# Symbolic integration of a Bessel function with a complex argument

• ocmaxwell
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.
ocmaxwell
Hello all

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

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

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

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

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

Any help would be greatly appreciated

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.

FactChecker
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

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?

You can use the option “Assumptions” to tell it to assume that it is real.

Thank you Sir, I will try that out.
Thank you for taking the time to respond.

## 1. What is a Bessel function with a complex argument?

A Bessel function with a complex argument is a special type of mathematical function that is used to describe oscillatory phenomena in various fields such as physics, engineering, and mathematics. It is defined as a solution to Bessel's differential equation and is denoted by the symbol J(z), where z is a complex number.

## 2. Why is symbolic integration of a Bessel function with a complex argument important?

Symbolic integration of a Bessel function with a complex argument is important because it allows us to find the exact analytical solution for integrals involving these functions. This is useful in many applications, such as in calculating physical quantities or solving differential equations in physics and engineering problems.

## 3. What are the challenges of symbolically integrating a Bessel function with a complex argument?

One of the main challenges of symbolically integrating a Bessel function with a complex argument is that it requires advanced mathematical techniques and knowledge of complex analysis. The integration process may also involve multiple steps and require the use of special functions such as the gamma function and hypergeometric functions.

## 4. Can a Bessel function with a complex argument be integrated numerically instead?

Yes, a Bessel function with a complex argument can also be integrated numerically using numerical integration methods such as the trapezoidal rule or Simpson's rule. However, these methods may not always give an accurate result and may require a large number of iterations.

## 5. Are there any software tools available for symbolic integration of Bessel functions with complex arguments?

Yes, there are various mathematical software tools such as Mathematica, Maple, and MATLAB that have built-in functions for symbolically integrating Bessel functions with complex arguments. These tools use advanced algorithms and techniques to compute the integrals accurately and efficiently.

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