Evaluation of Special function: A bug?

• To: mathgroup at christensen.cybernetics.net
• Subject: [mg1283] Evaluation of Special function: A bug?
• From: zhang at math.gwu.edu (Jun Zhang)
• Date: Wed, 31 May 1995 04:55:45 -0400
• Organization: GWU Department of Mathematics, Washington DC

```To Whom It May Share the Interest,

I'm using Mathematica Enhanced version 2.2.1 for windows
to evaluate the Bessel function of the second kind Y0(z).
The Mathematica built-in function is BesselY[0,z]. My problem
is when I specify the accuracy for 17 digits or more. Some
values of z can NOT be evaluated. For example:

z = 8 Exp[I 0] (this is real z), using
N[BesselY[0, 8 Exp[I 0]], 17]       is OK.
z = 8 Exp[I Pi/2] (purely imaginary,
N[BesselY[0, 8 Exp[I Pi/2]], 17]       is OK.
If z = 8 Exp[I Pi/4]   using
N[BesselY[0, 8 Exp[I Pi/4]], 17]  will not give any result,
Mathematica keeps evaluating it for hours.

The same is true for any complex number z = 8 Exp[i A], with A is
close to P4/4. Is there any reason for this?

Since I'm doing research on finding polynomial approximations for
special functions (especial Bessel functions Jn(z) and Yn(z)) with complex
argument. I'm interested in knowing what algorithm(s) is used in Mathematica
to evaluate BesselY[0,z] for arbitrary value z and to arbitrary accuracy.

By the way, N[BesselY[0,B Exp[i A]],17] is ok for small B.

Sincerely,

Jun Zhang

PS. I posted the same question on symbolic.net. Some users responded that
Maple has the some problem and it has drawn Maple developer's attention.
************************************************************************
* Jun Zhang                                                            *
* Department of Mathematics       *  e-mail: zhang at math.gwu.edu        *
* George Washington University    *  FAX: (202)994-6760 (Math Dept)    *
* Washington, DC 20052            *  Tel: (202)994-6886 (office)       *
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```

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