I am almost certain I understand the Bessel function expension correctly, but I just want to verify with you guys to be sure:(adsbygoogle = window.adsbygoogle || []).push({});

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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# I need to verify Bessel function expension.

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