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Re: bessel function with complex order

• To: mathgroup at smc.vnet.net
• Subject: [mg46841] Re: bessel function with complex order
• From: Paul Abbott <paul at physics.uwa.edu.au>
• Date: Wed, 10 Mar 2004 04:57:40 -0500 (EST)
• Organization: The University of Western Australia
• References: <c2lpqj$9du$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

In article <c2lpqj$9du$1 at smc.vnet.net>,
 cirella at fis.uniroma3.it (cirella antonella) wrote:

> I'm looking for the definition used by Mathematica 4.0 to evaluate
> the Bessel function of the first kind with COMPLEX order.
> Generally Bessel function of the first kind are defined only for REAL
> order, 

This is not correct. There standard definition of the Bessel function is 
valid for unrestricted \[Nu]. For example, in Abramowitz and Stegun, 
BesselJ[\[Nu],z], can by defined by (and directly computed using) 
9.1.10, 9.1.20, 9.1.22, 9.1.24, 9.1.26, 9.1.69 or 9.1.70 for complex 
order \[Nu].

Alternatively, at

  http://functions.wolfram.com/BesselAiryStruveFunctions/BesselJ/

you will see that the primary definition (which corresponds to 9.1.10) 
is valid for complex order \[Nu].

> but I have found that Mathematica can calculate Bessel function
> with COMPEX order.
>  I would know:
> 1)how Mathematica defines Bessel function of first kind with COMPLEX
> order?
> 2)how Mathematica estimates real and imaginary part of Bessel function
> with COMPLEX order?

Since I don't have access to the Mathematica internals, I cannot answer 
this question with certainty. However, using a hypergeometric form, e.g.,

 http://functions.wolfram.com/BesselAiryStruveFunctions/BesselJ/26/01/02/

your question is related to the more general question asking how does 
Mathematica compute hypergeometric functions involving complex 
parameters.

Anyway, here are three ways to compute BesselJ[n,z] with

  n = 1+I;
  z = 1.0;

Directly:
 
  BesselJ[n,z]

Using 9.1.10:

  (z/2)^n/Gamma[n + 1] Hypergeometric0F1[n + 1, -(z^2/4)]

Using 9.1.20:

  2 (z/2)^n NIntegrate[(1 - t^2)^(n - 1/2) Cos[z t], {t, 0, 1}]/
    (Gamma[n + 1/2] Sqrt[Pi])

The answers are consistent.

Cheers,
Paul 


> I have searched in the Wolfram web site and in the references:
> listed below (cited in the Mathematica help pages) but I have not
> found these informations.
> 
> http://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html
> 
> Abramowitz M., and Stegun I.A. (Eds.), 'Bessel Functions J and Y.'
> in 'Handbook of Mathematical functions with Formulas , Graphs , and
> Mathematical Tables' ,9th printing . New York : Dover ,pp.358 - 365.
> 
> Arfken G., 'Bessel Functions of the First Kind,J(x)',
> 3rd ed., Orlando , FL: Academic Press, pp.573-591 and 591-596,1985.
> 
> Watson G.N., 'A treatise on the Theory of Bessel Functions',
> 2nd ed. Cambridge, England.
> 
> Could you clarify me the Mathematica's definition od this function?
> I need this information as soon as in your possibility.
> So, I greatly appreciate your contribution and I thank you in advance
> for
> your help.
> 
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-- 
Paul Abbott                                   Phone: +61 8 9380 2734
School of Physics, M013                         Fax: +61 8 9380 1014
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Crawley WA 6009                      mailto:paul at physics.uwa.edu.au 
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