Integral over a Hankel function

• spainchaud
In summary, the author provides formulas and theorems for the special functions of mathematical physics. There is very little expository material, and the book is primarily a collection of formulas and theorems.
spainchaud
In Gradshteyn and Ryzhik there is a formula 6.616 number 3

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

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

$$r=0$$

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.

Doesn't Gradstein give a reference for the derivation? Usually that's a good place to learn how to evaluate this kind of integrals.

Very good question. Yes, there is a reference.

W. Magnus, F. Oberhettinger
Formeln und Sätzen für die speziellen Funktionen der Mathematischen Physik
Springer Verlag, Berlin 1948

There is a translated version:
Formulas and theorems for the special functions of mathematical physics
Chelsea Publishing Co. 1949

And this much expanded version
Formulas and Theorems for the Special Functions of Mathematical Physics.
By WILHELM MAGNUS, FRITZ OBERHETTINGER and RAZ PAL SONI. Springer-
Verlag New York Inc., New York, 1966.

So far I have been unable to get hold of a copy. I tried to buy a PDF copy of the expanded version from SIAM for $25. I was surprised to get a 1 page review of the book. There was nothing on the reciept to indicate it was a review and not a journal article. I didn't realize it was a book until I read the review. Still not worth$25.

This would be a great book to have in my library. However, from the review

"As in previous editions the book contains almost no expository material, merely
lists of formulas and theorems."

So I doubt that I would learn any new techniques.

BTW, I think I posted this thread in the wrong forum section, probably should be in the calculus section. Then again Hankel functions are solutions of PDEs. I didn't notice a special functions section.

1. What is an integral over a Hankel function?

An integral over a Hankel function is a mathematical operation that involves integrating a function of the form f(x) = x^nH_m(ax), where n and m are integers, x is the independent variable, and a is a constant. The Hankel function is a special type of mathematical function that is commonly used in physics and engineering to describe wave phenomena.

2. What is the significance of an integral over a Hankel function?

Integrals over Hankel functions are important in many areas of science and engineering, particularly in the study of diffraction, scattering, and wave propagation. They are also useful in solving certain types of differential equations and in the analysis of systems with cylindrical symmetry.

3. How is an integral over a Hankel function evaluated?

The evaluation of integrals over Hankel functions can be done using various techniques, such as contour integration, series expansion, or asymptotic approximations. The specific method used depends on the specific form of the integral and the desired level of accuracy.

4. What are some applications of integrals over Hankel functions?

Integrals over Hankel functions have a wide range of applications in physics, engineering, and mathematics. Some common uses include analyzing the diffraction patterns of electromagnetic and acoustic waves, solving boundary value problems in cylindrical coordinates, and calculating the scattering of particles in quantum mechanics.

5. Are there any special properties of integrals over Hankel functions?

Yes, there are several important properties of integrals over Hankel functions. For example, they satisfy certain recurrence relations, have specific symmetries, and are related to other special functions such as Bessel functions and Legendre functions. These properties make them valuable tools in many areas of science and engineering.

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