Integral involving Bessel function and exponential 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.
 A: 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. 
