{"version":"1.0","provider_name":"SOUL OF MATHEMATICS","provider_url":"https:\/\/soulofmathematics.com","author_name":"Rajarshi Dey","author_url":"https:\/\/soulofmathematics.com\/index.php\/author\/rajarshidey1729gmail-com\/","title":"JACOBI ELLIPTIC FUNCTIONS - SOUL OF MATHEMATICS","type":"rich","width":600,"height":338,"html":"<blockquote class=\"wp-embedded-content\" data-secret=\"wbCa7GROKk\"><a href=\"https:\/\/soulofmathematics.com\/index.php\/jacobi-elliptic-functions\/\">JACOBI ELLIPTIC FUNCTIONS<\/a><\/blockquote><iframe sandbox=\"allow-scripts\" security=\"restricted\" src=\"https:\/\/soulofmathematics.com\/index.php\/jacobi-elliptic-functions\/embed\/#?secret=wbCa7GROKk\" width=\"600\" height=\"338\" title=\"&#8220;JACOBI ELLIPTIC FUNCTIONS&#8221; &#8212; SOUL OF MATHEMATICS\" data-secret=\"wbCa7GROKk\" frameborder=\"0\" marginwidth=\"0\" marginheight=\"0\" scrolling=\"no\" class=\"wp-embedded-content\"><\/iframe><script type=\"text\/javascript\">\n\/* <![CDATA[ *\/\n\/*! This file is auto-generated *\/\n!function(d,l){\"use strict\";l.querySelector&&d.addEventListener&&\"undefined\"!=typeof URL&&(d.wp=d.wp||{},d.wp.receiveEmbedMessage||(d.wp.receiveEmbedMessage=function(e){var t=e.data;if((t||t.secret||t.message||t.value)&&!\/[^a-zA-Z0-9]\/.test(t.secret)){for(var s,r,n,a=l.querySelectorAll('iframe[data-secret=\"'+t.secret+'\"]'),o=l.querySelectorAll('blockquote[data-secret=\"'+t.secret+'\"]'),c=new RegExp(\"^https?:$\",\"i\"),i=0;i<o.length;i++)o[i].style.display=\"none\";for(i=0;i<a.length;i++)s=a[i],e.source===s.contentWindow&&(s.removeAttribute(\"style\"),\"height\"===t.message?(1e3<(r=parseInt(t.value,10))?r=1e3:~~r<200&&(r=200),s.height=r):\"link\"===t.message&&(r=new URL(s.getAttribute(\"src\")),n=new URL(t.value),c.test(n.protocol))&&n.host===r.host&&l.activeElement===s&&(d.top.location.href=t.value))}},d.addEventListener(\"message\",d.wp.receiveEmbedMessage,!1),l.addEventListener(\"DOMContentLoaded\",function(){for(var e,t,s=l.querySelectorAll(\"iframe.wp-embedded-content\"),r=0;r<s.length;r++)(t=(e=s[r]).getAttribute(\"data-secret\"))||(t=Math.random().toString(36).substring(2,12),e.src+=\"#?secret=\"+t,e.setAttribute(\"data-secret\",t)),e.contentWindow.postMessage({message:\"ready\",secret:t},\"*\")},!1)))}(window,document);\n\/* ]]> *\/\n<\/script>\n","description":"In the mathematical field of&nbsp;complex analysis&nbsp;ELLIPTIC FUNCTIONS&nbsp;are a special kind of&nbsp;meromorphic&nbsp;functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from&nbsp;elliptic integrals. Originally those integrals occurred at the calculation of the arc length of an&nbsp;ellipse. Important elliptic functions are&nbsp;Jacobi elliptic functions&nbsp;and the&nbsp;Weierstrass&nbsp;p-function. In&nbsp;mathematics,&nbsp; THETA FUNCTIONS &nbsp;are&nbsp;special functions&nbsp;of&nbsp;several complex variables. They are important in many areas, including the theories of&nbsp;Abelian varieties&nbsp;and&nbsp;moduli spaces, and of&nbsp;quadratic forms. They have also been applied to&nbsp;soliton&nbsp;theory. When generalized to a&nbsp;Grassmann algebra, they also appear in&nbsp;quantum field theory. The most common form of theta function is that occurring in the theory of&nbsp;elliptic functions. THE JACOBI ELLIPSE The cos \u03b8 and sin \u03b8 are defined on a unit circle, with radius = 1 and angle \u03b8 = arc length of the unit circle measured from the positive x-axis. Similarly, Jacobi elliptic functions are defined on the unit ellipse,&nbsp;with&nbsp;a&nbsp;=&nbsp;1. Let x2 + y2 \/ b2 = 1, b &gt; 1, m = 1 &#8211; 1\/b2, 0 &lt; m &lt; 1, x = r cos \u03b8 and y = r sin \u03b8 then, r ( \u03b8 ,m) = 1\/\u221a(1 &#8211; m sin2 \u03b8). For each angle \u03b8 the parameter u = u ( \u03b8 ,m) = 0\u222b\u03b8 r ( \u03b8 ,m) d\u03b8 is computed. On the unit circle a=b=1,&nbsp;u&nbsp;would be an arc length. While&nbsp;u&nbsp;does not carry a direct geometric interpretation in the elliptic case, it turns out to be the parameter that enters the definition of the elliptic functions. Indeed, let P=(x, y)=( r cos \u03b8 , r sin \u03b8) be a point on the ellipse, and let&nbsp;P&#8217;=(x&#8217;, y&#8217;)=(cos \u03b8, sin \u03b8) be the point where the unit circle intersects the line between&nbsp;P and the origin O. Then the familiar relations from the unit circle: x&#8217; = r cos \u03b8 and y&#8217; = r sin \u03b8 read for the ellipse as: x&#8217; = cn (u, m) and y&#8217; = sn (u, m). cn (u, m) = x \/ r ( \u03b8 ,m) , sn (u, m) = y \/ r ( \u03b8 ,m) and dn (u, m) = 1 \/ r ( \u03b8 ,m). DERIVATIVES d\/du { pq (u, m)} q c s n d c 0 -ds ns -dn sn -m&#8217; nd sd p s dc nc 0 cn dn cd nd n dc sc -cs ds 0 m cd sd d m&#8217; nc sc -cs ns -m cn sn 0 To be continued&#8230; INSPIRED BY AN ARTICLE FROM SCHOOL OF MATHEMATICS, UNIVERSITY OF LEEDS AND WIKIPEDIA. NO COPYRIGHT INFRINGEMENT INTENDED.","thumbnail_url":"https:\/\/soulofmathematics.com\/wp-content\/uploads\/2021\/05\/theta3.gif"}