Package 'jacobi' reference manual

Title:	Jacobi Theta Functions and Related Functions
Description:	Evaluation of the Jacobi theta functions and related functions: Weierstrass elliptic function, Weierstrass sigma function, Weierstrass zeta function, Klein j-function, Dedekind eta function, lambda modular function, Jacobi elliptic functions, Neville theta functions, Eisenstein series, lemniscate elliptic functions, elliptic alpha function, Rogers-Ramanujan continued fractions, and Dixon elliptic functions. Complex values of the variable are supported.
Authors:	Stéphane Laurent [aut, cre], Mikael Fremling [aut] (author of the original Fortran code for the theta functions)
Maintainer:	Stéphane Laurent <[email protected]>
License:	GPL-3
Version:	3.1.1
Built:	2024-11-02 04:43:12 UTC
Source:	https://github.com/stla/jacobi

Arithmetic-geometric mean

Description

Evaluation of the arithmetic-geometric mean of two complex numbers.

Usage

agm(x, y)
agm(x, y)

Arguments

x, y

complex numbers

Value

A complex number, the arithmetic-geometric mean of x and y.

Examples

agm(1, sqrt(2))
2*pi^(3/2)*sqrt(2) / gamma(1/4)^2
agm(1, sqrt(2))
2*pi^(3/2)*sqrt(2) / gamma(1/4)^2

Amplitude function

Description

Evaluation of the amplitude function.

Usage

am(u, m)
am(u, m)

Arguments

`u`	complex number
`m`	square of elliptic modulus, a complex number

Value

A complex number.

Examples

library(Carlson)
phi <- 1 + 1i
m <- 2
u <- elliptic_F(phi, m)
am(u, m) # should be phi
library(Carlson)
phi <- 1 + 1i
m <- 2
u <- elliptic_F(phi, m)
am(u, m) # should be phi

Costa surface

Description

Computes a mesh of the Costa surface.

Usage

CostaMesh(nu = 50L, nv = 50L)
CostaMesh(nu = 50L, nv = 50L)

Arguments

nu, nv

numbers of subdivisions

Value

A triangle rgl mesh (object of class mesh3d).

Examples

library(jacobi)
library(rgl)

mesh <- CostaMesh(nu = 250, nv = 250)
open3d(windowRect = c(50, 50, 562, 562), zoom = 0.9)
bg3d("#15191E")
shade3d(mesh, color = "darkred", back = "cull")
shade3d(mesh, color = "orange", front = "cull")

library(jacobi)
library(rgl)

mesh <- CostaMesh(nu = 250, nv = 250)
open3d(windowRect = c(50, 50, 562, 562), zoom = 0.9)
bg3d("#15191E")
shade3d(mesh, color = "darkred", back = "cull")
shade3d(mesh, color = "orange", front = "cull")

Disk to upper half-plane

Description

Conformal map from the unit disk to the upper half-plane. The function is vectorized.

Usage

disk2H(z)
disk2H(z)

Arguments

`z`	a complex number in the unit disk

Value

A complex number in the upper half-plane.

Examples

# map the disk to H and calculate kleinj
f <- function(x, y) {
  z <- complex(real = x, imaginary = y)
  K <- rep(NA_complex_, length(x))
  inDisk <- Mod(z) < 1
  K[inDisk] <- kleinj(disk2H(z[inDisk]))
  K
}
n <- 1024L
x <- y <- seq(-1, 1, length.out = n)
Grid <- expand.grid(X = x, Y = y)
K <- f(Grid$X, Grid$Y)
dim(K) <- c(n, n)
# plot
if(require("RcppColors")) {
  img <- colorMap5(K)
} else {
  img <- as.raster(1 - abs(Im(K))/Mod(K))
}
opar <- par(mar = c(0, 0, 0, 0))
plot(NULL, xlim = c(0, 1), ylim = c(0, 1), asp = 1, 
     axes = FALSE, xaxs = "i", yaxs = "i", xlab = NA, ylab = NA)
rasterImage(img, 0, 0, 1, 1)
par(opar)
# map the disk to H and calculate kleinj
f <- function(x, y) {
  z <- complex(real = x, imaginary = y)
  K <- rep(NA_complex_, length(x))
  inDisk <- Mod(z) < 1
  K[inDisk] <- kleinj(disk2H(z[inDisk]))
  K
}
n <- 1024L
x <- y <- seq(-1, 1, length.out = n)
Grid <- expand.grid(X = x, Y = y)
K <- f(Grid$X, Grid$Y)
dim(K) <- c(n, n)
# plot
if(require("RcppColors")) {
  img <- colorMap5(K)
} else {
  img <- as.raster(1 - abs(Im(K))/Mod(K))
}
opar <- par(mar = c(0, 0, 0, 0))
plot(NULL, xlim = c(0, 1), ylim = c(0, 1), asp = 1, 
     axes = FALSE, xaxs = "i", yaxs = "i", xlab = NA, ylab = NA)
rasterImage(img, 0, 0, 1, 1)
par(opar)

Disk to square

Description

Conformal map from the unit disk to the square $[-1,1] \times [-1,1]$ . The function is vectorized.

