How do you combine Bessel functions?

In summary, the conversation discusses a differential equation that has been solved, but includes an integral with two Bessel functions. The person is seeking help in finding a way to integrate the equation, specifically by combining the two Bessel functions. However, it is mentioned that the antiderivative may not have a closed form due to the presence of e^(-t*x^2). Eventually, the person finds a solution to the integral.
  • #1
renz
28
0
Hi,
I have been trying to solve this differential equation for a while now. Now I get to the point where I have the solution, but it includes an integral.

The integral is

[tex]\int x J_{1/4}(ax) J_{1/4}(bx) e^{-x^2t}dx[/tex]

, where a and b are constants, and the integral is from zero to infinity. I think I can figure out how to integrate this by using a table of integral, but I need to only have one Bessel function in it.
How can I combine the two Bessel functions?

Any help is much appreciated.
 
Physics news on Phys.org
  • #2
What do you mean by 'need to have one Bessel function in it?' If you mean the antiderivative, well I believe the antiderivative has no closed form. The problem is the e^(-t*x^2) in there.

Why don't you post the original DE?
 
  • #3
thank you for replying. I thought there's a way to make the product of two Bessel function become one function, or square of one function.

But never mind, I found the solution to the integral.
 

1. How do you combine Bessel functions of different orders?

To combine Bessel functions of different orders, you can use the addition theorem, which states that the sum of two Bessel functions with different orders can be expressed as a linear combination of Bessel functions with the same order but different arguments. Alternatively, you can use recurrence relations to express a Bessel function of a certain order in terms of Bessel functions of lower orders.

2. Can I combine Bessel functions with other types of functions?

Yes, you can combine Bessel functions with other types of functions, such as trigonometric functions or exponential functions, using appropriate mathematical operations. However, the resulting function may not have a closed form solution and may require numerical methods for evaluation.

3. Is there a general formula for combining Bessel functions?

Unfortunately, there is no general formula for combining Bessel functions. The specific combination method depends on the problem at hand and the types of Bessel functions involved.

4. How do I handle complex arguments when combining Bessel functions?

If the arguments of the Bessel functions are complex, you can use the properties of complex numbers to simplify the expressions before combining them. In some cases, you may need to use the generalized Bessel functions (also known as the Hankel functions) to handle complex arguments.

5. Can I use software to combine Bessel functions?

Yes, you can use mathematical software such as MATLAB, Mathematica, or Maple to combine Bessel functions. These software packages have built-in functions for evaluating and manipulating Bessel functions, making it easier to combine them. However, it is important to understand the underlying mathematics to ensure the accuracy of the results.

Similar threads

I Infinite product representation of Bessel's function of the 2nd kind
    • Differential Equations
Replies
2
Views
1K
A Bessel functions of imaginary order
    • Differential Equations
Replies
2
Views
2K
I Modified Bessel Equation
    • Differential Equations
Replies
3
Views
2K
Symbolic integration of a Bessel function with a complex argument
    • MATLAB, Maple, Mathematica, LaTeX
Replies
5
Views
1K
I Integral representation of incomplete gamma function
    • Calculus
Replies
3
Views
1K
MHB What are Bessel Functions and how can they help solve differential equations?
    • Differential Equations
Replies
1
Views
10K
I What is the indefinite integral of Bessel function of 1 order (first k
    • Calculus
Replies
5
Views
998
I Limit of spherical bessel function of the second kind
    • Differential Equations
Replies
6
Views
2K
A Integral -- Beta function, Bessel function?
    • Calculus
Replies
1
Views
842
A Bessel's Integrals with Cosine or Sine?
    • Differential Equations
Replies
1
Views
2K

Hot Threads

Recent Insights

Back
Top