- #1

spainchaud

- 2

- 0

[tex]

\int_{-\infty}^{+\infty} e^{itx} H_0^{(1)}(r\sqrt{\alpha^2 - t^2}) dt = -2i \frac{e^{i\alpha\sqrt{r^2 + x^2}}}{\sqrt{r^2 + x^2}}

[/tex]

I need to learn the techniques to evaluate this integral and similar integrals. I am not sure if I use recursion relations, contour integrals, or a combination of both.

Background:

The solution for the fields of a magnetic dipole in a stratified medium have been detailed by Tang, Electromagnetic Fields due to Dipole Antenna Embedded in Stratified Anisotropic Medium, and by Kong, Electromagnetic Fields due to Dipole Antennas over Stratified Anisotropic media. Kong references a variation of the equation above as an identity. I need to program a simulation from their formulation. To do this I will need solve many integrals similar to the one above. I could try a straight numerical integration but, a problem occurs when

[tex]

r=0

[/tex]

because the Hankel function is singular. For this case, and maybe some others, I will need to develop an analytic solution. Any hints would be appreciated.