# How Do You Solve Bessel Function Integrals?

• jayryu
In summary, the conversation discusses a question about solving Bessel function integrals, specifically the integral of J_{m-1}(ax)*J_{m+1} (ax)*x from 0 to R. The conversation suggests using an identity to simplify the expression, where each successive Bessel function can be defined in terms of its predecessor. The conversation also mentions the possibility of finding a solution by reading about the orthogonality of Bessel functions. Finally, the person asking the question confirms that they have found a solution.
jayryu
hello,everyone
i want to know how to solve this bessel function integrals:

\int_{0}^{R} J_m-1(ax)*J_m+1 (ax)*x dx
where J_m-1 and J_m+1 is the Bessel function of first kind, and a is a constant.

thanks.

Is this what you meant to post?

$$\int_{0}^{R} J_{m-1}(ax)*J_{m+1} (ax)*x \ dx$$

I am not an expert on Bessel functions, but isn't there an identity that you can use to simplify this expression? Something like

$$J_{m+1} = Some \ function \ of \ J_m$$

In other words, each successive Bessel function can be defined in terms of its predecessor. For example,

J1 = some function of J0,
J2 = some function of J1,
J3 = some function of J2,
J4 = some function of J3,
etc.

If you can find this identity, you should be able to simplify your integral.

Perhaps you will find what you are looking for reading about the orthogonality of the Bessel functions. If you don't already know about that, try looking at:

http://www.hit.ac.il/ac/files/shk_b/Differential.Equations/Orthogonality_of_Bessel_functions.htm

Last edited by a moderator:
LCKurtz said:
Perhaps you will find what you are looking for reading about the orthogonality of the Bessel functions. If you don't already know about that, try looking at:

http://www.hit.ac.il/ac/files/shk_b/Differential.Equations/Orthogonality_of_Bessel_functions.htm

thank you,i'll try that!

Last edited by a moderator:
DuncanM said:
Is this what you meant to post?

$$\int_{0}^{R} J_{m-1}(ax)*J_{m+1} (ax)*x \ dx$$

I am not an expert on Bessel functions, but isn't there an identity that you can use to simplify this expression? Something like

$$J_{m+1} = Some \ function \ of \ J_m$$

In other words, each successive Bessel function can be defined in terms of its predecessor. For example,

J1 = some function of J0,
J2 = some function of J1,
J3 = some function of J2,
J4 = some function of J3,
etc.

If you can find this identity, you should be able to simplify your integral.

yes,it is.thanks for your suggestions.i have solved it.

## What is a Bessel function integral?

A Bessel function integral is a mathematical function that is named after the German mathematician Friedrich Bessel. It is used to solve differential equations that arise in various physical problems, such as heat transfer, wave propagation, and vibration analysis.

## What is the formula for a Bessel function integral?

The formula for a Bessel function integral depends on the specific type of Bessel function being used. However, a general form of a Bessel function integral can be written as ∫0sup>θxνJν(x)dx, where ν is a constant and Jν(x) is a Bessel function of the first kind.

## What are the applications of Bessel function integrals?

Bessel function integrals have a wide range of applications in physics and engineering. They are used to solve problems related to heat transfer, wave propagation, and vibration analysis. They are also used in signal processing, optics, and electromagnetics.

## Can Bessel function integrals be evaluated analytically?

In general, Bessel function integrals cannot be evaluated analytically. However, there are certain special cases where closed-form solutions can be obtained. In most cases, numerical methods are used to evaluate Bessel function integrals.

## What are the properties of Bessel function integrals?

Bessel function integrals have several important properties, including orthogonality, recurrence relations, and generating functions. These properties are useful for solving differential equations and can be used to simplify complex calculations involving Bessel function integrals.

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