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<!DOCTYPE html><html lang="en"> <head> <meta http-equiv="content-type" content="text/html; charset=UTF-8"> <title>DLMF: Notices ‣ About the Project</title> <link rel="shortcut icon" href=".././style/DLMF-16.png" type="image/png"> <script type="text/javascript"></script> <script type="text/javascript" src=".././style/jquery.js"></script> <script type="text/javascript" src=".././style/jquery.leaveNotice.js"></script> <script type="text/javascript" src=".././style/DLMF.js"></script> <script async="" type="text/javascript" id="_fed_an_ua_tag" src="https://dap.digitalgov.gov/Universal-Federated-Analytics-Min.js?agency=DOC&subagency=NIST&pua=UA-37115410-44&yt=true&exts=ppsx,pps,f90,sch,rtf,wrl,txz,m1v,xlsm,msi,xsd,f,tif,eps,mpg,xml,pl,xlt,c"></script><script async="" src="https://www.googletagmanager.com/gtag/js?id=G-BZB64W4M0L"></script><script> window.dataLayer = window.dataLayer || []; function gtag(){dataLayer.push(arguments);} gtag('js', new Date()); gtag('config', 'G-BZB64W4M0L'); </script> <link rel="stylesheet" type="text/css" href=".././style/LaTeXML.css"> <link rel="stylesheet" type="text/css" href=".././style/DLMF.css"> <link rel="stylesheet" type="text/css" href=".././style/print.css" media="print"> <meta name="viewport" content="width=device-width, initial-scale=1"> <link rel="top" href=".././" title="DLMF"> <link rel="copyright" href=".././about/notices" title="Copyright"> <link rel="search" href=".././search/search" title="Search"> <link rel="help" class="ltx_help" href=".././help/" title="Help"> <link rel="up" href=".././about/." title="About the Project"> <link rel="start" href=".././" title=""> <link rel="prev" href=".././about/." title="About the Project"> <link rel="next" href=".././about/preface" title="Preface ‣ About the Project"> <link rel="document" href=".././about/preface" title="Preface ‣ About the Project"> <link rel="document" href=".././about/sponsors" title="Funding ‣ About the Project"> <link rel="document" href=".././about/news/" title="DLMF Project News ‣ About the Project"> <link rel="document" href=".././about/publications/" title="DLMF Related Publications ‣ About the Project"> <link rel="document" href=".././about/staff" title="Project Staff ‣ About the Project"> <link rel="document" href=".././about/bio/MJAblowitz" title="Profile Mark J. Ablowitz ‣ About the Project"> <link rel="document" href=".././about/bio/GEAndrews" title="Profile George E. Andrews ‣ About the Project"> <link rel="document" href=".././about/bio/BAntonishek" title="Profile Brian Antonishek ‣ About the Project"> <link rel="document" href=".././about/bio/TMApostol" title="Profile Tom M. Apostol ‣ About the Project"> <link rel="document" href=".././about/bio/RAAskey" title="Profile Richard A. Askey ‣ About the Project"> <link rel="document" href=".././about/bio/ARBarnett" title="Profile Antony Ross Barnett ‣ About the Project"> <link rel="document" href=".././about/bio/MVBerry" title="Profile Michael V. Berry ‣ About the Project"> <link rel="document" href=".././about/bio/AIBobenko" title="Profile Alexander I. Bobenko ‣ About the Project"> <link rel="document" href=".././about/bio/RFBoisvert" title="Profile Ronald F. Boisvert ‣ About the Project"> <link rel="document" href=".././about/bio/BLJBraaksma" title="Profile Boele L. J. 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Ismail ‣ About the Project"> <link rel="document" href=".././about/bio/AIts" title="Profile Alexander A. Its ‣ About the Project"> <link rel="document" href=".././about/bio/BRJudd" title="Profile Brian R. Judd ‣ About the Project"> <link rel="document" href=".././about/bio/RKoekoek" title="Profile Roelof Koekoek ‣ About the Project"> <link rel="document" href=".././about/bio/THKoornwinder" title="Profile Tom H. Koornwinder ‣ About the Project"> <link rel="document" href=".././about/bio/VBKuznetsov" title="Profile Vadim B. Kuznetsov ‣ About the Project"> <link rel="document" href=".././about/bio/DWLozier" title="Profile Daniel W. Lozier ‣ About the Project"> <link rel="document" href=".././about/bio/LCMaximon" title="Profile