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In the course of a complicated calculation, I encountered the following simple-looking integral $$ \Phi(\beta)=\int_\beta^\infty \frac{dz}{z}e^{-\beta z}I_1(z)\ , $$ where $I_1(z)$ is a Bessel function, and $\beta$ a real, positive number.

Mathematica is not able to solve it, and I could not find it easily in the tables. Does anyone know how to crack it? Many thanks.

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    $\begingroup$ Simple looking ??? This is worse than the Laplace transform of that special function, as the integral is incomplete. I would rather call it a nightmare. I wouldn't bet a cent on the existence of a closed-form. $\endgroup$ – Yves Daoust Mar 15 at 9:00
  • $\begingroup$ You may have a try by exploiting the differential equation, which allows you to express the function in terms of its derivatives, but I am not sure that will rescue you. $\endgroup$ – Yves Daoust Mar 15 at 9:39
  • $\begingroup$ The integral may be expressed in the form of a Kampe de Feriet function which is a hypergeometric function of two variables. For the derivation the integral has to be split in two from 0 to $\infty$ and 0 to $\beta$ $\endgroup$ – stocha Mar 15 at 13:53
  • $\begingroup$ @stocha Thanks very much. Could you please provide more details? $\endgroup$ – Pierpaolo Vivo Mar 15 at 13:54
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    $\begingroup$ I can't do the complete calculation because of a lack of time, but below you find a short post which gives you the hint. I did not found the paper in the internet, so I don't know how to exchange this. $\endgroup$ – stocha Mar 15 at 15:34
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First you have to split the integral: $$\int_{\beta }^{\infty }\frac{e^{-\beta ~z}I_{1}(z)}{z}\;dz=\int_{0}^{\infty }% \frac{e^{-\beta ~z}I_{1}(z)}{z}\;dz-\int_{0}^{1}\frac{e^{-\beta ^{2}~z}I_{1}(\beta ~z)}{z}\;dz$$ The first integral should be easy to be solved by Mathematica. The second one can be expressed by Kampé de Fériet Function, which is a hypergeometric function of two variables.

For the derivation one has to express the Bessel-function divided by z and the exponential function in the form of the hypergeometric function:

$$\frac{I_{1}(z)}{z}=\frac{\left( 2^{-\nu }z^{\nu -1}\right) \,_{0}F_{1}\left( ;\nu +1;\frac{z^{2}}{4}\right) }{\Gamma (\nu +1)}$$ and

$$\exp \left( {-\beta ~z}\right) =_{1}F_{1}(a;a;-\beta ~z)$$ The solution is given in a old paper I can't find in the internet:

"IntegraIs Involving Kampe de Feriet Function" by G. P. SRIVASTAVA and S. SARAN, 1966 therefore I give here the expressions:

\begin{equation} \int_{0}^{1}x^{s-1}\left( 1-x^{h}\right) ^{\sigma -1}\;_{\mu }F_{\rho }\left[ \left\vert \beta \right\vert _{\mu };\left\vert \delta \right\vert _{\rho };a~x^{nh}\right] _{\mu }F_{\rho }\left[ \left\vert \beta ^{\prime }\right\vert _{\mu };\left\vert \delta ^{\prime }\right\vert _{\rho };b~x^{nh}\right] dx= \end{equation} \begin{equation*} B_{h}F_{n,\rho }^{n,\mu }\left( a,b\left\vert \begin{array}{c} \left\{ s/h\right\} _{n};\left\vert \beta ,\beta ^{\prime }\right\vert _{\mu }; \\ \left\{ s/h+\sigma \right\} _{n};\left\vert \delta ,\delta ^{\prime }\right\vert _{\rho };% \end{array}% \right. \right) \end{equation*} and the symbols $\left\vert T\right\vert _{m}\;$and $\left\{ T\right\} _{n}$ stand for the sequences \begin{equation*} T_{m},\;\;m=1,2,...,m\;\;\text{and}\;\frac{T+r}{n},\;\;r=0,1,2,...,n-1 \end{equation*}

\begin{equation*} B_{h}=\frac{\Gamma \left( \sigma \right) \overset{n-1}{\underset{r=0}{\Pi }}% \Gamma \left( \frac{s/h+r}{n}\right) }{h~n^{\sigma }\overset{n-1}{\underset{% r=0}{\Pi }}\Gamma \left( \frac{s/h+\sigma +r}{n}\right) } \end{equation*} For the explizit calculation the paper and the definition of the Kampe de Feriet Function wikipedia is needed.

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