I need to verify Bessel function expension.

In summary: J_{p}(\alpha_{j}x) (where the \alpha_j are the roots of J_p(x) ) since the eigenvalue k^2 is nondegenerate (there is only one eigenfunction for each eigenvalue).In summary, the conversation discusses the Bessel function expansion, where equations 1 and 3 appear to be correct. Equation 2 defines a function f(x) as a weighted sum of Bessel functions of order p and may be used to prove the identity of an integral. The use of the gamma function is also mentioned and a link is provided for further reading. The conversation also delves into the orthogonality of B
  • #1
yungman
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I am almost certain I understand the Bessel function expension correctly, but I just want to verify with you guys to be sure:

1) [tex]J_{p}(\alpha_{j}x)=\sum_{n=0}^{\infty}\frac{(-1)^{n}\alpha_{j}^{2n+p}x^{2n+p}}{n!\Gamma(n+p+1)2^{2n+p}}[/tex]

2) [tex]f(x)=\sum_{j=1}^{\infty}A_{j}J_{p}(\alpha_{j}x)=\sum_{j=1}^{\infty}[A_{j}\sum_{n=0}^{\infty}\frac{(-1)^{n}\alpha_{j}^{2n+p}x^{2n+p}}{n!\Gamma(n+p+1)2^{2n+p}}][/tex]

3) [tex]\int_{0}^{R}xJ_{p}(\alpha_{j}x)J_{p}(\alpha_{k}x)dx=\int_{0}^{R}x[\sum_{n=0}^{\infty}\frac{(-1)^{n}\alpha_{j}^{2n+p}x^{2n+p}}{n!\Gamma(n+p+1)2^{2n+p}}][\sum_{n=0}^{\infty}\frac{(-1)^{n}\alpha_{k}^{2n+p}x^{2n+p}}{n!\Gamma(n+p+1)2^{2n+p}}]dx[/tex]

Please take a look and let me know if I am correct or not from studying the books.

Thanks

Alan
 
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  • #2
Any body please!
 
  • #3
Equations 1 and 3 look correct.

I'm not sure what one is trying to with equation 2. One is simply defining f(x) as a weighted sum of Jp(ajx), but it looks OK.

Curious about using [tex]\Gamma(n+p+1)[/tex] as opposed to (n+p)!

One can also check - http://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html
 
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  • #4
Astronuc said:
Equations 1 and 3 look correct.

I'm not sure what one is trying to with equation 2. One is simply defining f(x) as a weighted sum of Jp(ajx), but it looks OK.

Curious about using [tex]\Gamma(n+p+1)[/tex] as opposed to (n+p)!

One can also check - http://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html

Thanks for your help. 2) is just expand the [tex] J_{p}(\alpha_{j}x) and J_{p}(\alpha_{k}x)[/tex] out only. The reason I put this in because I have problem doing integration following this logic to prove the identity of integral on [0,a] of [tex] J^{2}_{p}(\alpha_{j}x)[/tex]

I want to verify 2) so I can continue to figure out what I do wrong. With 2), I can pull out all the constant term, gamma function and all, then I only have to integrate[tex]x^{4n+2p+1}[/tex] on [0,a].


Regarding to the gamma function [tex]\Gamma(n+p+1)=(n+p)! [/tex] only if p is an integer. I have to read more on the link that you provide, I cannot pull it out of my head right at the moment, I have to do a little reading before I can answer that. All I know if p=1/2, your will have some number times [tex]\sqrt{\pi}[/tex]. I don't think you can get the answer using (n+p)!.
 
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  • #5
Astronuc said:
I'm not sure what one is trying to with equation 2. One is simply defining f(x) as a weighted sum of Jp(ajx), but it looks OK.

Equation 2 is taking an arbitrary function f(x) and writing it as an expansion of Bessel functions of order p.

In order to do this, f(x) can't blow up at the origin (or else you would have to include Bessel functions of the 2nd kind in your expansion). The Bessel functions of any order 'p' form a complete orthogonal set with weight x (and if your function doesn't blow up at the origin, then your expansion will only involve Bessel functions of the 1st kind of order 'p'), where the orthogonal set is [tex]J_p (\alpha_i x) [/tex] where [tex]\alpha_i=\frac{r_i}{R} [/tex] where the ri are the roots of the bessel function of order p, and x=R is a boundary where the function vanishes.

The orthogonality follows from the fact that [tex]J_m(kx)[/tex] is a solution of the Bessel equation (written here in Sturm-Lioville form):

[tex](xy')'+k^2xy-\frac{m^2}{x}y=0 [/tex]

which is a Hermitian linear differential operator with eigenvalue k^2 of weight x:

[tex]\mathcal Ly=[x\frac{d^2}{dx^2}+\frac{d}{dx}-\frac{m^2}{x}]y=-k^2xy [/tex]

when appropriate boundary conditions are applied, which is done by discretizing or quantizing the allowed eigenvalues k to [tex]\frac{r_i}{R} [/tex]
 

Related to I need to verify Bessel function expension.

1. What is a Bessel function expansion?

A Bessel function expansion is a mathematical technique used to express a function as a sum of Bessel functions. Bessel functions are a special type of solution to the Bessel differential equation, which arises in many physical and engineering problems.

2. Why do I need to verify a Bessel function expansion?

Verifying a Bessel function expansion is important to ensure the accuracy of the solution and to check for any potential errors or mistakes in the calculation. It also allows for a better understanding of the behavior of the function and its relationship to other mathematical concepts.

3. How do I verify a Bessel function expansion?

To verify a Bessel function expansion, you can use the properties and identities of Bessel functions to simplify the expression and compare it to known values or other alternative expressions. You can also use numerical methods and computer software to evaluate the expansion and check for consistency.

4. What are some applications of Bessel function expansion?

Bessel function expansion has many applications in various fields, such as physics, engineering, and mathematics. It is commonly used in solving problems involving wave phenomena, such as sound waves and electromagnetic waves. It also has applications in heat transfer, quantum mechanics, and signal processing.

5. Are there any limitations to using Bessel function expansion?

While Bessel function expansion is a powerful mathematical tool, it may not be suitable for all types of functions and problems. It is most effective for functions that exhibit cylindrical symmetry, and it may not provide accurate results for functions with complicated or irregular shapes. In addition, the convergence of the expansion may be slow, so alternate methods may be needed in some cases.

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