Solving this Integral with Bessel Functions Any suggestions on solving this (J0,J1 Bessel function of first kind, 0th and 1st order, respectively) :
$$
T = \int_0^a \int_0^\infty J_0(\lambda r) J_1(\lambda a) e^{-\sqrt{\frac{s}{\alpha}+\lambda^2} z} \frac{1}{\sqrt{\frac{s}{\alpha}+\lambda^2} }  d\lambda dr\
$$
Maple gives the result as
$$
{\frac {a \alpha\,{{ J}_{1}\left(\lambda\,a\right)} \left( {
{ J}_{1}\left(\lambda\,a\right)}\pi \,{{  H}_{0}\left(\lambda
\,a\right)}-{{ J}_{0}\left(\lambda\,a\right)}\pi \,{{  H}_{1
}\left(\lambda\,a\right)}+2\,{{ J}_{0}\left(\lambda\,a\right)}
 \right) }{2({\lambda}^{2}\alpha+s)}}
$$
where $H_0$ and $H_1$ are Struve functions.

But did you really mean what you wrote?  Note that the $J_1(\lambda a)$ is just a constant, the integral over $z$ is easy, and the only nontrivial part is integrating
$J_0(\lambda r)$ from $r=0$ to $a$.

EDIT: With the integral over $\lambda$ rather than $z$, you can still do the integral over $r$, with a result involving Struve functions, but 
the integral over $\lambda$ seems to be difficult.
Maple gives the result as
$$
{\frac {a \alpha\,{{ J}_{1}\left(\lambda\,a\right)} \left( {
{ J}_{1}\left(\lambda\,a\right)}\pi \,{{  H}_{0}\left(\lambda
\,a\right)}-{{ J}_{0}\left(\lambda\,a\right)}\pi \,{{  H}_{1
}\left(\lambda\,a\right)}+2\,{{ J}_{0}\left(\lambda\,a\right)}
 \right) }{2({\lambda}^{2}\alpha+s)}}
$$
where $H_0$ and $H_1$ are Struve functions.

But did you really mean what you wrote?  Note that the $J_1(\lambda a)$ is just a constant, the integral over $z$ is easy, and the only nontrivial part is integrating
$J_0(\lambda r)$ from $r=0$ to $a$.

EDIT: With the integral over $\lambda$ rather than $z$, you can still do the integral over $r$, with a result involving Struve functions, but 
the integral over $\lambda$ seems to be difficult.