When finding the coefficients of a Fourier-Bessel series, the Bessel functions satisfies, for [itex]k_1[/itex]and [itex]k_2[/itex] both zeroes of [itex]J_n(t)[/itex], the orthogonality relation given by:(adsbygoogle = window.adsbygoogle || []).push({});

$$\int_0^1 J_n(k_1r)J_n(k_2r)rdr = 0, (k_1≠k_2)$$

and for [itex]k_1 = k_2 = k[/itex]:

$$\int_0^1 J_n^2(kr)rdr = \frac12J_n^{'2}(k)$$

I understand how to get the first result since the Bessel's equation can be interpreted as a Sturm-Liouville problem, but how can I show the second one?

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# Coefficients of a Fourier-Bessel series

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