...nice calculus... calculate ::: Ω=^(???) ∫_0 ^( ∞) e^( −t) t^( 2) j_0 ( t )dt where : j_((v)) (x)=x^v Σ_(n=0) ^( ∞) (((−1)^n x^(2n) )/(2^(2n+v) n!Γ(n+v+1))) ::: Bessel function of the first type of order v ... j


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Question Number 126349 by mnjuly1970 last updated on 19/Dec/20

$. . . n i c e c a l c u l u s . . .$ $c a l c u l a t e :::$ $Ω \overset{? ? ?}{=} \int_{0}^{\infty} e^{- t} t^{2} j_{0} (t) d t$ $w h e r e : j_{(v)} (x) = x^{v} \underset{n = 0}{\sum^{\infty}} \frac{{(- 1)}^{n} x^{2 n}}{2^{2 n + v} n! Γ (n + v + 1)}$ $::: B e s s e l f u n c t i o n o f$ $t h e f i r s t t y p e o f o r d e r v . . .$ $j_{v} (x) i s c o n v e r g e n t (w h y ?) : \forall x \in R . . .$
Commented by Dwaipayan Shikari last updated on 20/Dec/20

$\frac{1}{2}$
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