I already prove the orthogonality condition of Bessel functions for discrete case [imath](0,b)[/imath].
[math]\int_0^{b}\rho J_{\nu}(\chi_{\nu l}\rho/b)J_{\nu}(\chi_{\nu l'}\rho/b)d\rho = \frac{b^2}{2}[J_{\nu+1}(\chi_{\nu l})]^2\delta_{ll'}[/math]
Now, I need to prove that the orthogonality condition of Bessel Functions in the continuous case [imath](0,\infty)[/imath]. can be written as:
[math]\int_0^{\infty}\rho J_{\nu}(k\rho)J_{\nu}(k'\rho)d\rho = \frac{1}{k}\delta(k'-k)[/math]
With [imath]\chi_{\nu l} = kb[/imath] and [imath]\chi_{\nu l´} = k'b[/imath]. But I don't know how to do it.
Thank you for your help!!!
[math]\int_0^{b}\rho J_{\nu}(\chi_{\nu l}\rho/b)J_{\nu}(\chi_{\nu l'}\rho/b)d\rho = \frac{b^2}{2}[J_{\nu+1}(\chi_{\nu l})]^2\delta_{ll'}[/math]
Now, I need to prove that the orthogonality condition of Bessel Functions in the continuous case [imath](0,\infty)[/imath]. can be written as:
[math]\int_0^{\infty}\rho J_{\nu}(k\rho)J_{\nu}(k'\rho)d\rho = \frac{1}{k}\delta(k'-k)[/math]
With [imath]\chi_{\nu l} = kb[/imath] and [imath]\chi_{\nu l´} = k'b[/imath]. But I don't know how to do it.
Thank you for your help!!!