm{0} \\ $$ $$\left.\mathrm{2}\right){calculate}\:\Gamma\left({n}\right)\:{for}\:{n}\:\in\:{N}^{\bigstar} \\ $$ $$\left.\mathrm{3}\right){calculate}\:\Gamma\left(\frac{\mathrm{3}}{\mathrm{2}}\right)\:. \\ $$
Answered by tanmay.chaudhury50@gmail.com last updated on 27/Apr/18
$$\lceil\left(\mathrm{3}/\mathrm{2}\right)=\lceil\left(\mathrm{1}/\mathrm{2}+\mathrm{1}\right) \\ $$ $$=\mathrm{1}/\mathrm{2}\lceil\left(\mathrm{1}/\mathrm{2}\right) \\ $$ $$=\mathrm{1}/\mathrm{2}\sqrt{\Pi} \\ $$
Answered by tanmay.chaudhury50@gmail.com last updated on 27/Apr/18
$$\mathrm{1}.\lceil\left({x}+\mathrm{1}\right)={x}.\lceil\left({x}\right) \\ $$
Answered by tanmay.chaudhury50@gmail.com last updated on 27/Apr/18
$$\mathrm{2}.\lceil\left({n}\right)=\lceil\left({n}−\mathrm{1}+\mathrm{1}\right) \\ $$ $$\:\:\:\:\:\:\:\:\:\:\:\:\:=\left({n}−\mathrm{1}\right)\lceil\left({n}−\mathrm{2}\right) \\ $$ $$\:\:\:\:\:\:\:\:\:\:\:\:\:=\left({n}−\mathrm{1}\right)\left({n}−\mathrm{2}\right)\lceil\left({n}−\mathrm{3}\right) \\ $$ $$\:\:\:\:\:\:\:\:\:\:\:\:{thus}\:{the}\:{value}\:{is}\:{factorial}\left({n}−\mathrm{1}\right) \\ $$ $$\:\:\:\:\:\:\:\:\:\:\:\:\lceil\left({n}\right)=\left({n}−\mathrm{1}\right)! \\ $$