Leonard C. Maximon ‣ About the Project"> <link rel="document" href=".././about/bio/MAMcClain" title="Profile Marjorie A. McClain ‣ About the Project"> <link rel="document" href=".././about/bio/BRMiller" title="Profile Bruce R. Miller ‣ About the Project"> <link rel="document" href=".././about/bio/VHMoll" title="Profile Victor Moll ‣ About the Project"> <link rel="document" href=".././about/bio/RJMuirhead" title="Profile Robb J. Muirhead ‣ About the Project"> <link rel="document" href=".././about/bio/GNemes" title="Profile Gergő Nemes ‣ About the Project"> <link rel="document" href=".././about/bio/ENeuman" title="Profile Edward Neuman ‣ About the Project"> <link rel="document" href=".././about/bio/MNewman" title="Profile Morris Newman ‣ About the Project"> <link rel="document" href=".././about/bio/IOlkin" title="Profile Ingram Olkin ‣ About the Project"> <link rel="document" href=".././about/bio/FWJOlver" title="Profile Frank W. J. Olver ‣ About the Project"> <link rel="document" href=".././about/bio/RBParis" title="Profile Richard B. Paris ‣ About the Project"> <link rel="document" href=".././about/bio/PPaule" title="Profile Peter Paule ‣ About the Project"> <link rel="document" href=".././about/bio/JWPitman" title="Profile Jim Pitman ‣ About the Project"> <link rel="document" href=".././about/bio/WPReinhardt" title="Profile William P. Reinhardt ‣ About the Project"> <link rel="document" href=".././about/bio/DSPRichards" title="Profile Donald St. P. Richards ‣ About the Project"> <link rel="document" href=".././about/bio/RRoy" title="Profile Ranjan Roy ‣ About the Project"> <link rel="document" href=".././about/bio/SRuijsenaars" title="Profile Simon Ruijsenaars ‣ About the Project"> <link rel="document" href=".././about/bio/BVSaunders" title="Profile Bonita V. Saunders ‣ About the Project"> <link rel="document" href=".././about/bio/BISchneider" title="Profile Barry I. Schneider ‣ About the Project"> <link rel="document" href=".././about/bio/JJSegura" title="Profile Javier Segura ‣ About the Project"> <link rel="document" href=".././about/bio/BDSleeman" title="Profile Brian D. Sleeman ‣ About the Project"> <link rel="document" href=".././about/bio/RFSwarttouw" title="Profile René F. Swarttouw ‣ About the Project"> <link rel="document" href=".././about/bio/NMTemme" title="Profile Nico M. Temme ‣ About the Project"> <link rel="document" href=".././about/bio/IJThompson" title="Profile Ian J. Thompson ‣ About the Project"> <link rel="document" href=".././about/bio/HVolkmer" title="Profile Hans Volkmer ‣ About the Project"> <link rel="document" href=".././about/bio/PLWalker" title="Profile Peter L. Walker ‣ About the Project"> <link rel="document" href=".././about/bio/GWiersma" title="Profile Gloria Wiersma ‣ About the Project"> <link rel="document" href=".././about/bio/SMWatt" title="Profile Stephen M. Watt ‣ About the Project"> <link rel="document" href=".././about/bio/QWang" title="Profile Qiming Wang ‣ About the Project"> <link rel="document" href=".././about/bio/GWolf" title="Profile Gerhard Wolf ‣ About the Project"> <link rel="document" href=".././about/bio/RWong" title="Profile Roderick S. C. Wong ‣ About the Project"> <link rel="document" href=".././about/bio/AYoussef" title="Profile Abdou Youssef ‣ About the Project"> <link rel="document" href=".././about/bio/JVanderJeugt" title="Profile Joris Van der Jeugt ‣ About the Project"> <link rel="document" href=".././about/bio/WGroenevelt" title="Profile Wolter Groenevelt ‣ About the Project"> <link rel="document" href=".././about/bio/WZudilin" title="Profile Wadim Zudilin ‣ About the Project"> <link rel="document" href=".././front/foreword" title="Foreword"> <link rel="document" href=".././front/preface" title="Preface"> <link rel="document" href=".././front/introduction" title="Mathematical Introduction"> <link rel="chapter" href=".././1" title="Chapter 1 Algebraic and Analytic Methods"> <link rel="chapter" href=".././2" title="Chapter 2 Asymptotic Approximations"> <link rel="chapter" href=".././3" title="Chapter 3 Numerical Methods"> <link rel="chapter" href=".././4" title="Chapter 