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Zbl 0697.33003
Sala, Kenneth L.
Transformations of the Jacobian amplitude function and its calculation via the arithmetic-geometric mean. (English)
[J] SIAM J. Math. Anal. 20, No.6, 1514-1528 (1989). ISSN 0036-1410; ISSN 1095-7154/e

Three members of a family of Jacobian elliptic functions are given below: $$ sn(z,m)=\sin [am(z;m)]=\frac{\theta\sb 3}{\theta\sb 2}\frac{\theta\sb 1(z/\theta\sp 2\sb 3;q)}{\theta\sb 4(z/(\theta\sp 2\sb 3;q)}, $$ $$ cn(z,m)=\cos [am(z;m)]=\frac{\theta\sb 4}{\theta\sb 2}\frac{\theta\sb 2(z/\theta\sp 2\sb 3;q)}{\theta\sb 4(z/\theta\sp 2\sb 3;q)}, $$ $$ dn(z,m)=\frac{d}{dz}am(z,m)=\frac{\theta\sb 4}{\theta\sb 3}\frac{\theta\sb 3(z/\theta\sp 2\sb 3,q)}{\theta\sb 4(z/\theta\sp 2\sb 3,q)} $$ where am(z;m) is the Jacobian amplitude function, m is the Jacobian parameter $(\kappa =+m\sp{1/2}$ is the modulus), $q=\exp [-\pi \kappa '(m)/\kappa (m)]$ is the nome with $\kappa$ (m) and $\kappa '(m)=\kappa (1-m)$ the Jacobian quarter periods, and $\theta\sb i(z,q)$, $k=1,2,3,4$ are the theta functions with $\theta\sb i$ denoting $\theta\sb i(z=0;q)$. With the aid of the Poisson summation formula, expressions for the Jacobian amplitude function, am(z;m) along with the complete set of Jacobian elliptic functions are given that, aside from their branchpoints and poles, respectively, are convergent throughout the complex plane for arbitrary parameter m. By usilizing the expression for am(z;m), its periodicity properties are determined in each of the regions $m<0$, $0<m<1$, and $m>1$. Novel yet fundamental identities are presented describing various linear and quadratic transformations of the Jacobian amplitude function. Finally, that method based on the arithmetic- geometric mean and most widely employed for calculating the Jacobian elliptic functions is shown to be, when interpreted explicitly in terms of am(z,m) and its transformation properties, a method first and foremost for the calculation of the Jacobian amplitude and coamplitude functions from which the elliptic functions themselves are subsequently evaluated by means of simple, trigonometric identities.
[R.S.Dahiya]

MSC 2000:: *33E05 Elliptic functions and integrals

Keywords: elliptic functions

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