vivishek
Jun27-08, 05:00 AM
This one's from mathematical physics. Given the hankel functions
H_m^(1)(x) of the first kind and H_m^(2)(x) of the second kind as
functions of x and order m, what is the weight function under which
they are orthogonal in the domain [0,\infty) over the order m i.e what
is w(x) such that \int_0^{\infty} w(x) H_m^(2)(x) H_n^(2)(x) dx =
\delta_{mn}? What is the similar relation for the hankel function of
the first kind? And are there orthogonalities or closure relations for
a mixture of both?
How do these results, of they exist, carry over to the spherical
hankel functions?
H_m^(1)(x) of the first kind and H_m^(2)(x) of the second kind as
functions of x and order m, what is the weight function under which
they are orthogonal in the domain [0,\infty) over the order m i.e what
is w(x) such that \int_0^{\infty} w(x) H_m^(2)(x) H_n^(2)(x) dx =
\delta_{mn}? What is the similar relation for the hankel function of
the first kind? And are there orthogonalities or closure relations for
a mixture of both?
How do these results, of they exist, carry over to the spherical
hankel functions?