# Integral solving request

Dear all,
please help me solve the following integral.
I need to solve this integral for one of my problems. $$(\frac{1}{2\pi})^2\int_0^\infty\int_{-\infty}^\infty \frac{J_0(\rho R_0)J_0(\rho r)L*Sinc(\frac{Lk_z}{2})e^{jk_zz}}{(\frac{\omega_0}{c})^2-(\rho^2+k_z^2)}\rho\operatorname{d}k_z\operatorname{d}\rho$$ $J_0(x)$ is first kind bessel function and $Sinc(x)=sin(x)/x$.
Is this integral have an analytical solution?
I tried to solve it with Mathematica and Matlab but they even can not solve the simplest orthogonal equation: $$\int_0^\infty J_0(ux)J_0(vx)x\operatorname{d}x=\frac{1}{u}\delta(u-v)$$

Please help me solve this integral.
Thanks.

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When you say 'one of your problems', what do you mean? Homework? –  David Roberts Feb 8 '13 at 4:00
No. It is a physical problem that I encounter at work. –  Vadim Winebrand Feb 8 '13 at 13:09

## 1 Answer

I found out an identity: $$\int_0^\infty\frac{\Phi(x)}{x^2-k^2}J_n(xr)x\operatorname{d}x=\frac{i\pi}{2}\Phi(k)H_n^{(1)}(kr) \text{ -> } Im(k)>0$$ where $H_n$ is Hankel function of the first kind function
Using this identity I can reduce the above integral to following $$-i\frac{L}{8\pi}\int_{-\infty}^\infty J_0(R_0\sqrt{(\frac{\omega_0}{c})^2-k_z^2})H_0^{(1)}(r\sqrt{(\frac{\omega_0}{c})^2-k_z^2})Sinc(\frac{Lk_z}{2})e^{jk_zz}\operatorname{d}k_z \text{ } r>R_0$$

Therefore the problem now is much simpler.
Please help me solve this inverse fourie transform

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