A typical BVP of Bessel function is approximation of f(x) by a Bessel series expansion with y(0)=0 and y(a)=0, 0<x<a.(adsbygoogle = window.adsbygoogle || []).push({});

For example if we use [tex]J_{\frac{1}{2}}[/tex] to approximate f(x) on 0<x<1. Part of the answer contain

[tex]J_{\frac{1}{2}}=\sqrt{\frac{2}{\pi x}}sin(\alpha_{j}x), j=1,2,3....[/tex]

This will give [tex]\alpha_{j}=\pi,2\pi,3\pi.....[/tex]for j=1,2,3......

But in the books, they always have a table of [tex]\alpha_{j}[/tex] for each order of J. For example for [tex]J_{1},\alpha_{1}=3.83171,\alpha_{2}=7.01559,\alpha_{3}=10.1735[/tex] etc. This mean [tex]\alpha_{j}[/tex] is a constant for each order of the Bessel series. THis mean the zeros are a constant on the x-axis.

You see the above two example is contradicting each other. The series expansion show [tex]\alpha_{j}[/tex] depends on the boundaries, the second show [tex]\alpha_{j}[/tex] are constants!!!

Please tell me what did I miss.

Thanks

Alan

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# Question on zeros of a Bessel function.

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