Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

share|improve this question
1  
When you say 'one of your problems', what do you mean? Homework? –  David Roberts Feb 8 '13 at 4:00
2  
No. It is a physical problem that I encounter at work. –  Vadim Winebrand Feb 8 '13 at 13:09

1 Answer 1

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

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.