Integral of first order (first kind) bessel function

In summary, the conversation is about a problem involving an integral with J1 as the first order Bessel function. The question is whether there is an analytical solution for the integral and if it is possible to integrate it numerically. The answer is that the anti-derivative of the Bessel function can be expressed as another Bessel function and the behavior of the integral at zero and infinity needs to be studied. The person seeking help is grateful for the clarification and mentions that it has been a while since they have dealt with Bessel functions.
  • #1
Pratyush
3
0
hello,

while working on a problem i encountered the following integral :(limits are zero and infinity)

Integral[J1(kR)dk]

J1 is the first order bessel function..cudnt put 1 in subscripts..

Is there an analytical solution for this?? also is it possible to integrate it numerically?

please help. thanks.
 
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  • #2
The answer is 1/R.

Check some basic properties of Bessel function, you will see that anti-derivatives of Bessel function can be expressed again as Bessel functions, then you have to study the asymptotic behavior of those ...
 
  • #3
hey thanks a lot...

i got daunted by the zero and the infinity...been long since i saw bessel functions...thanks
 

1. What is the definition of the integral of first order Bessel function?

The integral of first order Bessel function is a mathematical operation that calculates the area under the curve of a Bessel function of the first kind. It is represented by the symbol ∫J0(x)dx and is used in various fields of science and engineering.

2. What is the significance of the integral of first order Bessel function?

The integral of first order Bessel function is used to solve problems involving oscillatory systems, such as in acoustics, electromagnetism, and quantum mechanics. It is also used in the analysis of cylindrical and spherical geometries.

3. How is the integral of first order Bessel function calculated?

The integral of first order Bessel function can be calculated using various methods, such as numerical integration techniques or by using special functions such as the incomplete gamma function. It can also be approximated using series expansions or asymptotic expansions.

4. What is the relationship between the integral of first order Bessel function and the Bessel function itself?

The integral of first order Bessel function is closely related to the Bessel function itself. In fact, the integral can be expressed in terms of the Bessel function using a special property known as the Laplace transform. This relationship is useful in solving certain types of differential equations.

5. Can the integral of first order Bessel function be generalized to higher order Bessel functions?

Yes, the integral of first order Bessel function can be generalized to higher order Bessel functions, such as the second order Bessel function (J1(x)), third order Bessel function (J2(x)), and so on. However, the calculation methods may differ for each order of Bessel function.

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