JacobiEllipticFunctions.h File Reference

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Classes
class	CJacobiEllipticFunctions
	Class that contains methods for computing Jacobi elliptic functions related to complex analysis. These functions are inverse of the elliptic integral of first kind, i.e. \[ u(k,m)=\int_{0}^{k}\frac{dt}{\sqrt{(1-t^{2})(1-m^{2}t^{2})}} =\int_{0}^{\varphi}\frac{d\theta}{\sqrt{(1-m^{2}sin^{2}\theta)}} \] where \(k=sin\varphi\), \(t=sin\theta\) and parameter \(m, 0\le m \le 1\) is called modulus. Three main Jacobi elliptic functions are defined as \(sn(u,m)=k=sin\theta\), \(cn(u,m)=cos\theta=\sqrt{1-sn(u,m)^{2}}\) and \(dn(u,m)=\sqrt{1-m^{2}sn(u,m)^{2}}\). For \(k=1\), i.e. \(\varphi=\frac{\pi}{2}\), \(u(1,m)=K(m)\) is known as the complete elliptic integral of first kind. Similarly, \(u(1,m'))= K'(m')\), \(m'=\sqrt{1-m^{2}}\) is called the complementary complete elliptic integral of first kind. Jacobi functions are double periodic with quardratic periods \(K\) and \(K'\). More...