4 Elementary Functions"> <link rel="chapter" href=".././5" title="Chapter 5 Gamma Function"> <link rel="chapter" href=".././6" title="Chapter 6 Exponential, Logarithmic, Sine, and Cosine Integrals"> <link rel="chapter" href=".././7" title="Chapter 7 Error Functions, Dawson’s and Fresnel Integrals"> <link rel="chapter" href=".././8" title="Chapter 8 Incomplete Gamma and Related Functions"> <link rel="chapter" href=".././9" title="Chapter 9 Airy and Related Functions"> <link rel="chapter" href=".././10" title="Chapter 10 Bessel Functions"> <link rel="chapter" href=".././11" title="Chapter 11 Struve and Related Functions"> <link rel="chapter" href=".././12" title="Chapter 12 Parabolic Cylinder Functions"> <link rel="chapter" href=".././13" title="Chapter 13 Confluent Hypergeometric Functions"> <link rel="chapter" href=".././14" title="Chapter 14 Legendre and Related Functions"> <link rel="chapter" href=".././15" title="Chapter 15 Hypergeometric Function"> <link rel="chapter" href=".././16" title="Chapter 16 Generalized Hypergeometric Functions and Meijer 𝐺-Function"> <link rel="chapter" href=".././17" title="Chapter 17 𝑞-Hypergeometric and Related Functions"> <link rel="chapter" href=".././18" title="Chapter 18 Orthogonal Polynomials"> <link rel="chapter" href=".././19" title="Chapter 19 Elliptic Integrals"> <link rel="chapter" href=".././20" title="Chapter 20 Theta Functions"> <link rel="chapter" href=".././21" title="Chapter 21 Multidimensional Theta Functions"> <link rel="chapter" href=".././22" title="Chapter 22 Jacobian Elliptic Functions"> <link rel="chapter" href=".././23" title="Chapter 23 Weierstrass Elliptic and Modular Functions"> <link rel="chapter" href=".././24" title="Chapter 24 Bernoulli and Euler Polynomials"> <link rel="chapter" href=".././25" title="Chapter 25 Zeta and Related Functions"> <link rel="chapter" href=".././26" title="Chapter 26 Combinatorial Analysis"> <link rel="chapter" href=".././27" title="Chapter 27 Functions of Number Theory"> <link rel="chapter" href=".././28" title="Chapter 28 Mathieu Functions and Hill’s Equation"> <link rel="chapter" href=".././29" title="Chapter 29 Lamé Functions"> <link rel="chapter" href=".././30" title="Chapter 30 Spheroidal Wave Functions"> <link rel="chapter" href=".././31" title="Chapter 31 Heun Functions"> <link rel="chapter" href=".././32" title="Chapter 32 Painlevé Transcendents"> <link rel="chapter" href=".././33" title="Chapter 33 Coulomb Functions"> <link rel="chapter" href=".././34" title="Chapter 34 3⁢𝑗,6⁢𝑗,9⁢𝑗 Symbols"> <link rel="chapter" href=".././35" title="Chapter 35 Functions of Matrix Argument"> <link rel="chapter" href=".././36" title="Chapter 36 Integrals with Coalescing Saddles"> <link rel="bibliography" href=".././bib/" title="Bibliography"> <link rel="index" href=".././idx/" title="Index"> <link rel="glossary" href=".././not/" title="Notations"> <link rel="document" href=".././lof/" title="List of Figures"> <link rel="document" href=".././lot/" title="List of Tables"> <link rel="document" href=".././software/" title="Software Index"> <link rel="document" href=".././errata/" title="Errata"> <link rel="document" href=".././help/" title="Need Help?"> <link rel="document" href=".././search/" title="Advanced Search"> <link rel="document" href=".././about/." title="About the Project"> </head> <body class="color_default textfont_default titlefont_default fontsize_default navbar_default"> <div class="ltx_page_navbar"> <div class="ltx_page_navlogo"><a href=".././" title="Digital Library of Mathematical Functions"><span>DLMF</span></a></div> <div class="ltx_page_navitems"> <form method="get" action=".././search/search"> <ul> <li><a href=".././idx/">Index</a></li> <li><a href=".././not/">Notations</a></li> <li><small><input type="text" name="q" value="" size="6" class="ltx_page_navitem_search"><button type="submit">Search</button></small></li> <li><a href=".././help/" class="ltx_help">Help?