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Question Number 33896 by math khazana by abdo last updated on 26/Apr/18
let Γ(x)=∫_0 ^∞ t^(x−1) e^(−t) dt 1) find Γ(x+1) interms of Γ(x) with x>0 2)calculate Γ(n) for n ∈ N^★ 3)calculate Γ((3/2)) .
$${let}\:\Gamma\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:\:{t}^{{x}−\mathrm{1}} \:{e}^{−{t}} {dt} \\ $$$$\left.\mathrm{1}\right)\:{find}\:\Gamma\left({x}+\mathrm{1}\right)\:{interms}\:{of}\:\Gamma\left({x}\right)\:\:\:{with}\:{x}>\mathrm{0} \\ $$$$\left.\mathrm{2}\right){calculate}\:\Gamma\left({n}\right)\:{for}\:{n}\:\in\:{N}^{\bigstar} \\ $$$$\left.\mathrm{3}\right){calculate}\:\Gamma\left(\frac{\mathrm{3}}{\mathrm{2}}\right)\:. \\ $$
Answered by tanmay.chaudhury50@gmail.com last updated on 27/Apr/18
⌈(3/2)=⌈(1/2+1) =1/2⌈(1/2) =1/2(√Π)
$$\lceil\left(\mathrm{3}/\mathrm{2}\right)=\lceil\left(\mathrm{1}/\mathrm{2}+\mathrm{1}\right) \\ $$$$=\mathrm{1}/\mathrm{2}\lceil\left(\mathrm{1}/\mathrm{2}\right) \\ $$$$=\mathrm{1}/\mathrm{2}\sqrt{\Pi} \\ $$
Answered by tanmay.chaudhury50@gmail.com last updated on 27/Apr/18
1.⌈(x+1)=x.⌈(x)
$$\mathrm{1}.\lceil\left({x}+\mathrm{1}\right)={x}.\lceil\left({x}\right) \\ $$
Answered by tanmay.chaudhury50@gmail.com last updated on 27/Apr/18
2.⌈(n)=⌈(n−1+1) =(n−1)⌈(n−2) =(n−1)(n−2)⌈(n−3) thus the value is factorial(n−1) ⌈(n)=(n−1)!
$$\mathrm{2}.\lceil\left({n}\right)=\lceil\left({n}−\mathrm{1}+\mathrm{1}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:=\left({n}−\mathrm{1}\right)\lceil\left({n}−\mathrm{2}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:=\left({n}−\mathrm{1}\right)\left({n}−\mathrm{2}\right)\lceil\left({n}−\mathrm{3}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:{thus}\:{the}\:{value}\:{is}\:{factorial}\left({n}−\mathrm{1}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\lceil\left({n}\right)=\left({n}−\mathrm{1}\right)! \\ $$

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