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<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01//EN" "http://www.w3.org/TR/html4/strict.dtd"> <html lang="en"> <head> <meta http-equiv="content-type" content="text/html; charset=UTF-8"> <title>DLMF: NIST Digital Library of Mathematical Functions</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 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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_content"> <div class="ltx_document mainpage"> <div class="ltx_titlepage ltx_centering"> <div class="ltx_block"> <img src="./style/banner.png" class="ltx_graphics" width="459" height="114" alt="Riemann zeta function"> </div> <h1 class="ltx_title ltx_title_mainpage ltx_title_titlepage">NIST Digital Library of Mathematical Functions</h1> </div> <div class="ltx_recentnews"> <h6>Project News</h6> <ul class="ltx_description ltx_news"> <li class="ltx_newsitem">2025-03-15 <a href="about/news/#S1.I1.i1"><span class="ltx_font_italic">DLMF Update; Version 1.2.4</span></a> </li> <li class="ltx_newsitem">2024-12-15 <a href="about/news/#S1.I1.i2"><span class="ltx_font_italic">DLMF Update; Version 1.2.3</span></a> </li> <li class="ltx_newsitem">2024-09-15 <a href="about/news/#S1.I1.i3"><span class="ltx_font_italic">DLMF Update; Version 1.2.2</span></a> </li> <li class="ltx_newsitem">2024-09-15 <a href="about/news/#S1.I1.i4"><span class="ltx_font_italic">Barry I. Schneider, General Editor of the DLMF, dies at age 83</span></a> </li> <li class="ltx_newsitem"><span class="ltx_font_italic"><a href="about/news/">More news</a></span></li> </ul> </div> <div class="ltx_TOC"> <div class="ltx_page_columns"> <div class="ltx_page_column1"> <ul class="ltx_toclist ltx_toclist_mainpage"> <li class="ltx_tocentry"><a href="./front/foreword" title="" class="ltx_ref"><span class="ltx_text ltx_ref_title">Foreword</span></a></li> <li class="ltx_tocentry"><a href="./front/preface" title="" class="ltx_ref"><span class="ltx_text ltx_ref_title">Preface</span></a></li> <li class="ltx_tocentry"><a href="./front/introduction" title="" class="ltx_ref"><span class="ltx_text ltx_ref_title">Mathematical Introduction</span></a></li> <li class="ltx_tocentry"><a href="./1" title="" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1 </span>Algebraic and Analytic Methods</span></a></li> <li class="ltx_tocentry"><a href="./2" title="" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2 </span>Asymptotic Approximations</span></a></li> <li class="ltx_tocentry"><a href="./3" title="" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3 </span>Numerical Methods</span></a></li> <li class="ltx_tocentry"><a href="./4" title="" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4 </span>Elementary Functions</span></a></li> <li class="ltx_tocentry"><a href="./5" title="" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5 </span>Gamma Function</span></a></li> <li class="ltx_tocentry"><a href="./6" title="" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">6 </span>Exponential, Logarithmic, Sine, and <span class="ltx_text"> </span>Cosine Integrals</span></a></li> <li class="ltx_tocentry"><a href="./7" title="" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">7 </span>Error Functions, Dawson’s and Fresnel Integrals</span></a></li> <li class="ltx_tocentry"><a href="./8" title="" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">8 </span>Incomplete Gamma and Related <span class="ltx_text"> </span>Functions</span></a></li> <li class="ltx_tocentry"><a href="./9" title="" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">9 </span>Airy and Related Functions</span></a></li> <li class="ltx_tocentry"><a href="./10" title="" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">10 </span>Bessel Functions</span></a></li> <li class="ltx_tocentry"><a href="./11" title="" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">11 </span>Struve and Related Functions</span></a></li> <li class="ltx_tocentry"><a href="./12" title="" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">12 </span>Parabolic Cylinder Functions</span></a></li> <li class="ltx_tocentry"><a href="./13" title="" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">13 </span>Confluent Hypergeometric Functions</span></a></li> <li class="ltx_tocentry"><a href="./14" title="" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">14 </span>Legendre and Related Functions</span></a></li> <li class="ltx_tocentry"><a href="./15" title="" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">15 </span>Hypergeometric Function</span></a></li> <li class="ltx_tocentry"><a href="./16" title="" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">16 </span>Generalized Hypergeometric Functions & Meijer <span class="ltx_text ltx_font_italic">G</span>-Function</span></a></li> <li class="ltx_tocentry"><a href="./17" title="" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">17 </span><span class="ltx_text