# Weierstrass and Jacobi Elliptic Functions

## Documentation for package ‘elliptic’ version 1.4-0

- DESCRIPTION file.
- User guides, package vignettes and other documentation.
- Code demos. Use demo() to run them.

## Help Pages

A C D E F G H I J K L M N P R S T U V W Z misc

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