Usage

disk2square(z)
disk2square(z)

Arguments

`z`	a complex number in the unit disk

Value

A complex number in the square $[-1,1] \times [-1,1]$ .

Examples

n <- 70L
r <- seq(0, 1, length.out = n)
theta <- seq(0, 2*pi, length.out = n+1L)[-1L]
Grid <- transform(
  expand.grid(R = r, Theta = theta),
  Z = R*exp(1i*Theta)
)
s <- vapply(Grid$Z, disk2square, complex(1L))
plot(Re(s), Im(s), pch = ".", asp = 1, cex = 2)
#
# a more insightful plot ####
r_ <- seq(0, 1, length.out = 10L)
theta_ <- seq(0, 2*pi, length.out = 33)[-1L]
plot(
  NULL, xlim = c(-1, 1), ylim = c(-1, 1), asp = 1, xlab = "x", ylab = "y"
)
for(r in r_) {
  theta <- sort(
    c(seq(0, 2, length.out = 200L), c(1/4, 3/4, 5/4, 7/4))
  )
  z <- r*(cospi(theta) + 1i*sinpi(theta))
  s <- vapply(z, disk2square, complex(1L))
  lines(Re(s), Im(s), col = "blue", lwd = 2)
}
for(theta in theta_) {
  r <- seq(0, 1, length.out = 30L)
  z <- r*exp(1i*theta)
  s <- vapply(z, disk2square, complex(1L))
  lines(Re(s), Im(s), col = "green", lwd = 2)
}
n <- 70L
r <- seq(0, 1, length.out = n)
theta <- seq(0, 2*pi, length.out = n+1L)[-1L]
Grid <- transform(
  expand.grid(R = r, Theta = theta),
  Z = R*exp(1i*Theta)
)
s <- vapply(Grid$Z, disk2square, complex(1L))
plot(Re(s), Im(s), pch = ".", asp = 1, cex = 2)
#
# a more insightful plot ####
r_ <- seq(0, 1, length.out = 10L)
theta_ <- seq(0, 2*pi, length.out = 33)[-1L]
plot(
  NULL, xlim = c(-1, 1), ylim = c(-1, 1), asp = 1, xlab = "x", ylab = "y"
)
for(r in r_) {
  theta <- sort(
    c(seq(0, 2, length.out = 200L), c(1/4, 3/4, 5/4, 7/4))
  )
  z <- r*(cospi(theta) + 1i*sinpi(theta))
  s <- vapply(z, disk2square, complex(1L))
  lines(Re(s), Im(s), col = "blue", lwd = 2)
}
for(theta in theta_) {
  r <- seq(0, 1, length.out = 30L)
  z <- r*exp(1i*theta)
  s <- vapply(z, disk2square, complex(1L))
  lines(Re(s), Im(s), col = "green", lwd = 2)
}

Dixon elliptic functions

Description

The Dixon elliptic functions.

Usage

sm(z)

cm(z)
sm(z)

cm(z)

Arguments

`z`	a real or complex number

Value

A complex number.

Examples

# cubic Fermat curve x^3+y^3=1
pi3 <- beta(1/3, 1/3)
epsilon <- 0.7
t_ <- seq(-pi3/3 + epsilon, 2*pi3/3 - epsilon, length.out = 100)
pts <- t(vapply(t_, function(t) {
  c(Re(cm(t)), Re(sm(t)))
}, FUN.VALUE = numeric(2L)))
plot(pts, type = "l", asp = 1)
# cubic Fermat curve x^3+y^3=1
pi3 <- beta(1/3, 1/3)
epsilon <- 0.7
t_ <- seq(-pi3/3 + epsilon, 2*pi3/3 - epsilon, length.out = 100)
pts <- t(vapply(t_, function(t) {
  c(Re(cm(t)), Re(sm(t)))
}, FUN.VALUE = numeric(2L)))
plot(pts, type = "l", asp = 1)

Eisenstein series

Description

Evaluation of Eisenstein series with weight 2, 4 or 6.

Usage

EisensteinE(n, q)
EisensteinE(n, q)

Arguments

`n`	the weight, can be 2, 4 or 6
`q`	nome, complex number with modulus smaller than one

Value

A complex number, the value of the Eisenstein series.

Elliptic alpha function

Description

Evaluates the elliptic alpha function.

Usage

ellipticAlpha(z)
ellipticAlpha(z)

Arguments

`z`	a complex number

Value

A complex number.

References

Weisstein, Eric W. "Elliptic Alpha Function".

Elliptic invariants

Description

Elliptic invariants from half-periods.

Usage

ellipticInvariants(omega1omega2)
ellipticInvariants(omega1omega2)

Arguments

omega1omega2

the half-periods, a vector of two complex numbers

Value

The elliptic invariants, a vector of two complex numbers.

Dedekind eta function

Description

Evaluation of the Dedekind eta function.

