CINXE.COM

<!DOCTYPE html><html lang="en"> <head> <meta http-equiv="content-type" content="text/html; charset=UTF-8"> <title>DLMF: Chapter 22 Jacobian Elliptic 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 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="start" href="./" title=""> <link rel="prev" href="./21.11" title="§21.11 Software ‣ Computation ‣ Chapter 21 Multidimensional Theta Functions"> <link rel="next" href="./22.1" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions"> <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="./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"> <link rel="section" href="./22.1" title="§22.1 Special Notation ‣ Notation ‣ Chapter 22 Jacobian Elliptic Functions"> <link rel="section" href="./22.2" title="§22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions"> <link rel="section" href="./22.3" title="§22.3 Graphics ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions"> <link rel="section" href="./22.4" title="§22.4 Periods, Poles, and Zeros ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions"> <link rel="section" href="./22.5" title="§22.5 Special Values ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions"> <link rel="section" href="./22.6" title="§22.6 Elementary Identities ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions"> <link rel="section" href="./22.7" title="§22.7 Landen Transformations ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions"> <link rel="section" href="./22.8" title="§22.8 Addition Theorems ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions"> <link rel="section" href="./22.9" title="§22.9 Cyclic Identities ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions"> <link rel="section" href="./22.10" title="§22.10 Maclaurin Series ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions"> <link rel="section" href="./22.11" title="§22.11 Fourier and Hyperbolic Series ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions"> <link rel="section" href="./22.12" title="§22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions"> <link rel="section" href="./22.13" title="§22.13 Derivatives and Differential Equations ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions"> <link rel="section" href="./22.14" title="§22.14 Integrals ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions"> <link rel="section" href="./22.15" title="§22.15 Inverse Functions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions"> <link rel="section" href="./22.16" title="§22.16 Related Functions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions"> <link rel="section" href="./22.17" title="§22.17 Moduli Outside the Interval [0,1] ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions"> <link rel="section" href="./22.18" title="§22.18 Mathematical Applications ‣ Applications ‣ Chapter 22 Jacobian Elliptic Functions"> <link rel="section" href="./22.19" title="§22.19 Physical Applications ‣ Applications ‣ Chapter 22 Jacobian Elliptic Functions"> <link rel="section" href="./22.20" title="§22.20 Methods of Computation ‣ Computation ‣ Chapter 22 Jacobian Elliptic Functions"> <link rel="section" href="./22.21" title="§22.21 Tables ‣ Computation ‣ Chapter 22 Jacobian Elliptic Functions"> <link rel="section" href="./22.22" title="§22.22 Software ‣ Computation ‣ Chapter 22 Jacobian Elliptic Functions"> <link rel="sidebar" href="./22.SB1" title="Sidebar 22.SB1: Decay of a Soliton in a Bose–Einstein Condensate ‣ Chapter 22 Jacobian Elliptic Functions"> <link rel="author" href="./about/bio/WPReinhardt" title="W. P. Reinhardt"> <link rel="author" href="./about/bio/PLWalker" title="P. L. Walker"> </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> <li id="showinfo"><a href="" onclick="showInfo(); return false;" title="Show Annotations">Annotate</a></li> <li id="hideinfo"><a href="" onclick="hideInfo(); return false;">UnAnnotate</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"> <section class="ltx_chapter ltx_leqno"> <h1 class="ltx_title ltx_title_chapter"> <span class="ltx_tag ltx_tag_chapter">Chapter 22 </span>Jacobian Elliptic Functions</h1> <div class="ltx_authors"> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname"><a href="./about/bio/WPReinhardt">W. P. Reinhardt</a> </span><span class="ltx_author_notes"> <span class="ltx_contact ltx_role_affiliation">University of Washington, Seattle, Washington. </span></span></span> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname"><a href="./about/bio/PLWalker">P. L. Walker</a> </span><span class="ltx_author_notes"> <span class="ltx_contact ltx_role_affiliation">American University of Sharjah, Sharjah, United Arab Emirates. </span></span></span> </div> <div id="info" class="ltx_metadata ltx_info"> <span href="" class="ltx_infoicon ltx_icon" title="Show Annotations"><span class="ltx_font_bold">ⓘ</span></span> <div class="ltx_infocontent"> <dl> <dt>Acknowledgements:</dt> <dd class="ltx_acknowledgements">This chapter is based in part on <cite class="ltx_cite ltx_citemacro_citet">Abramowitz and Stegun (<a href="./bib/#bib24" title="Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables" class="ltx_ref">1964</a>, Chapters 16,18)</cite> by L. M. Milne-Thomson and T. H. Southard respectively.