ltx_font_italic">q</span>-Hypergeometric and Related Functions</span></a></li> <li class="ltx_tocentry"><a href="./18" title="" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">18 </span>Orthogonal Polynomials</span></a></li> <li class="ltx_tocentry"><a href="./19" title="" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">19 </span>Elliptic Integrals</span></a></li> </ul> </div> <div class="ltx_page_column2"> <ul class="ltx_toclist ltx_toclist_mainpage"> <li class="ltx_tocentry"><a href="./20" title="" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">20 </span>Theta Functions</span></a></li> <li class="ltx_tocentry"><a href="./21" title="" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">21 </span>Multidimensional Theta Functions</span></a></li> <li class="ltx_tocentry"><a href="./22" title="" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">22 </span>Jacobian Elliptic Functions</span></a></li> <li class="ltx_tocentry"><a href="./23" title="" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">23 </span>Weierstrass Elliptic and Modular <span class="ltx_text"> </span>Functions</span></a></li> <li class="ltx_tocentry"><a href="./24" title="" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">24 </span>Bernoulli and Euler Polynomials</span></a></li> <li class="ltx_tocentry"><a href="./25" title="" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">25 </span>Zeta and Related Functions</span></a></li> <li class="ltx_tocentry"><a href="./26" title="" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">26 </span>Combinatorial Analysis</span></a></li> <li class="ltx_tocentry"><a href="./27" title="" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">27 </span>Functions of Number Theory</span></a></li> <li class="ltx_tocentry"><a href="./28" title="" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">28 </span>Mathieu Functions and Hill’s Equation</span></a></li> <li class="ltx_tocentry"><a href="./29" title="" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">29 </span>Lamé Functions</span></a></li> <li class="ltx_tocentry"><a href="./30" title="" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">30 </span>Spheroidal Wave Functions</span></a></li> <li class="ltx_tocentry"><a href="./31" title="" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">31 </span>Heun Functions</span></a></li> <li class="ltx_tocentry"><a href="./32" title="" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">32 </span>Painlevé Transcendents</span></a></li> <li class="ltx_tocentry"><a href="./33" title="" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">33 </span>Coulomb Functions</span></a></li> <li class="ltx_tocentry"><a href="./34" title="" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">34 </span>3<span class="ltx_text ltx_font_italic">j</span>, 6<span class="ltx_text ltx_font_italic">j</span>, 9<span class="ltx_text ltx_font_italic">j</span> Symbols</span></a></li> <li class="ltx_tocentry"><a href="./35" title="" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">35 </span>Functions of Matrix Argument</span></a></li> <li class="ltx_tocentry"><a href="./36" title="" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">36 </span>Integrals with Coalescing Saddles</span></a></li> <li class="ltx_tocentry backmatter"><a href="./bib/" title="" class="ltx_ref">Bibliography</a></li> <li class="ltx_tocentry backmatter"><a href="./idx/" title="Index" class="ltx_ref">Index</a></li> <li class="ltx_tocentry backmatter"><a href="./not/" title="Notations" class="ltx_ref">Notations</a></li> <li class="ltx_tocentry backmatter"><a href="./lof/" title="List of Figures" class="ltx_ref">List of Figures</a></li> <li class="ltx_tocentry backmatter"><a href="./lot/" title="List of Tables" class="ltx_ref">List of Tables</a></li> <li class="ltx_tocentry backmatter"><a href="./software/" title="Software Index" class="ltx_ref">Software</a></li> <li class="ltx_tocentry backmatter"><a href="./errata/" title="Errata" class="ltx_ref">Errata</a></li> </ul> </div> </div> </div> <div class="ltx_para" style="clear:both;;"> <p class="ltx_p ltx_subtitle">Companion to the <a href="http://www.cambridge.org/catalogue/catalogue.asp?isbn=9780521192255" title="" class="ltx_ref external external">NIST Handbook of Mathematical Functions <img src="./style/hmf-small.png" class="ltx_graphics" width="140" height="123" alt="Handbook of Mathematical Functions"></a></p> </div> </div> </div> <div class="ltx_page_footer"> <div class="ltx_siblings"><a href="./front/foreword" title="" class="ltx_ref" rel="next"><span class="ltx_text ltx_ref_title">Foreword</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> <div class="ltx_footer_links ltx_centering">If you have difficulties viewing this site, please consult our <a href="help/">Help</a> pages.</div> <span id="pagesettings"></span> <div class="ltx_nist_logo"><a href="http://www.nist.gov/"><img src="./style/nistidc.png" 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