# Weierstrass and Jacobi Elliptic Functions

## Help Pages

 elliptic-package Weierstrass and Jacobi Elliptic Functions

## -- A --

 amn matrix a on page 637 as.primitive Converts basic periods to a primitive pair

## -- C --

 cc Jacobi form of the elliptic functions cd Jacobi form of the elliptic functions ck Coefficients of Laurent expansion of Weierstrass P function cn Jacobi form of the elliptic functions congruence Solves mx+by=1 for x and y coqueraux Fast, conceptually simple, iterative scheme for Weierstrass P functions cs Jacobi form of the elliptic functions

## -- D --

 dc Jacobi form of the elliptic functions dd Jacobi form of the elliptic functions divisor Number theoretic functions dn Jacobi form of the elliptic functions ds Jacobi form of the elliptic functions

## -- E --

 e16.1.1 quarter period K e16.27.1 Jacobi theta functions 1-4 e16.27.2 Jacobi theta functions 1-4 e16.27.3 Jacobi theta functions 1-4 e16.27.4 Jacobi theta functions 1-4 e16.28.1 Numerical verification of equations 16.28.1 to 16.28.5 e16.28.2 Numerical verification of equations 16.28.1 to 16.28.5 e16.28.3 Numerical verification of equations 16.28.1 to 16.28.5 e16.28.4 Numerical verification of equations 16.28.1 to 16.28.5 e16.28.5 Numerical verification of equations 16.28.1 to 16.28.5 e16.28.6 Derivative of theta1 e16.31.1 Jacobi theta functions 1-4 e16.31.2 Jacobi theta functions 1-4 e16.31.3 Jacobi theta functions 1-4 e16.31.4 Jacobi theta functions 1-4 e16.36.3 Jacobi form of the elliptic functions e16.36.6 Neville's form for the theta functions e16.36.6a Neville's form for the theta functions e16.36.6b Neville's form for the theta functions e16.36.7 Neville's form for the theta functions e16.36.7a Neville's form for the theta functions e16.36.7b Neville's form for the theta functions e16.37.1 Neville's form for the theta functions e16.37.2 Neville's form for the theta functions e16.37.3 Neville's form for the theta functions e16.37.4 Neville's form for the theta functions e16.38.1 Neville's form for the theta functions e16.38.2 Neville's form for the theta functions e16.38.3 Neville's form for the theta functions e16.38.4 Neville's form for the theta functions e18.1.1 Calculates the invariants g2 and g3 e18.10.1 Weierstrass P and related functions e18.10.10 Numerical checks of equations 18.10.9-11, page 650 e18.10.10a Numerical checks of equations 18.10.9-11, page 650 e18.10.10b Numerical checks of equations 18.10.9-11, page 650 e18.10.11 Numerical checks of equations 18.10.9-11, page 650 e18.10.11a Numerical checks of equations 18.10.9-11, page 650 e18.10.11b Numerical checks of equations 18.10.9-11, page 650 e18.10.12 Numerical checks of equations 18.10.9-11, page 650 e18.10.12a Numerical checks of equations 18.10.9-11, page 650 e18.10.12b Numerical checks of equations 18.10.9-11, page 650 e18.10.2 Weierstrass P and related functions e18.10.3 Weierstrass P and related functions e18.10.4 Weierstrass P and related functions e18.10.5 Weierstrass P and related functions e18.10.6 Weierstrass P and related functions e18.10.7 Weierstrass P and related functions e18.10.9 Numerical checks of equations 18.10.9-11, page 650 e18.10.9a Numerical checks of equations 18.10.9-11, page 650 e18.10.9b Numerical checks of equations 18.10.9-11, page 650 e18.3.1 Calculate e1, e2, e3 from the invariants e18.3.3 Parameters for Weierstrass's P function e18.3.37 Parameters for Weierstrass's P function e18.3.38 Parameters for Weierstrass's P function e18.3.39 Parameters for Weierstrass's P function e18.3.5 Parameters for Weierstrass's P function e18.3.7 Calculate e1, e2, e3 from the invariants e18.3.8 Calculate e1, e2, e3 from the invariants e18.5.1 Laurent series for elliptic and related functions e18.5.16 Coefficients of Laurent expansion of Weierstrass P function e18.5.2 Coefficients of Laurent expansion of Weierstrass P function e18.5.3 Coefficients of Laurent expansion of Weierstrass P function e18.5.4 Laurent series for elliptic and related functions e18.5.5 Laurent series for elliptic and related functions e18.5.6 Laurent series for elliptic and related functions e18.7.4 Parameters for Weierstrass's P function e18.7.5 Parameters for Weierstrass's P function e18.7.7 Parameters for Weierstrass's P function e18f.5.3 Laurent series for elliptic and related functions e1e2e3 Calculate e1, e2, e3 from the invariants eee.cardano Calculate e1, e2, e3 from the invariants elliptic Weierstrass and Jacobi Elliptic Functions equianharmonic Special cases of the Weierstrass elliptic function eta Dedekind's eta function eta.series Dedekind's eta function

