Can the Divergence of a Bessel Integral Be Prevented?

In summary, the conversation discusses an integral involving a second order bessel function and its convergence. It is determined that the integral is divergent for a lower boundary of r=0, but can be made finite by choosing a higher boundary.
  • #1
tx213
7
0
Hi,

I would like to confirm my intuition about a bessel integral from you guys.

The integral is: Integrate[ (1/r) * J[2,2*pi*phi*r] ] from 0 → ∞ with respect to r.

J[2,2*pi*phi*r] is a second order bessel. Integrals with 1/x from 0 to Inf are divergent. Sure enough, this one is going to diverge so mathematica says. But is there anything I might be able to do to stop this integral from diverging?

Thanks in advance for any insight!
 
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  • #2
Integrate[ (1/r) * J[2,2*pi*phi*r] ] = -J[1,2*pi*phi*r]/(2*pi*phi*r)
limit (r -> 0) = -1/2
limit (r -> infinity) = 0
The integral is convergent = -1/2
 
  • #3
JJacquelin said:
Integrate[ (1/r) * J[2,2*pi*phi*r] ] = -J[1,2*pi*phi*r]/(2*pi*phi*r)
limit (r -> 0) = -1/2
limit (r -> infinity) = 0
The integral is convergent = -1/2

Ah, oops I'm sorry. I had meant a 0th order bessel, not 2nd order! It's J[0,2*pi*phi*r]. =(
 
  • #4
In this case, the integral is divergent when the lower boundary for r is =0. That is a fact and you can do nothing against a fact.
If you want to obtain a finite value, you have to chose a boundary higher than 0.
 

What is a Bessel integral divergence?

A Bessel integral divergence is a mathematical concept that involves the integration of a Bessel function, which is a special type of mathematical function used in many areas of physics and engineering. It is used to solve problems involving wave phenomena, such as sound and heat transfer.

What is the importance of Bessel integral divergence?

Bessel integral divergence is important because it helps us understand and solve problems related to wave phenomena in various fields, such as acoustics, electromagnetics, and fluid dynamics. It is also used in the study of differential equations and in the development of mathematical models.

How is Bessel integral divergence calculated?

Bessel integral divergence is calculated by using the properties of Bessel functions and applying them to the integral expression. This involves a series of mathematical steps and techniques, such as integration by parts and substitution, to evaluate the integral and obtain a solution.

What are some real-world applications of Bessel integral divergence?

Bessel integral divergence has many applications in various fields of science and engineering. For example, it is used in the analysis of sound waves in musical instruments, the study of heat transfer in solids, and the design of antennas in telecommunications.

What are some other names for Bessel integral divergence?

Bessel integral divergence is also known as Bessel function integral, Bessel transform, or Bessel-Hankel integral. These terms are often used interchangeably and refer to the same mathematical concept and its various applications.

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