- #1
yungman
- 5,723
- 242
I am almost certain I understand the Bessel function expension correctly, but I just want to verify with you guys to be sure:
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
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