Coefficients of a Fourier-Bessel series

In summary, the coefficients of a Fourier-Bessel series are used to represent a function on a domain by breaking it down into a series of components or "harmonics". These coefficients are calculated by taking the inner product of the function with the corresponding basis function, which in this case are the Bessel functions. The resulting series can then be used to approximate the original function and study its behavior and properties. This method is particularly useful for functions that are defined on a circular or spherical domain.
  • #1
ebernardes
2
0
When finding the coefficients of a Fourier-Bessel series, the Bessel functions satisfies, for [itex]k_1[/itex]and [itex]k_2[/itex] both zeroes of [itex]J_n(t)[/itex], the orthogonality relation given by:
$$\int_0^1 J_n(k_1r)J_n(k_2r)rdr = 0, (k_1≠k_2)$$
and for [itex]k_1 = k_2 = k[/itex]:

$$\int_0^1 J_n^2(kr)rdr = \frac12J_n^{'2}(k)$$

I understand how to get the first result since the Bessel's equation can be interpreted as a Sturm-Liouville problem, but how can I show the second one?
 
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  • #2
This paper might offer a clue:
 

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  • #3
SteamKing said:
This paper might offer a clue:

But how exactly i can evaluate the integral and get the result given by: [tex] I = \frac{R^2}{\alpha_m^2 - \alpha_n^2}[\alpha_mJ_0(\alpha_n)J_1(\alpha_m) - \alpha_nJ_0(\alpha_m)J_1(\alpha_n)][/tex] or another more general formula for Bessel functions of different order?
 
  • #4
I believe an integration by parts is called for, using certain relations of Bessel functions to get over the tricky bits:

http://home.comcast.net/~rmorelli146/U3150/Bessel.pdf
 
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1. What is a Fourier-Bessel series?

A Fourier-Bessel series is a type of mathematical series used in the field of Fourier analysis. It is an infinite sum of sine and cosine functions multiplied by a set of coefficients, known as the Bessel coefficients. This series is commonly used to represent functions that are periodic in nature.

2. How is a Fourier-Bessel series different from a Fourier series?

While both Fourier and Fourier-Bessel series are used to represent periodic functions, the main difference is in the choice of basis functions. A Fourier series uses trigonometric functions (sine and cosine), while a Fourier-Bessel series uses Bessel functions. This makes the Fourier-Bessel series more suitable for functions that are not necessarily periodic, but have a finite range of values.

3. What are the applications of Fourier-Bessel series?

Fourier-Bessel series are commonly used in fields such as physics, engineering, and signal processing. They are particularly useful in solving partial differential equations, as well as representing functions with cylindrical or spherical symmetry, such as in electromagnetism and quantum mechanics.

4. How are the coefficients of a Fourier-Bessel series calculated?

The coefficients of a Fourier-Bessel series can be calculated using an integral formula or by solving a system of linear equations. The integral formula involves multiplying the function being represented by the appropriate Bessel function and integrating over the interval of interest. Solving a system of linear equations involves using orthogonality properties of Bessel functions.

5. Are there any limitations to using Fourier-Bessel series?

While Fourier-Bessel series can be used to represent a wide range of functions, they do have some limitations. They are not effective for functions that have discontinuities or sharp corners, as they may result in Gibbs phenomenon. They are also not suitable for functions that have an infinite range of values, as they may not converge in those cases.

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