Integrating product of bessel function,

Another option is to use a computer program, such as MATLAB, to calculate the integral numerically. This will give you a more accurate result than attempting to find an analytical solution. In summary, the integral in question cannot be calculated directly and requires the use of numerical integration or a computer program to approximate its value.
  • #1
thomitsu
6
0
Hallo there. I m trying to integrate a bessel function but with no great success... I thing it can't be calculated..
I m trying to simulate the airy pattern of a certain aperture radius and wavelength in matlab.

the integral is : int (besselj(1,16981.9*sin(x)))^2/ sin(x) dx

where you can change 1/sinx to csc(x) so the integral becomes

int csc(x)*(besselj(1,16981.9*sin(x)))^2 dx x from 0 to 0.0002257

the problem is that there is dx and not d(sinx) ( it whould be easy then..)

any ideas how can I do this??


thanks in advance!
 
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  • #2
Unfortunately, it is not possible to integrate this expression directly. The best way to approach this problem is to use numerical integration methods such as the trapezoidal rule or Simpson's rule. These methods allow you to approximate the value of the integral without needing to find an exact solution.
 

1. What is a Bessel function?

A Bessel function is a mathematical function that is used to describe certain physical phenomena, such as heat flow and sound waves. It is named after the mathematician Friedrich Bessel and is commonly denoted by the symbol J.

2. How is the product of Bessel functions integrated?

The product of Bessel functions is integrated using a technique known as integration by parts. This involves splitting the integral into two parts and then using the product rule to simplify the integration.

3. What is the significance of integrating the product of Bessel functions?

Integrating the product of Bessel functions is important in many areas of science and engineering. It can be used to solve differential equations that arise in physics, such as in heat conduction and electromagnetism. It is also used in signal processing and image reconstruction techniques.

4. Are there any special cases when integrating the product of Bessel functions?

Yes, there are several special cases when integrating the product of Bessel functions. One example is when the product of Bessel functions is integrated over the entire real line, resulting in a special function known as the Dirichlet kernel. Another special case is when one of the Bessel functions is replaced by its modified form, resulting in a modified Bessel function integral.

5. Can the integration of the product of Bessel functions be generalized to higher dimensions?

Yes, the integration of the product of Bessel functions can be extended to higher dimensions. In this case, the Bessel functions are replaced by their higher-dimensional equivalents, such as the spherical Bessel functions. The integration technique remains the same, but the resulting integrals may have different forms and properties.

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