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Question Number 218043 by SdC355 last updated on 27/Mar/25

\int e^{- s t} J_{ν} (t) d t = \frac{{(s + \sqrt{s^{2} + 1})}^{- ν}}{\sqrt{s^{2} + 1}} . ν \in R^{+}

\frac{d}{d s} \int e^{- s t} J_{ν} (t) d t = \frac{{(s + \sqrt{s^{2} + 1})}^{- ν} (s + ν \sqrt{s^{2} + 1})}{\sqrt{{(s^{2} + 1)}^{3}}}

and \frac{d}{d s} \int e^{- s t} J_{ν} (t) d t = - \int {t e}^{- s t} J_{ν} (t) d t

\int {t e}^{- t} J_{- \frac{1}{2}} (t) d t = - \frac{{(1 + \sqrt{2})}^{\frac{1}{2}} (1 - \frac{1}{2} \sqrt{2})}{\sqrt{2^{3}}}

Answered by SdC355 last updated on 27/Mar/25

Q 218025

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