- #1
ebernardes
- 2
- 0
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:
$$\int_0^1 J_n(k_1r)J_n(k_2r)rdr = 0, (k_1≠k_2)$$
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?
$$\int_0^1 J_n(k_1r)J_n(k_2r)rdr = 0, (k_1≠k_2)$$
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?