nice-calculus-calculate-0-e-t-t-2-j-0-t-dt-where-j-v-x-x-v-n-0-1-n-x-2n-2-2n

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

\dots n i c e c a l c u l u s \dots

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 \dots

j_{v} (x) i s c o n v e r g e n t (w h y ?) : \forall x \in R \dots

Commented by Dwaipayan Shikari last updated on 20/Dec/20

\frac{1}{2}