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

               ...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_v (x) is convergent (why?): ∀x∈R...

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:...{nice}\:\:{calculus}... \\ $$$$\:\:\:\:\:{calculate}\:::: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\Omega\overset{???} {=}\int_{\mathrm{0}} ^{\:\:\infty} {e}^{\:−{t}} \:{t}^{\:\mathrm{2}} \:{j}_{\mathrm{0}} \left(\:{t}\:\right){dt} \\ $$$$\:\:\:\:\:{where}\::\:\:{j}_{\left({v}\right)} \left({x}\right)={x}^{{v}} \underset{{n}=\mathrm{0}} {\overset{\:\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} {x}^{\mathrm{2}{n}} }{\mathrm{2}^{\mathrm{2}{n}+{v}} {n}!\Gamma\left({n}+{v}+\mathrm{1}\right)}\: \\ $$$$\:\:\:\:\:\:\:\:\::::\:{Bessel}\:{function}\:{of}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{the}\:{first}\:{type}\:{of}\:{order}\:{v}\:...\: \\ $$$$\:\:\:\:\:\:\:\:\:{j}_{{v}} \left({x}\right)\:{is}\:{convergent}\:\left({why}?\right):\:\forall{x}\in\mathbb{R}... \\ $$

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

(1/2)

$$\frac{\mathrm{1}}{\mathrm{2}} \\ $$

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