</dd> <dt>Notes:</dt> <dd>The references used for the mathematical properties in this chapter are <cite class="ltx_cite ltx_citemacro_citet">Armitage and Eberlein (<a href="./bib/#bib2736" title="Elliptic Functions" class="ltx_ref">2006</a>)</cite>, <cite class="ltx_cite ltx_citemacro_citet">Bowman (<a href="./bib/B#bib325" title="Introduction to Elliptic Functions with Applications" class="ltx_ref">1953</a>)</cite>, <cite class="ltx_cite ltx_citemacro_citet">Copson (<a href="./bib/C#bib580" title="An Introduction to the Theory of Functions of a Complex Variable" class="ltx_ref">1935</a>)</cite>, <cite class="ltx_cite ltx_citemacro_citet">Lawden (<a href="./bib/L#bib1385" title="Elliptic Functions and Applications" class="ltx_ref">1989</a>)</cite>, <cite class="ltx_cite ltx_citemacro_citet">McKean and Moll (<a href="./bib/M#bib1582" title="Elliptic Curves" class="ltx_ref">1999</a>)</cite>, <cite class="ltx_cite ltx_citemacro_citet">Walker (<a href="./bib/W#bib2359" title="Elliptic Functions. A Constructive Approach" class="ltx_ref">1996</a>)</cite>, <cite class="ltx_cite ltx_citemacro_citet">Whittaker and Watson (<a href="./bib/W#bib2404" title="A Course of Modern Analysis" class="ltx_ref">1927</a>)</cite>, and for physical applications <cite class="ltx_cite ltx_citemacro_citet">Drazin and Johnson (<a href="./bib/D#bib684" title="Solitons: An Introduction" class="ltx_ref">1993</a>)</cite>, <cite class="ltx_cite ltx_citemacro_citet">Lawden (<a href="./bib/L#bib1385" title="Elliptic Functions and Applications" class="ltx_ref">1989</a>)</cite>, <cite class="ltx_cite ltx_citemacro_citet">Walker (<a href="./bib/W#bib2359" title="Elliptic Functions. A Constructive Approach" class="ltx_ref">1996</a>)</cite>.The references used for the mathematical properties in this chapter are <cite class="ltx_cite ltx_citemacro_citet">Armitage and Eberlein (<a href="./bib/#bib2736" title="Elliptic Functions" class="ltx_ref">2006</a>)</cite>, <cite class="ltx_cite ltx_citemacro_citet">Bowman (<a href="./bib/B#bib325" title="Introduction to Elliptic Functions with Applications" class="ltx_ref">1953</a>)</cite>, <cite class="ltx_cite ltx_citemacro_citet">Copson (<a href="./bib/C#bib580" title="An Introduction to the Theory of Functions of a Complex Variable" class="ltx_ref">1935</a>)</cite>, <cite class="ltx_cite ltx_citemacro_citet">Lawden (<a href="./bib/L#bib1385" title="Elliptic Functions and Applications" class="ltx_ref">1989</a>)</cite>, <cite class="ltx_cite ltx_citemacro_citet">McKean and Moll (<a href="./bib/M#bib1582" title="Elliptic Curves" class="ltx_ref">1999</a>)</cite>, <cite class="ltx_cite ltx_citemacro_citet">Walker (<a href="./bib/W#bib2359" title="Elliptic Functions. A Constructive Approach" class="ltx_ref">1996</a>)</cite>, <cite class="ltx_cite ltx_citemacro_citet">Whittaker and Watson (<a href="./bib/W#bib2404" title="A Course of Modern Analysis" class="ltx_ref">1927</a>)</cite>, and for physical applications <cite class="ltx_cite ltx_citemacro_citet">Drazin and Johnson (<a href="./bib/D#bib684" title="Solitons: An Introduction" class="ltx_ref">1993</a>)</cite>, <cite class="ltx_cite ltx_citemacro_citet">Lawden (<a href="./bib/L#bib1385" title="Elliptic Functions and Applications" class="ltx_ref">1989</a>)</cite>, <cite class="ltx_cite ltx_citemacro_citet">Walker (<a href="./bib/W#bib2359" title="Elliptic Functions. A Constructive Approach" class="ltx_ref">1996</a>)</cite>.