</a></li> <li><a href=".././help/cite">Citing</a></li> <li><a href=".././help/customize" class="ltx_customize">Customize</a></li> </ul> </form> </div> <div class="ltx_page_navsponsors"> <div><a href=".././about/" class="ltx_page_navabout">About the Project</a></div> </div> </div> <div class="ltx_page_main"> <div class="ltx_page_header"> <a href=".././about/." title="" class="ltx_ref" rel="up"><span class="ltx_text ltx_ref_title">About the Project</span></a><a href=".././about/." title="" class="ltx_ref" rel="prev"><span class="ltx_text ltx_ref_title">About the Project</span></a><a href=".././about/preface" title="In About the Project" class="ltx_ref" rel="next"><span class="ltx_text ltx_ref_title">Preface</span></a> </div> <div class="ltx_page_content"> <article class="ltx_document ltx_leqno"> <h1 class="ltx_title ltx_title_document">Notices</h1> <div class="ltx_toc_compact"> ♦ <a href="#S1">Copyright</a> ♦ <a href="#S2">Disclaimer</a> ♦ <a href="#S3">Software Indexing Policy</a> ♦ </div> <section id="S1" class="ltx_section"> <h2 class="ltx_title ltx_title_section">Copyright</h2> <div id="S1.p1" class="ltx_para"> <p class="ltx_p">Pursuant to Title 17 USC 105, the National Institute of Standards and Technology (NIST), United States Department of Commerce, is authorized to receive and hold copyrights transferred to it by assignment or otherwise. Authors of the works appearing in the Digital Library of Mathematical Functions (DLMF) have assigned copyright to the works to NIST, United States Department of Commerce, as represented by the Secretary of Commerce. All materials on this website are owned by NIST. </p> </div> <div id="S1.p2" class="ltx_para"> <p class="ltx_p">Limited copying and internal distribution of the content of these pages is permitted for research and teaching. Reproduction, copying, or distribution for any commercial purpose is strictly prohibited. Bulk copying, reproduction, or redistribution in any form is not permitted. Excessive downloading which results in bandwidth degradation for other users is not permitted and may result in having offending subnets blocked. Questions regarding this copyright policy should be directed to <a href="mailto:dlmf-feedback@nist.gov" title="" class="ltx_ref">dlmf-feedback</a>.</p> </div> </section> <section id="S2" class="ltx_section"> <h2 class="ltx_title ltx_title_section">Disclaimer</h2> <div id="S2.p1" class="ltx_para"> <p class="ltx_p">While NIST has made every effort to ensure the accuracy and reliability of the information in the DLMF, the DLMF is expressly provided “AS-IS.” NIST makes NO WARRANTY OF ANY TYPE, including no warranties of merchantability or fitness for a particular purpose. NIST makes no warranties or representations as to the correctness, accuracy, or reliability of the DLMF. As a condition of using the DLMF, you explicitly release NIST from any and all liabilities for any damage of any type that may result from errors or omissions in the DLMF.</p> </div> <div id="S2.p2" class="ltx_para"> <p class="ltx_p">Certain products, commercial and otherwise, are mentioned in the DLMF. These mentions are for informational purposes only, and do not imply recommendation or endorsement by NIST.</p> </div> <div id="S2.p3" class="ltx_para"> <p class="ltx_p">If you feel you have found an error in DLMF, please see <a href=".././help/errors" title="In Need Help?" class="ltx_ref"><span class="ltx_text ltx_ref_title">Possible Errors in DLMF</span></a>.</p> </div> </section> <section id="S3" class="ltx_section"> <h2 class="ltx_title ltx_title_section">Software Indexing Policy</h2> <div id="S3.p1" class="ltx_para"> <p class="ltx_p">The DLMF wishes to provide users of special functions with essential reference information related to the use and application of special functions in research, development, and education. Using special functions in applications often requires computing them. Thus, we seek to provide DLMF users with links to sources of such software. We do this in two ways.