From mboxrd@z Thu Jan 1 00:00:00 1970 Return-Path: Received: (qmail 108774 invoked by alias); 27 Mar 2017 19:04:01 -0000 Mailing-List: contact gsl-discuss-help@sourceware.org; run by ezmlm Precedence: bulk List-Id: List-Subscribe: List-Archive: List-Post: List-Help: , Sender: gsl-discuss-owner@sourceware.org Received: (qmail 108749 invoked by uid 89); 27 Mar 2017 19:04:00 -0000 Authentication-Results: sourceware.org; auth=none X-Virus-Found: No X-Spam-SWARE-Status: No, score=-1.8 required=5.0 tests=AWL,BAYES_00,NO_RDNS_DOTCOM_HELO,RCVD_IN_DNSWL_LOW,RP_MATCHES_RCVD,SPF_PASS autolearn=ham version=3.3.2 spammy=gamma, HTo:U*gsl-discuss, Hx-spam-relays-external:ESMTPA X-HELO: vms173021pub.verizon.net Received: from vms173021pub.verizon.net (HELO vms173021pub.verizon.net) (206.46.173.21) by sourceware.org (qpsmtpd/0.93/v0.84-503-g423c35a) with ESMTP; Mon, 27 Mar 2017 19:03:59 +0000 Received: from vz-proxy-l006.mx.aol.com ([64.236.82.153]) by vms173021.mailsrvcs.net (Oracle Communications Messaging Server 7.0.5.32.0 64bit (built Jul 16 2014)) with ESMTPA id <0ONH00BQ8MA45QC0@vms173021.mailsrvcs.net> for gsl-discuss@sourceware.org; Mon, 27 Mar 2017 14:03:41 -0500 (CDT) X-CMAE-Score: 0 X-CMAE-Analysis: v=2.2 cv=JcfMlQCV c=1 sm=1 tr=0 a=FJ1kTJ0/xm5uTekQe8vMdQ==:117 a=hTZS6uLwhHpx5LOQAS/v3KalXBI=:19 a=IkcTkHD0fZMA:10 a=3RG-kA92mEAA:10 a=6Iz7jQTuP9IA:10 a=H1Z7X4ggLe3Cw73SRagA:9 a=QEXdDO2ut3YA:10 Received: by 173.67.5.14 with SMTP id 16938f63; Mon, 27 Mar 2017 19:03:40 GMT Subject: Re: Debye functions. To: gsl-discuss@sourceware.org References: <213cda3e-0b6c-d4bb-ca4c-f868245b791b@verizon.net> <70579fe6-ef32-5dac-9918-8f7f21a929a9@gmail.com> From: Ed Smith-Rowland <3dw4rd@verizon.net> Message-id: <59d21e6b-f304-8e8f-cc36-b6445a6e44b2@verizon.net> Date: Mon, 27 Mar 2017 19:04:00 -0000 User-Agent: Mozilla/5.0 (X11; Linux x86_64; rv:45.0) Gecko/20100101 Thunderbird/45.8.0 MIME-version: 1.0 In-reply-to: <70579fe6-ef32-5dac-9918-8f7f21a929a9@gmail.com> Content-type: text/plain; charset=utf-8; format=flowed Content-transfer-encoding: 7bit X-SW-Source: 2017-q1/txt/msg00002.txt.bz2 On 03/25/2017 05:24 PM, maxgacode wrote: > Il 24/03/2017 10:19, Ed Smith-Rowland ha scritto: >> Greetings, >> >> I've been looking at the Debye integrals >> >> D_n(x) = \frac{n}{x^n}\int_{0}^{x} \frac{t^n}{e^t - 1}dt >> >> The integrand is everywhere positive. >> >> The definite integral must be zero for x=0. > > But the 1/x factor goes to zero and so you get a 0/0 indeterminate > ratio. Computing the limit to zero returns 1.0! > >> >> The values returned by gsl debye functions start at one for x=0 and >> monotonically decrease. > > > Please note the factor > > \frac{n}{x^n} > > That factor is the responsible of the observed behavior. > > >> >> The definite integral of a positive functions must start at zero and >> monotonically increase. >> >> Is it possible that we have a complementary Debye integral? Perhaps >> scaled? >> >> In any case, the functions can't match the formulas in the manual. >> > > I don't think so. Please try to multiply the result of > gsl_sf_debye_n(x) by n/x^n and see. > > Moreover the Chapter 27 of Abramowitz and Stegun (page 998 of my ninth > edition) is listing the values of the Debye functions, you can easily > verify that GSL implementation is correct. > > > Hope this helps > > Max > Ah. This is just a convention. Wolfram and others lose the n/x^n. So the thins look sigmoid and level off at \Gamma(n+1)\zeta(n+1). Sorry for the noise. Ed