Bessel functions in vector field

[Enialis]

I need to solve this general problem. Let's consider the following vector field in cylindrical coordinates:
\vec{A}=-J'_m(kr)\cos(\phi)\hat{\rho}+\frac{m^2}{k}\frac{J_m(kr)}{r}\sin(\phi)\hat{\phi}+0\hat{z}

where m is an integer, and k could satisfy to:
J_m(ka)=0 or J_m'(ka)=0 with a real.
(the apex is used for the derivative respect to r)

In this case I need to express the divergence of the field and the following integral:
\int_0^a\int_0^{2\pi}(|A_r|^2+|A_\phi|^2)r\,dr\,d\phi

Is it possible to found simpler analytical formulas for general m for the results?
Where can I found useful relationships for the bessel integrals involved? thank you.

 

[Enialis]

For the divergence of the field I found:

\nabla\cdot\vet{A}= kJ_m(kr)\cos(\phi)

using recursive formulas for the Bessel derivatives.

I still need help for the integrals, in particular does the bessel functions have particular orthogonality relation that admit to solve:

\int J_m(x) J_n(x) x\,dx = ?

 

[JJacquelin]

Bessel functions are orthogonal in particular domains. See Eq.(53) in :
http://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html

 

[Enialis]

Ok, but that relation does not help me so much. In this example I need to solve an integral of the form:
\int_0^a J_{m+1}(kr)J_{m-1}(kr) r\,dr

I found only two useful formulas (that include the result of eq. 53):

\int J_m(kx)J_m(lx)x dx=\frac{x}{k^2-l^2}[kJ_m(lx)J_{m+1}(kx)-lJ_m(kx)J_{m+1}(lx)]

\int J_m^2(kx)x dx=\frac{x^2}{2}[J_m'^2(kx)+(1-\frac{m^2}{k^2x^2})J_m^2(kx)]

Now my question is: there are other relations for Bessel functions of different order? (Jm, Jn). I do not know how these formulas are obtained...

 

[JJacquelin]

I don't know if the attached formula could help you :