> From: Billinghurst, David
> > 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)
>
I forgot to say that Bessel functions of real order and argument
are real, so that for real x
realpart(bessel_j(0,%i*x)) = bessel_i(0,x)
imagpart(bessel_j(0,%i*x)) = 0
realpart(bessel_j(1,%i*x)) = 0
imagpart(bessel_j(1,%i*x)) = bessel_i(1,x)
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