Bessel function and integral I am  working on a  problem. Solving the PDE for my problem, this Bessel integral arises: 
$$\int^\infty_0 e^{-ax}x^m(J_0(bx))^2dx,\quad \int^\infty_0 e^{-ax}x^m(J_1(bx))^2dx\qquad \text{and} \qquad\int^\infty_0 e^{-ax}x^mJ_0(bx)J_1(bx)dx$$
where $~J_0~$ and $~J_1~$ are the Bessel functions of first kind. 
I haven't found the solution in any table or book, and due to my limited background in applied mathematics I don't know how to integrate it by myself. 
Does anybody know the solution? 
Thanks a lot in advance
 A: Sorry this is not a clear answer. But I myself is in the process of following the derivations of some of the well known integrals involving Bessel functions, and you may find your answers in Luke's, "Integrals of Bessel Functions", p.314 and after. The pity is that you can't find in detail how they are derived. Maybe you should consult the original papers if you are interested in the derivations. It seems to me that the multiplications of the Bessel functions sre transformed to the single one by the additional theorem, which then is followed by transforming into elliptic integrals.
A: I have no access to Luke's book "Integrals of Bessel Functions" that 萬雄彦 recommended in his/her answer.
Using a CAS, I got some results. Naming
$$I_{i,j}=\int_{0}^\infty e^{-ax}\, x^m\, J_i(bx)\, J_j(bx)\,dx$$
$$\color{blue}{I_{0,0}=a^{-(m+1)} \Gamma (m+1) \,
   _3F_2\left(\frac{1}{2},\frac{m+1}{2},\frac{m+2}{2};1,1;-\frac{4b^2}{a^2}\right)}$$ provided
$b\geq 0\land \Re(m)+1>0\land ((\Re(a)=0\land \Re(m)<1)\lor \Re(a)>0)$
$$\color{blue}{I_{1,1}=\frac{b^2}{4}  a^{-(m+3)} \Gamma (m+3) \,
   _3F_2\left(\frac{3}{2},\frac{m+3}{2},\frac{m+4}{2};2,3;-\frac{4b^2}{a^2}\right)}$$ provided
$b\geq 0\land \Re(m)+1>0\land ((\Re(a)=0\land \Re(m)<1)\lor \Re(a)>0)$
$$\color{blue}{I_{0,1}=\frac{b}{2} a^{-(m+2)} \Gamma (m+2) \,
   _3F_2\left(\frac{3}{2},\frac{m+2}{2},\frac{m+3}{2};2,2;-\frac{4b^2}{a^2}\right)}$$
provided
$b\geq 0\land \Re(m)+2>0\land ((\Re(a)=0\land \Re(m)<1)\lor \Re(a)>0)$
A few numerical checks have been done.
Edit
Making the problem more general, it seems that 
$$J_{i,j}= \frac{ 2^{(i+j)}\, \Gamma (i+1)\, \Gamma (j+1) \, a^{(m+i+j+1)}}{\Gamma (m+i+j+1)\, b^{(i+j)}} \, I_{i,j}$$ can write
$$\color{blue}{J_{i,j}=\,
   _4F_3\left(\frac{i+j+1}{2},\frac{i+j+2}{2},\frac{m+i+j+1
   }{2},\frac{m+i+j+2}{2};i
   +1,j+1,i+j+1;-\frac{4 b^2}{a^2}\right)}$$
provided $\Re(m+i+j)>-1\land b\geq 0\land ((\Re(a)=0\land \Re(m)<1)\lor \Re(a)>0)$