</p> </div> <div id="S3.p2" class="ltx_para"> <dl id="S3.I1" class="ltx_description"> <dt id="S3.I1.ix1" class="ltx_item"><span class="ltx_tag ltx_tag_item"><span class="ltx_text ltx_font_bold">Index of Selected Software Within the DLMF Chapters</span></span></dt> <dd class="ltx_item"> <div id="S3.I1.ix1.p1" class="ltx_para"> <p class="ltx_p">Within each of the DLMF chapters themselves we will provide a list of <span class="ltx_text ltx_font_italic">research software</span> for the functions discussed in that chapter. The purpose of these listings is to provide references to the research literature on the engineering of software for special functions. To qualify for listing, the development of the software must have been the subject of a research paper published in the peer-reviewed literature. If such software is available online for free download we will provide a link to the software.</p> </div> <div id="S3.I1.ix1.p2" class="ltx_para"> <p class="ltx_p">In general, we will not index other software within DLMF chapters unless the software is unique in some way, such as being the only known software for computing a particular function.</p> </div> </dd> <dt id="S3.I1.ix2" class="ltx_item"><span class="ltx_tag ltx_tag_item"><span class="ltx_text ltx_font_bold">Master Software Index</span></span></dt> <dd class="ltx_item"> <div id="S3.I1.ix2.p1" class="ltx_para"> <p class="ltx_p">In association with the DLMF we will provide an index of all software for the computation of special functions covered by the DLMF. It is our intention that this will become an exhaustive list of sources of software for special functions. In each case we will maintain a single link where readers can obtain more information about the listed software. We welcome requests from software authors (or distributors) for new items to list.</p> </div> <div id="S3.I1.ix2.p2" class="ltx_para"> <p class="ltx_p">Note that here we will only include software with capabilities that go beyond the computation of elementary functions in standard precisions since such software is nearly universal in scientific computing environments.</p> </div> </dd> </dl> </div> <div id="S3.p3" class="ltx_para"> <p class="ltx_p">These software indexes are provided as a service to the user community. NIST expressly does not endorse or recommend any specific product or service. The absence of a listing does not imply that software is unsuitable for its intended purpose. NIST does not provide support of any kind for software indexed in the DLMF. Please <a href="mailto:dlmf-feedback@nist.gov" title="" class="ltx_ref">contact us</a> if you have software that you believe should be listed.</p> </div> </section> </article> </div> <div class="ltx_page_footer"> <div class="ltx_siblings"> <a href=".././about/." title="" class="ltx_ref" rel="prev"><span class="ltx_text ltx_ref_title">About the Project</span></a><a href=".././about/preface" title="In About the Project" class="ltx_ref" rel="next"><span class="ltx_text ltx_ref_title">Preface</span></a> </div> <div class="ltx_footer_links ltx_centering"> <a href=".././about/notices">© 2010–2025 NIST</a> / <a href=".././about/notices#S2">Disclaimer</a> / <a href="mailto:DLMF-feedback@nist.gov">Feedback</a>; Version 1.2.4; Release date 2025-03-15.</div> <span id="pagesettings"></span> <div class="ltx_nist_logo"><a href="http://www.nist.gov/"><img src=".././style/nistidc.png" width="500" alt="NIST"></a></div> <div class="ltx_nist_links ltx_align_right"> <a href="https://www.nist.gov/privacy-policy">Site Privacy</a> <a href="https://www.nist.gov/oism/accessibility">Accessibility</a> <a href="https://www.nist.gov/privacy">Privacy Program</a> <a href="https://www.nist.gov/oism/copyrights">Copyrights</a> <a href="https://www.commerce.gov/vulnerability-disclosure-policy">Vulnerability Disclosure</a> <a href="https://www.nist.gov/no-fear-act-policy">No Fear Act Policy</a> <a href="https://www.nist.gov/foia">FOIA</a> <a href="https://www.nist.gov/environmental-policy-statement">Environmental Policy</a> <a href="https://www.nist.gov/summary-report-scientific-integrity">Scientific Integrity</a> <a href="https://www.nist.gov/nist-information-quality-standards">Information Quality Standards</a> <a href="https://www.commerce.gov/">Commerce.gov</a> <a href="http://www.science.gov/">Science.gov</a> <a href="http://www.usa.gov/">USA.gov</a> </div> </div> </div> </body> </html>