- #1
hmhm696
- 3
- 0
Is there somebody who can help me how to solve this integral
[tex]
\int_{0}^{+\infty} dr r^{^{n+1}} e^{-\alpha r} j_l(kr)
[/tex]
[tex]
\int_{0}^{+\infty} dr r^{^{n+1}} e^{-\alpha r} j_l(kr)
[/tex]
Last edited:
Integral over spherical Bessel function
- Thread starter hmhm696
- Start date
In summary, the conversation revolves around finding the solution to the integral \int_{0}^{+\infty} dr r^{^{n+1}} e^{-\alpha r} j_l(kr)I using Mathematica and relating the spherical Bessel function to the normal Bessel function. The answer given by Mathematica involves using the Hypergeometric2F1Regularized function and the formula from the Gradshteyn-Ryzhik book. However, there is some discussion about potential errors in the formula and the importance of understanding the process of obtaining the solution.
- #2
- 13,353
- 3,212
I have a saying for this type of questions. If it's not in one of the Gradshteyn-Rytzhik editions of their famous book, then it must be discovered. :)
- #3
Dickfore
- 2,987
- 5
This is the answer Mathematica gives:
- #4
- 13,353
- 3,212
I think you can relate the spherical Besselfunction to the normal J Bessel function by the definition:
http://functions.wolfram.com/Bessel-TypeFunctions/SphericalBesselJ/02/
and then use formula attached below which is taken from G & R
http://functions.wolfram.com/Bessel-TypeFunctions/SphericalBesselJ/02/
and then use formula attached below which is taken from G & R
- #5
hmhm696
- 3
- 0
Thanks guys.
I think that given formula has error in the part where it derivative by alpha instead of betha.
For me is important a process, how i can get it.
I think that given formula has error in the part where it derivative by alpha instead of betha.
For me is important a process, how i can get it.
Related to Integral over spherical Bessel function
1. What is an integral over spherical Bessel function?
An integral over spherical Bessel function is a mathematical calculation that involves integrating a spherical Bessel function over a certain range of values. Spherical Bessel functions are a type of special function that appear in many areas of physics and engineering, particularly in problems involving spherical symmetry.
2. How is an integral over spherical Bessel function solved?
The solution to an integral over spherical Bessel function involves using integration techniques, such as integration by parts or substitution, to simplify the integral. In some cases, the integral can be solved analytically, while in others it may require the use of numerical methods.
3. What is the significance of an integral over spherical Bessel function?
An integral over spherical Bessel function is often used in solving problems in physics and engineering, specifically in situations involving spherical symmetry. It is also used in the solution of differential equations, particularly in problems with cylindrical or spherical symmetry.
4. What are some applications of an integral over spherical Bessel function?
Some common applications of an integral over spherical Bessel function include solving problems involving electromagnetic fields, acoustic waves, and heat transfer in spherical geometries. It is also used in the solution of problems in quantum mechanics and fluid dynamics.
5. Are there any special properties of an integral over spherical Bessel function?
Yes, an integral over spherical Bessel function has several special properties, such as orthogonality and recursion. These properties make it a useful tool in solving differential equations and in the analysis of physical systems with spherical symmetry.
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