## -- F --

 factorize Number theoretic functions farey Farey sequences fpp Fundamental period parallelogram

## -- G --

 g.fun Calculates the invariants g2 and g3 g2.fun Calculates the invariants g2 and g3 g2.fun.direct Calculates the invariants g2 and g3 g2.fun.divisor Calculates the invariants g2 and g3 g2.fun.fixed Calculates the invariants g2 and g3 g2.fun.lambert Calculates the invariants g2 and g3 g2.fun.vectorized Calculates the invariants g2 and g3 g3.fun Calculates the invariants g2 and g3 g3.fun.direct Calculates the invariants g2 and g3 g3.fun.divisor Calculates the invariants g2 and g3 g3.fun.fixed Calculates the invariants g2 and g3 g3.fun.lambert Calculates the invariants g2 and g3 g3.fun.vectorized Calculates the invariants g2 and g3 GP Wrappers for PARI functions Gp Wrappers for PARI functions gp Wrappers for PARI functions

## -- H --

 H Jacobi theta functions 1-4 H1 Jacobi theta functions 1-4 half.periods Calculates half periods in terms of e

## -- I --

 Im<- Manipulate real or imaginary components of an object integrate.contour Complex integration integrate.segments Complex integration is.primitive Converts basic periods to a primitive pair

## -- J --

 J Various modular functions

## -- K --

 K.fun quarter period K

## -- L --

 lambda Various modular functions latplot Plots a lattice of periods on the complex plane lattice Lattice of complex numbers lemniscatic Special cases of the Weierstrass elliptic function limit Limit the magnitude of elements of a vector liouville Number theoretic functions

## -- M --

 massage Massages numbers near the real line to be real mn Fundamental period parallelogram mob Moebius transformations mobius Number theoretic functions myintegrate Complex integration

## -- N --

 nc Jacobi form of the elliptic functions nd Jacobi form of the elliptic functions near.match Are two vectors close to one another? Newton_Raphson Newton Raphson iteration to find roots of equations Newton_raphson Newton Raphson iteration to find roots of equations newton_Raphson Newton Raphson iteration to find roots of equations newton_raphson Newton Raphson iteration to find roots of equations nn Jacobi form of the elliptic functions nome Nome in terms of m or k nome.k Nome in terms of m or k ns Jacobi form of the elliptic functions

## -- P --

 P Weierstrass P and related functions P.laurent Laurent series for elliptic and related functions P.pari Wrappers for PARI functions p1.tau Does the right thing when calling g2.fun() and g3.fun() parameters Parameters for Weierstrass's P function PARI Wrappers for PARI functions pari Wrappers for PARI functions Pdash Weierstrass P and related functions Pdash.laurent Laurent series for elliptic and related functions primes Number theoretic functions pseudolemniscatic Special cases of the Weierstrass elliptic function

## -- R --

 Re<- Manipulate real or imaginary components of an object residue Complex integration

## -- S --

 sc Jacobi form of the elliptic functions sd Jacobi form of the elliptic functions sigma Weierstrass P and related functions sigma.laurent Laurent series for elliptic and related functions sigmadash.laurent Laurent series for elliptic and related functions sn Jacobi form of the elliptic functions sqrti Generalized square root ss Jacobi form of the elliptic functions

## -- T --

 Theta Jacobi theta functions 1-4 theta Jacobi theta functions 1-4 theta.00 Jacobi theta functions 1-4 theta.01 Jacobi theta functions 1-4 theta.10 Jacobi theta functions 1-4 theta.11 Jacobi theta functions 1-4 theta.c Neville's form for the theta functions theta.d Neville's form for the theta functions theta.n Neville's form for the theta functions theta.neville Neville's form for the theta functions theta.s Neville's form for the theta functions Theta1 Jacobi theta functions 1-4 theta1 Jacobi theta functions 1-4 theta1.dash.zero Derivative of theta1 theta1.dash.zero.q Derivative of theta1 theta1dash Derivatives of theta functions theta1dashdash Derivatives of theta functions theta1dashdashdash Derivatives of theta functions theta2 Jacobi theta functions 1-4 theta3 Jacobi theta functions 1-4 theta4 Jacobi theta functions 1-4 totient Number theoretic functions

## -- U --

 unimodular Unimodular matrices unimodularity Unimodular matrices

## -- V --

 view Visualization of complex functions

## -- W --

 WeierstrassP Weierstrass P and related functions

## -- Z --

 zeta Weierstrass P and related functions zeta.laurent Laurent series for elliptic and related functions

## -- misc --

 %mob% Moebius transformations 18.5.7 matrix a on page 637 18.5.8 matrix a on page 637