Bill Allombert on Fri, 10 Feb 2012 15:26:44 +0100

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bessel function algorithm

Hello PARI users,

I forward this post from Marcel Bezerra about the Bessel function
which was misfilled.

From: Marcel Bezerra <>
Date: Fri, 10 Feb 2012 11:38:57 -0200


I'm working with cylindrical electromagnetic waveguides. The conditions I'm
studying now need solutions using Bessel functions with complex order. I
need to calculate them numerically, so I started to look for software
capable to calculate such functions.

I need some info about the algorithms used to calculate bessel functions. I
want to know what algorithm is used to calculate bessel functions with
complex order and argument in PARI/GP.

I found some papers regarding this subject. The first one is the Algorithm
644 from Amos. But his paper says his method is applied for nonnegative
order. The second one is from Masao Kodama, that is exactly an algorithim
about complex order and argument.

I used a software called Mathematica, that showed in an example the bessel
function of order 7.3 + 1.0*I and argument 4.5-1*I. But they don't make any
reference about the algorithm they use. I found out the PARI/GP software
also is capable to deal with complex order, and tested the same example and
I got the same result compared to Mathematica.

I need the info of the algorithm used in PARI to calculate bessel functions
in order to know if there are any limitations of order, argument, precision
and so on.

Thank you in advance!

Regards from Brazil!

Marcel Bezerra