</dd> <dt>Referenced by:</dt> <dd> <a href="./32.2#i.p5" title="§32.2(i) Introduction ‣ §32.2 Differential Equations ‣ Properties ‣ Chapter 32 Painlevé Transcendents" class="ltx_ref"><span class="ltx_text ltx_ref_tag">§32.2(i)</span></a>, <a href="./about/news/#S1.I1.i5.p1" title="In Recent News ‣ DLMF Project News ‣ About the Project" class="ltx_ref"><span class="ltx_text ltx_ref_tag">New Associate Editors for the DLMF</span></a>, <a href="./about/bio/HVolkmer#S0.I1.ix1.p1" title="Profile Hans Volkmer ‣ About the Project" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_text ltx_font_italic">Profile<span class="ltx_text"> </span></span>Hans Volkmer</span></a>, <a href="./about/bio/PLWalker#S0.I1.ix2.p1" title="Profile Peter L. Walker ‣ About the Project" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_text ltx_font_italic">Profile<span class="ltx_text"> </span></span>Peter L. Walker</span></a>, <a href="./about/bio/WPReinhardt#S0.I1.ix2.p1" title="Profile William P. Reinhardt ‣ About the Project" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_text ltx_font_italic">Profile<span class="ltx_text"> </span></span>William P. Reinhardt</span></a>, <a href="./about/bio/WPReinhardt#p7" title="Profile William P. Reinhardt ‣ About the Project" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_text ltx_font_italic">Profile<span class="ltx_text"> </span></span>William P. Reinhardt</span></a>, <a href="./about/staff#S2.I1.ix19.p1" title="Chapter Authors ‣ Project Staff ‣ About the Project" class="ltx_ref"><span class="ltx_text ltx_ref_tag">§</span> ‣ <span class="ltx_text ltx_ref_title">Chapter Authors</span></a>, <a href="./about/staff#S2.I1.ix27.p1" title="Chapter Authors ‣ Project Staff ‣ About the Project" class="ltx_ref"><span class="ltx_text ltx_ref_tag">§</span> ‣ <span class="ltx_text ltx_ref_title">Chapter Authors</span></a>, <a href="./about/staff#S4.I1.ix17.p1" title="Associate Editors ‣ Project Staff ‣ About the Project" class="ltx_ref"><span class="ltx_text ltx_ref_tag">§</span> ‣ <span class="ltx_text ltx_ref_title">Associate Editors</span></a>, <a href="./about/staff#S4.I1.ix26.p1" title="Associate Editors ‣ Project Staff ‣ About the Project" class="ltx_ref"><span class="ltx_text ltx_ref_tag">§</span> ‣ <span class="ltx_text ltx_ref_title">Associate Editors</span></a>, <a href="./about/staff#S4.I1.ix27.p1" title="Associate Editors ‣ Project Staff ‣ About the Project" class="ltx_ref"><span class="ltx_text ltx_ref_tag">§</span> ‣ <span class="ltx_text ltx_ref_title">Associate Editors</span></a>, <a href="./errata/#V1.0.1.I4.ix3.p1" title="In Other Changes ‣ Version 1.0.1 (June 27, 2011) ‣ Errata" class="ltx_ref"><span class="ltx_text ltx_ref_tag">Erratum (V1.0.1) for References</span></a>, <a href="./software/#S1.t1.r80" title="Software Cross Index ‣ Software Index" class="ltx_ref"><span class="ltx_text ltx_ref_tag">§</span> ‣ <span class="ltx_text ltx_ref_title">Software Cross Index</span></a> </dd> <dt>Permalink:</dt> <dd><a href="./22" title="" class="ltx_ref">http://dlmf.nist.gov/22</a></dd> <dt>Addition (effective with 1.0.1):</dt> <dd> <cite class="ltx_cite ltx_citemacro_citet">Armitage and Eberlein (<a href="./bib/#bib2736" title="Elliptic Functions" class="ltx_ref">2006</a>)</cite> was added as a general reference for this chapter. </dd> </dl> </div> </div> <div class="ltx_gallery"> <a href="./22.SB1" title="Sidebar 22.SB1: Decay of a Soliton in a Bose–Einstein Condensate ‣ Chapter 22 Jacobian Elliptic Functions"><img src="./22/g1.png" width="139" height="139" alt="Sidebar 22.SB1: Decay of a Soliton in a Bose–Einstein Condensate"></a> </div> <div class="ltx_page_columns"> <div class="ltx_page_column1"> <ul class="ltx_toclist ltx_toclist_chapter"> <li id="PT1" class="ltx_tocentry"> <span class="ltx_text">Notation</span> <ol class="ltx_toclist ltx_toclist_part"> <li class="ltx_tocentry"><a href="./22.1" title="In Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref ltx_toc"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">22.1 </span>Special Notation</span></a></li> </ol> </li> <li id="PT2" class="ltx_tocentry"> <span class="ltx_text">Properties</span> <ol class="ltx_toclist ltx_toclist_part"> <li class="ltx_tocentry"><a href="./22.2" title="In Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref ltx_toc"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">22.2 </span>Definitions</span></a></li> <li class="ltx_tocentry"><a href="./22.3" title="In Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref ltx_toc"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">22.3 </span>Graphics</span></a></li> <li class="ltx_tocentry"><a href="./22.4" title="In Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref ltx_toc"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">22.4 </span>Periods, Poles, and Zeros</span></a></li> <li class="ltx_tocentry"><a href="./22.5" title="In Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref ltx_toc"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">22.5 </span>Special Values</span></a></li> <li class="ltx_tocentry"><a href="./22.6" title="In Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref ltx_toc"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">22.6 </span>Elementary Identities</span></a></li> <li class="ltx_tocentry"><a href="./22.7" title="In Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref ltx_toc"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">22.7 </span>Landen Transformations</span></a></li> <li class="ltx_tocentry"><a href="./22.8" title="In Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref ltx_toc"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">22.8 </span>Addition Theorems</span></a></li> <li class="ltx_tocentry"><a href="./22.9" title="In Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref ltx_toc"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">22.9 </span>Cyclic Identities</span></a></li> <li class="ltx_tocentry"><a href="./22.10" title="In Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref ltx_toc"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">22.10 </span>Maclaurin Series</span></a></li> <li class="ltx_tocentry"><a href="./22.11" title="In Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref ltx_toc"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">22.11 </span>Fourier and Hyperbolic Series</span></a></li> <li class="ltx_tocentry"><a href="./22.12" title="In Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref ltx_toc"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">22.12 </span>Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series</span></a></li> <li class="ltx_tocentry"><a href="./22.13" title="In Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref ltx_toc"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">22.13 </span>Derivatives and Differential Equations</span></a></li> <li class="ltx_tocentry"><a href="./22.14" title="In Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref ltx_toc"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">22.14 </span>Integrals</span></a></li> <li class="ltx_tocentry"><a href="./22.15" title="In Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref ltx_toc"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">22.15 </span>Inverse Functions</span></a></li> <li class="ltx_tocentry"><a href="./22.16" title="In Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref ltx_toc"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">22.16 </span>Related Functions</span></a></li> <li class="ltx_tocentry"><a href="./22.17" title="In Properties ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref ltx_toc"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">22.17 </span>Moduli Outside the Interval [0,1]</span></a></li> </ol> </li> </ul> </div> <div class="ltx_page_column2"> <ul class="ltx_toclist ltx_toclist_chapter"> <li id="PT3" class="ltx_tocentry"> <span class="ltx_text">Applications</span> <ol class="ltx_toclist ltx_toclist_part"> <li class="ltx_tocentry"><a href="./22.18" title="In Applications ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref ltx_toc"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">22.18 </span>Mathematical Applications</span></a></li> <li class="ltx_tocentry"><a href="./22.19" title="In Applications ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref ltx_toc"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">22.19 </span>Physical Applications</span></a></li> </ol> </li> <li id="PT4" class="ltx_tocentry"> <span class="ltx_text">Computation</span> <ol class="ltx_toclist ltx_toclist_part"> <li class="ltx_tocentry"><a href="./22.20" title="In Computation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref ltx_toc"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">22.20 </span>Methods of Computation</span></a></li> <li class="ltx_tocentry"><a href="./22.21" title="In Computation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref ltx_toc"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">22.21 </span>Tables</span></a></li> <li class="ltx_tocentry"><a href="./22.22" title="In Computation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref ltx_toc"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">22.22 </span>Software</span></a></li> </ol> </li> </ul> </div> </div> </section> </div> <div class="ltx_page_footer"> <div class="ltx_siblings"> <a href="./21.11" title="In Computation ‣ Chapter 21 Multidimensional Theta Functions" class="ltx_ref" rel="prev"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">21.11 </span>Software</span></a><a href="./22.1" title="In Notation ‣ Chapter 22 Jacobian Elliptic Functions" class="ltx_ref" rel="next"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">22.1 </span>Special Notation</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>