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I-0-e-x-J-1-2-x-dx-J-v-x-x-v-n-0-1-n-x-2n-2-n-v-n-n-v-1-




Question Number 146054 by mnjuly1970 last updated on 10/Jul/21
            I := ∫_0 ^(  ∞) e^( −x)  . J_(1/2)  (x ) dx            J_(v )  (x ) = x^( v)  Σ_(n=0) ^( ∞) (((− 1 )^( n)  x^( 2n) )/(2^( n + v)  n ! Γ ( n + v +1 )))     ....
$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathrm{I}\::=\:\int_{\mathrm{0}} ^{\:\:\infty} {e}^{\:−{x}} \:.\:\mathrm{J}_{\frac{\mathrm{1}}{\mathrm{2}}} \:\left({x}\:\right)\:{dx} \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathrm{J}_{{v}\:} \:\left({x}\:\right)\:=\:{x}^{\:{v}} \:\underset{{n}=\mathrm{0}} {\overset{\:\infty} {\sum}}\frac{\left(−\:\mathrm{1}\:\right)^{\:{n}} \:{x}^{\:\mathrm{2}{n}} }{\mathrm{2}^{\:{n}\:+\:{v}} \:{n}\:!\:\Gamma\:\left(\:{n}\:+\:{v}\:+\mathrm{1}\:\right)} \\ $$$$\:\:\:…. \\ $$

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