Symbolic integration of a Bessel function with a complex argument

In summary, Mr Simpson is trying to solve an integral involving a Bessel function of first kind and order 0, but is having difficulty because of the complex arguments. He has found that r1 is r' in the integral, and is wondering if there is a way to solve the integral symbolically.
  • #1
ocmaxwell
8
1
Hello all

I am trying to solve the following integral with Mathematica and I'm having some issues with it.

1676210485824.png

where Jo is a Bessel Function of first kind and order 0. Notice that k is a complex number given by

1676210684948.png


Where delta is a coefficient.

Due to the complex arguments I'm integrating the absolute value of the Bessel function.

I would like to solve the integral symbolically to get the result as a function or r, and not a number, so I can plot f(r) later on. See below what I did

1676211023657.png
notice that r1 is r' in the integral above
and the result

1676211136130.png


I am totally sure there should be a way in which it can be done.

Any help would be greatly appreciated

Thank you in advance
 
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  • #2
ocmaxwell said:
I am totally sure there should be a way in which it can be done
Usually if Mathematica doesn’t know a way then there is no known way.
 
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  • #3
This is certainly not a proof, but if you Plot3D the Re and the Im of either your Bessel or the Abs of your Bessel, say for {delta,-5,5} and {r1,-5,5} then that blowing up to infinity in lots of odd ways strongly hints to me that finding that integral might be really challenging. If you could restrict delta to being an integer, or even a specific integer, then the graphs hint that might possibly be a simpler problem. Without giving Mathematica any information about delta it will proceed imagining that delta might even be complex
 
  • #4
You are very right Mr. Simpson
For this case delta is real, how could I enforce this condition so mathematica undertands it and still gives me the result of the integral in terms of delta and r?

Thank you for your response
 
  • #5
You can use the option “Assumptions” to tell it to assume that it is real.
 
  • #6
Thank you Sir, I will try that out.
Thank you for taking the time to respond.
 

FAQ: Symbolic integration of a Bessel function with a complex argument

What is a Bessel function?

A Bessel function is a type of special function that often appears in solutions to differential equations with cylindrical or spherical symmetry. They are named after the German mathematician Friedrich Bessel and come in several varieties, including the Bessel functions of the first kind (J_n) and the second kind (Y_n).

What does it mean to integrate a Bessel function with a complex argument?

Integrating a Bessel function with a complex argument means performing an integral where the Bessel function's input is a complex number. This can involve both real and imaginary parts, making the integration process more intricate due to the properties of complex functions.

Why is symbolic integration of Bessel functions with complex arguments challenging?

Symbolic integration of Bessel functions with complex arguments is challenging because Bessel functions themselves are complex and have intricate behaviors. When the argument is complex, the integral can involve complex analysis techniques, making it difficult to find closed-form solutions.

Are there any known closed-form solutions for the symbolic integration of Bessel functions with complex arguments?

There are some specific cases where closed-form solutions are known, but in general, the symbolic integration of Bessel functions with complex arguments does not yield simple closed-form expressions. Often, numerical methods or approximations are used instead.

What methods can be used to perform symbolic integration of Bessel functions with complex arguments?

Several methods can be employed, including contour integration in the complex plane, series expansions, and the use of special functions like the Gamma function or hypergeometric functions. In many cases, computer algebra systems like Mathematica or Maple are used to handle the complexity of these integrals.

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