> From:Schirmacher, Rolf
>
> For my actual problem, all I need (at the moment, at least)
> is bessel_j(0,z) and bessel_j(1,z) for purely real and purely
> imaginary arguments z. Now, according to A&S 9.1.10, the
> series for bessel_j(0,z) contains only even powers of z, so
> it would be purely real for purely real or purely imaginary
> arguments. The series for bessel_j(1,z) contains an
> additional factor z, so it would be purely real for purely
> real z and purely imag for purely imag z.
>
Try modified Bessel functions.
A&S 9.6.3:
bessel_i(v,z)=exp(-v*%pi*%i/2)*bessel_j(v,z*exp(%pi*%i/2))
(-%pi<arg(z)<=%pi/2)
bessel_i(v,z)=exp(3*v*%pi*%i/2)*bessel_j(v,z*exp(-3*%pi*%i/2))
(%pi/2<arg(z)<=%pi)
therefore for x real (but check for sign errors)
bessel_j(0,%i*x) = bessel_i(0,x)
bessel_j(1,%i*x) = %i*bessel_i(1,x)
David
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