Integral of spherical bessel function (first kind), first order

In summary, the integral of spherical bessel function (first kind), first order is a mathematical operation used to calculate the area under a spherical bessel function curve with a given order. It is calculated using numerical methods or analytically for specific values. This integral is significant in various areas of physics and has practical applications in antenna design, scattering and diffraction analysis, and solving problems in fluid mechanics and elasticity. Alternative functions and methods, such as the integral of spherical bessel function (second kind) and other spherical functions, can also be used for problems involving spherical symmetry.
  • #1
efrenfer
2
0
Hello,
I am trying to solve the following integral (limits from 0 to inf).

∫j_1(kr) dr

where j_1 is the first order SPHERICAL Bessel function of the first kind, of argument (k*r). Unfortunately, I cannot find it in the tables, nor manage to solve it... Can anybody help?

Thanks a lot! Any help will be very appreciated.

P.S: I came across this integral trying to apply analytically a direct filtering operation (convolution) in space domain to a point source (in a 3D space).
 
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  • #2
Thanks to WolframAlpha :
 

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  • #3
Alright! Thanks a lot!
 

1. What is the definition of the integral of spherical bessel function (first kind), first order?

The integral of spherical bessel function (first kind), first order is a mathematical operation that calculates the area under the curve of a spherical bessel function (first kind) with a given order. It is often used in physics and engineering to solve problems involving spherical symmetry.

2. How is the integral of spherical bessel function (first kind), first order calculated?

The integral of spherical bessel function (first kind), first order can be calculated using various numerical methods, such as the trapezoidal rule or Simpson's rule. It can also be calculated analytically for specific values of the order and arguments.

3. What is the significance of the integral of spherical bessel function (first kind), first order?

The integral of spherical bessel function (first kind), first order is important in many areas of physics, such as electromagnetism, quantum mechanics, and acoustics. It is used to solve boundary value problems and to analyze the behavior of waves and fields in spherical systems.

4. What are the applications of the integral of spherical bessel function (first kind), first order?

The integral of spherical bessel function (first kind), first order has many practical applications, such as in the design of antennas, analysis of scattering and diffraction phenomena, and calculation of electric and magnetic fields in spherical systems. It is also used in solving problems in fluid mechanics, heat transfer, and elasticity.

5. Are there any alternative functions or methods to the integral of spherical bessel function (first kind), first order?

Yes, there are alternative functions and methods that can be used to solve problems that involve spherical symmetry. These include the integral of spherical bessel function (second kind), first and second order, as well as other types of spherical functions such as spherical harmonics. Additionally, there are numerical methods such as finite difference and finite element methods that can be used to solve problems in spherical systems.

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