Integrals of the Bessel functions of the first kind

In summary, the two functions f(x,a) and g(x,a) defined by the given integrals are known as K_0(ax) and fall into a class of integrals solved in Watson's A Treatise on the Theory of Bessel Functions. They are also expressible as hypergeometric functions.
  • #1
Wuberdall
34
0
Hi Physics Forums.

I am wondering if I can be so lucky that any of you would know, if these two functions -- defined by the bellow integrals -- have a "name"/are well known. I have sporadically sought through the entire Abramowitz and Stegun without any luck.

[itex]f(x,a) = \int_0^\infty\frac{t\cdot J_0(at)}{t^2 + x^2}\,\mathrm{d}t [/itex]

and

[itex]g(x,a) = \int_0^\infty\frac{t^2\cdot J_0(at)}{t^2 + x^2}\,\mathrm{d}t [/itex]

where [itex]J_0(x)[/itex] is the Bessel function of the first kind.
 
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  • #2
Wuberdall said:
[itex]f(x,a) = \int_0^\infty\frac{t\cdot J_0(at)}{t^2 + x^2}\,\mathrm{d}t [/itex]

This one is 6.532.4 in Gradshteyn and Ryzhik, equal to ##K_0(ax)## for ##a>0, \text{Re}~x>0##.

[itex]g(x,a) = \int_0^\infty\frac{t^2\cdot J_0(at)}{t^2 + x^2}\,\mathrm{d}t [/itex]

I think this one falls into a class of integrals solved in Watson's A Treatise on the Theory of Bessel Functions, expressible as hypergeometric functions. I attached the relevant page:

bessel.JPG
 

1. What are Bessel functions of the first kind?

Bessel functions of the first kind are a type of special function in mathematics that are used to solve differential equations with cylindrical symmetry. They are named after the mathematician Friedrich Bessel and are denoted by the symbol Jn(x), where n is the order of the Bessel function and x is the variable.

2. What is the significance of integrals of Bessel functions of the first kind?

The integrals of Bessel functions of the first kind are important in many areas of physics and engineering, particularly in the fields of electromagnetism, fluid mechanics, and quantum mechanics. They are also used in the analysis of vibrations, heat transfer, and signal processing.

3. How are integrals of Bessel functions of the first kind calculated?

There are several techniques for calculating integrals of Bessel functions of the first kind, depending on the specific form of the integral. These include using recurrence relations, integral representations, and power series expansions. In some cases, numerical methods may also be used for more complex integrals.

4. Are there any applications of integrals of Bessel functions of the first kind?

Yes, there are many applications of integrals of Bessel functions of the first kind in various fields of science and engineering. Some examples include calculating the magnetic field of a circular current loop, solving the heat equation in cylindrical coordinates, and analyzing the diffraction patterns of light passing through a circular aperture.

5. Can integrals of Bessel functions of the first kind be solved analytically?

In general, integrals of Bessel functions of the first kind cannot be solved analytically. However, there are some special cases where closed-form solutions exist, such as for certain specific values of the order and variable. In most cases, numerical methods or approximations are used to evaluate these integrals.

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