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. > > --------------------------------------------------- > > > > > > > > > > > > -- Paul Abbott Phone: +61 8 9380 2734 School of Physics, M013 Fax: +61 8 9380 1014 The University of Western Australia (CRICOS Provider No 00126G) 35 Stirling Highway Crawley WA 6009 mailto:paul at physics.uwa.edu.au AUSTRALIA http://physics.uwa.edu.au/~paul