1.0SOUL OF MATHEMATICShttps://soulofmathematics.comRajarshi Deyhttps://soulofmathematics.com/index.php/author/rajarshidey1729gmail-com/rich600338<blockquote class="wp-embedded-content" data-secret="KDrUSVeY6e"><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=KDrUSVeY6e" width="600" height="338" title="“JACOBI ELLIPTIC FUNCTIONS” — SOUL OF MATHEMATICS" data-secret="KDrUSVeY6e" frameborder="0" marginwidth="0" marginheight="0" scrolling="no" class="wp-embedded-content"></iframe><script type="text/javascript"> /* <![CDATA[ */ /*! This file is auto-generated */ !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); /* ]]> */ </script> In the mathematical field of complex analysis ELLIPTIC FUNCTIONS are a special kind of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Originally those integrals occurred at the calculation of the arc length of an ellipse. Important elliptic functions are Jacobi elliptic functions and the Weierstrass p-function. In mathematics,  THETA FUNCTIONS  are special functions of several complex variables. They are important in many areas, including the theories of Abelian varieties and moduli spaces, and of quadratic forms. They have also been applied to soliton theory. When generalized to a Grassmann algebra, they also appear in quantum field theory. The most common form of theta function is that occurring in the theory of elliptic functions. THE JACOBI ELLIPSE The cos θ and sin θ are defined on a unit circle, with radius = 1 and angle θ = arc length of the unit circle measured from the positive x-axis. Similarly, Jacobi elliptic functions are defined on the unit ellipse, with a = 1. Let x2 + y2 / b2 = 1, b > 1, m = 1 – 1/b2, 0 < m < 1, x = r cos θ and y = r sin θ then, r ( θ ,m) = 1/√(1 – m sin2 θ). For each angle θ the parameter u = u ( θ ,m) = 0∫θ r ( θ ,m) dθ is computed. On the unit circle a=b=1, u would be an arc length. While u 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 θ , r sin θ) be a point on the ellipse, and let P’=(x’, y’)=(cos θ, sin θ) be the point where the unit circle intersects the line between P and the origin O. Then the familiar relations from the unit circle: x’ = r cos θ and y’ = r sin θ read for the ellipse as: x’ = cn (u, m) and y’ = sn (u, m). cn (u, m) = x / r ( θ ,m) , sn (u, m) = y / r ( θ ,m) and dn (u, m) = 1 / r ( θ ,m). DERIVATIVES d/du { pq (u, m)} q c s n d c 0 -ds ns -dn sn -m’ nd sd p s dc nc 0 cn dn cd nd n dc sc -cs ds 0 m cd sd d m’ nc sc -cs ns -m cn sn 0 To be continued… INSPIRED BY AN ARTICLE FROM SCHOOL OF MATHEMATICS, UNIVERSITY OF LEEDS AND WIKIPEDIA. NO COPYRIGHT INFRINGEMENT INTENDED.https://soulofmathematics.com/wp-content/uploads/2021/05/theta3.gif