let Γ(x)=∫_0 ^∞ t^(x−1) e^(−t) dt with x>0 1) find Γ^((n)) (x) with n∈ N^★ 2) calculate Γ(n +(3/2)) for n integr.


Question and Answers Forum

All Questions Topic List
Integration Questions
Previous in All Question Next in All Question
Previous in Integration Next in Integration
Question Number 33915 by prof Abdo imad last updated on 27/Apr/18

$l e t Γ (x) = \int_{0}^{\infty} t^{x - 1} e^{- t} d t w i t h x > 0$ $1) f i n d Γ^{(n)} (x) w i t h n \in N^{★}$ $2) c a l c u l a t e Γ (n + \frac{3}{2}) f o r n i n t e g r .$
Commented byprof Abdo imad last updated on 29/Apr/18

$1) w e h a v e Γ (x) = \int_{0}^{\infty} e^{(x - 1) l n t} e^{- t} d t \Rightarrow$ $Γ^{^{'}} (x) = \int_{0}^{\infty} l n (t) e^{(x - 1) l n (t)} e^{- t} d t$ $Γ^{^{″}} (x) = \int_{0}^{\infty} {(l n t)}^{2} e^{(x - 1) l n (t)} e^{- t} d t a n d i t s e a s y$ $t o p r o v e b y r e c u r r e n c e t h a t$ $Γ^{(n)} (x) = \int_{0}^{\infty} {(l n t)}^{n} t^{x - 1} e^{- t} d t \forall n \in N (Γ^{(0)} = Γ)$
Commented byprof Abdo imad last updated on 29/Apr/18

$w e k n e w t h a t Γ (x + 1) = x Γ (x) \Rightarrow$ $Γ (n + \frac{3}{2}) = Γ ((n + \frac{1}{2}) + 1) = (n + \frac{1}{2}) Γ (n + \frac{1}{2})$ $= (n + \frac{1}{2}) Γ (n - \frac{1}{2} + 1) = (n + \frac{1}{2}) (n - \frac{1}{2}) Γ (n - \frac{1}{2})$ $= (n + \frac{1}{2}) (n - \frac{1}{2}) (n - \frac{3}{2}) Γ (n - \frac{3}{2})$ $= (n + \frac{1}{2}) (n - \frac{1}{2}) (n - \frac{3}{2}) . . . . (n - \frac{2 n - 1}{2}) Γ (n - \frac{2 n - 1}{2})$ $= (n + \frac{1}{2}) (n - \frac{1}{2}) (n - \frac{3}{2}) . . . \frac{1}{2} Γ (\frac{1}{2})$ $b u t$ $Γ (\frac{1}{2}) = \int_{0}^{\infty} t^{- \frac{1}{2}} e^{- t} d t = \int_{0}^{\infty} \frac{e^{- t}}{\sqrt{t}} d t$ $=_{\sqrt{t} = x} \int_{0}^{\infty} \frac{e^{- x^{2}}}{x} 2 x d c = 2 \int_{0}^{\infty} e^{- x^{2}} d x = 2 \frac{\sqrt{π}}{2} = \sqrt{π}$ $Γ (n + \frac{3}{2}) = \sqrt{π} (n + \frac{1}{2}) (n - \frac{1}{2}) (n - \frac{3}{2}) . . . . \frac{3}{2} \frac{1}{2}$ $a n d t h i s q u a n t i t y c a n b e g i v e n b y f a c t o r i e l s .$
Commented byprof Abdo imad last updated on 29/Apr/18

$Γ (n + \frac{3}{2}) = \frac{\sqrt{π}}{2} \prod_{k = 1}^{n} \frac{2 k + 1}{2}$ $= \frac{\sqrt{π}}{2} \frac{1}{2^{n}} (3.5.7 . . . . . (2 n + 1))$ $= \frac{\sqrt{π}}{2^{n + 1}} (2.3.4.5.6 . . . . . (2 n) (2 n + 1)) {(2.4.6 . . . . (2 n))}^{- 1}$ $= \frac{\sqrt{π}}{2^{n + 1}} \frac{(2 n + 1)!}{2^{n} n!} = \frac{\sqrt{π}}{2^{2 n + 1}} \frac{(2 n + 1)!}{n!} .$
	Answered by tanmay.chaudhury50@gmail.com last updated on 27/Apr/18

	$2 . ⌈ (n + 3 / 2)$ $= ⌈ (n + 1 + 1 / 2)$ $= (n + 1 / 2) ⌈ (n + 1 / 2)$ $= (n + 1 / 2) ⌈ (n - 1 / 2 + 1)$ $= (n + 1 / 2) (n - 1 / 2) ⌈ (n - 1 / 2)$ $= (n + 1 / 2) (n - 1 / 2) (n - 3 / 2) ⌈ (n - 3 / 2)$ $= (n + 1 / 2) (n - 1 / 2) (n - 3 / 2) (n - 5 / 2) ⌈ (n - 5 / 2$ $t h e e x p r e s s i o n c a n n o t b e i n f a c t o r i a l f o r m$ $s i n c e e x t r e m e r i g h t f a c t o r c a n n o t b e 1$
	Answered by MJS last updated on 27/Apr/18

	$1 . Γ^{(n)} (x) = \underset{0}{\int^{\infty}} t^{x - 1} \ln {(t)}^{n} e^{- t} d t$ $2 . \frac{\sqrt{π}}{2} \times \underset{i = 1}{\prod^{n}} \frac{2 i + 1}{2}$
Terms of Service
Privacy Policy
Contact: info@tinkutara.com

Question and Answers Forum