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Mathematics LibreTexts

10.8: Fourier-Bessel Series

( \newcommand{\kernel}{\mathrm{null}\,}\)

So how can we determine in general the coefficients in the Fourier-Bessel series

f(ρ)=j=1CjJν(αjρ)?

The corresponding self-adjoint version of Bessel’s equation is easily found to be (with Rj(ρ)=Jν(αjρ))

(ρRj)+(α2jρν2ρ)Rj=0.

Where we assume that f and R satisfy the boundary condition

b1f(c)+b2f(c)=0b1Rj(c)+b2Rj(c)=0

From Sturm-Liouville theory we do know that c0ρJν(αiρ)Jν(αjρ)=0if ij,

but we shall also need the values when i=j!

Let us use the self-adjoint form of the equation, and multiply with 2ρR, and integrate over ρ from 0 to c,

c0[(ρRj)+(α2jρν2ρ)Rj]2ρRjdρ=0.

This can be brought to the form (integrate the first term by parts, bring the other two terms to the right-hand side)

c0ddρ(ρRj)2dρ=2ν2c0RjRjdρ2α2jc0ρ2RjRjdρ(ρRj)2|c0=ν2R2j|c02α2jc0ρ2RjRjdρ.

The last integral can now be done by parts: 2c0ρ2RjRjdρ=2c0ρR2jdρ+ρR2j|c0.

So we finally conclude that

2α2jc0ρR2jdρ=[(α2jρ2ν2)R2j+(ρRj)2|c0.

In order to make life not too complicated we shall only look at boundary conditions where f(c)=R(c)=0. The other cases (mixed or purely f(c)=0) go very similar! Using the fact that Rj(r)=Jν(αjρ), we find Rj=αjJν(αjρ).

We conclude that

2α2jc0ρ2R2jdρ=[(ραjJν(αjρ))2|c0=c2α2j(Jν(αjc))2=c2α2j(ναjcJν(αjc)Jν+1(αjc))2=c2α2j(Jν+1(αjc))2,

We thus finally have the result c0ρ2R2jdρ=c22J2ν+1(αjc).

Example 10.8.1:

Consider the function

f(x)={x30<x<100x>10

Expand this function in a Fourier-Bessel series using J3.

Solution

From our definitions we find that

f(x)=j=1AjJ3(αjx),

with

Aj=2100J4(10αj)2100x3J3(αjx)dx=2100J4(10αj)21α5j10αj0s4J3(s)ds=2100J4(10αj)21α5j(10αj)4J4(10αj)ds=200αjJ4(10αj).

Using αj=, we find that the first five values of Aj are 1050.95,821.503,703.991,627.577,572.301. The first five partial sums are plotted in Fig. 10.8.1.

Besselx3.png
Figure 10.8.1: A graph of the first five partial sums for x3 expressed in J3.

This page titled 10.8: Fourier-Bessel Series is shared under a CC BY-NC-SA 2.0 license and was authored, remixed, and/or curated by Niels Walet via source content that was edited to the style and standards of the LibreTexts platform.

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