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What did I do wrong on this Bessel expansion?

  1. Dec 26, 2009 #1
    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


    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

  2. jcsd
  3. Dec 26, 2009 #2
    Anyone please!!
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