(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I cannot get the answer given by the book. The question is:

Using Bessel function of order = 2 to represent f(x):

f(x)=0 for 0<x<1/2 and f(x)=1 for 1/2<x<1.

The Answer given by the book is [tex]-2\sum_{j=1}^{\infty} \frac{J_{1}(\alpha_{2,j})-2J_{1}(\frac{\alpha_{2,j}}{2})}{\alpha_{2,j}J_{1}(\alpha_{2,j})^{2}}J_{2}(\alpha_{2,j}x)[/tex]

I got everything correct except the deminator where I have

[tex]\alpha_{2,j}J_{3}(\alpha_{j})^{2}[/tex]

I know there is a way to reduce the order if it start with order of 1, I cannot reduce the order of 3 to 1.

2. Relevant equations

[tex]\int x^{-p+1}J_{p}(x)dx=-x^{-p+1}J_{p-1}(x)+C[/tex] for [tex]p=2\Rightarrow \int x^{-1}J_{2}(x)dx=-x^{-1}J_{1}(x)+C[/tex]

3. The attempt at a solution

We let [tex] s=\alpha_{j}x \Rightarrow \frac{ds}{\alpha_{j}}=dx, x=\frac{s}{\alpha_{j}},a=1[/tex]

[tex]A_{j}=\frac{\int_{0}^{a}xf(x)J_{2}(\lambda_{j}x)dx}{\int_{0}^{a}xJ_{2}(\lambda_{j}x)^{2}dx}=\frac{\int_{\frac{1}{2}}^{1}x \frac{1}{x^{2}} J_{2}(\alpha_{j}x)dx}{\frac{a^{2}}{2}J_{3}(\alpha_{j})^{2}dx}=\frac{2\int_{\frac{\alpha_{j}}{2}}^{\alpha_{j}}\alpha_{j}s^{-1}J_{2}(s)\frac{ds}{\alpha_{J}}}{J_{3}(\alpha_{j})^{2}} = \frac{-2}{J_{3}(\alpha_{j})^{2}}[\frac{J_{1}(s)}{s}]_{\frac{\alpha_{j}}{2}}^{\alpha_{j}}[/tex]

[tex]A_{j}=\frac{-2}{J_{3}(\alpha_{j})^{2}}[\frac{J_{1}(\alpha_{j})}{\alpha_{j}}-\frac{2J_{1}(\frac{\alpha_{j}}{2})}{\alpha_{j}}] = \frac{-2[J_{1}(\alpha_{j})-2J_{1}(\frac{\alpha_{j}}{2})]}{\alpha_{j}J_{3}(\alpha_{j})^{2}}[/tex]

[tex] f(x)=\sum_{j=1}^{\infty}A_{j}J_{2}(\lambda_{j}x) = -2\sum_{j=1}^{\infty}\frac{[J_{1}(\alpha_{j})-2J_{1}(\frac{\alpha_{j}}{2})]}{\alpha_{j}J_{3}(\alpha_{j})^{2}}J_{2}(\alpha_{j}x)[/tex]

Thanks for your time and Merry Christmas

Alan

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: What did I do wrong on this Bessel expansion?

**Physics Forums | Science Articles, Homework Help, Discussion**