Usage

eta(tau)
eta(tau)

Arguments

tau

a vector of complex numbers with strictly positive imaginary parts

Value

A vector of complex numbers.

Examples

eta(2i)
gamma(1/4) / 2^(11/8) / pi^(3/4)
eta(2i)
gamma(1/4) / 2^(11/8) / pi^(3/4)

Half-periods

Description

Half-periods from elliptic invariants.

Usage

halfPeriods(g2g3)
halfPeriods(g2g3)

Arguments

g2g3

the elliptic invariants, a vector of two complex numbers

Value

The half-periods, a vector of two complex numbers.

Jacobi elliptic functions

Description

Evaluation of the Jacobi elliptic functions.

Usage

jellip(kind, u, tau = NULL, m = NULL)
jellip(kind, u, tau = NULL, m = NULL)

Arguments

`kind`	a string with two characters among `"s"`, `"c"`, `"d"` and `"n"`; this string specifies the function: the two letters respectively denote the basic functions $sn$ , $cn$ , $dn$ and $1$ , and the string specifies the ratio of two such functions, e.g. $ns = 1/sn$ and $cd = cn/dn$
`u`	a complex number, vector or matrix
`tau`	complex number with strictly positive imaginary part; it is related to `m` and only one of them must be supplied
`m`	the "parameter", square of the elliptic modulus; it is related to `tau` and only one of them must be supplied

Value

A complex number, vector or matrix.

Examples

u <- 2 + 2i
tau <- 1i
jellip("cn", u, tau)^2 + jellip("sn", u, tau)^2 # should be 1
u <- 2 + 2i
tau <- 1i
jellip("cn", u, tau)^2 + jellip("sn", u, tau)^2 # should be 1

Jacobi theta function with characteristics

Description

Evaluates the Jacobi theta function with characteristics.

Usage

jtheta_ab(a, b, z, tau = NULL, q = NULL)
jtheta_ab(a, b, z, tau = NULL, q = NULL)

Arguments

`a`, `b`	the characteristics, two complex numbers
`z`	complex number, vector, or matrix
`tau`	lattice parameter, a complex number with strictly positive imaginary part; the two complex numbers `tau` and `q` are related by `q = exp(1ipitau)`, and only one of them must be supplied
`q`	the nome, a complex number whose modulus is strictly less than one, but not zero

Details

The Jacobi theta function with characteristics generalizes the four Jacobi theta functions. It is denoted by 𝜃[a,b](z|τ). One gets the four Jacobi theta functions when a and b take the values 0 or 0.5:

if a=b=0.5: then one gets -𝜗₁(z|τ)
if a=0.5 and b=0: then one gets 𝜗₂(z|τ)
if a=b=0: then one gets 𝜗₃(z|τ)
if a=0 and b=0.5: then one gets 𝜗₄(z|τ)

Both 𝜃[a,b](z+π|τ) and 𝜃[a,b](z+π×τ|τ) are equal to 𝜃[a,b](z|τ) up to a factor - see the examples for the details.

Value

A complex number, vector or matrix, like z.

Note

Different conventions are used in the book cited as reference.

References

Hershel M. Farkas, Irwin Kra. Theta Constants, Riemann Surfaces and the Modular Group: An Introduction with Applications to Uniformization Theorems, Partition Identities and Combinatorial Number Theory. Graduate Studies in Mathematics, volume 37, 2001.

Examples

a   <- 2 + 0.3i
b   <- 1 - 0.6i
z   <- 0.1 + 0.4i
tau <- 0.2 + 0.3i
jab <- jtheta_ab(a, b, z, tau) 
# first property ####
jtheta_ab(a, b, z + pi, tau) # is equal to:
jab * exp(2i*pi*a)
# second property ####
jtheta_ab(a, b, z + pi*tau, tau) # is equal to:
jab * exp(-1i*(pi*tau + 2*z + 2*pi*b))
a   <- 2 + 0.3i
b   <- 1 - 0.6i
z   <- 0.1 + 0.4i
tau <- 0.2 + 0.3i
jab <- jtheta_ab(a, b, z, tau) 
# first property ####
jtheta_ab(a, b, z + pi, tau) # is equal to:
jab * exp(2i*pi*a)
# second property ####
jtheta_ab(a, b, z + pi*tau, tau) # is equal to:
jab * exp(-1i*(pi*tau + 2*z + 2*pi*b))

Jacobi theta function one

Description

Evaluates the first Jacobi theta function.

Usage

jtheta1(z, tau = NULL, q = NULL)

ljtheta1(z, tau = NULL, q = NULL)
jtheta1(z, tau = NULL, q = NULL)

ljtheta1(z, tau = NULL, q = NULL)

Arguments

`z`	complex number, vector, or matrix
`tau`	lattice parameter, a complex number with strictly positive imaginary part; the two complex numbers `tau` and `q` are related by `q = exp(1ipitau)`, and only one of them must be supplied
`q`	the nome, a complex number whose modulus is strictly less than one, but not zero

Value

A complex number, vector or matrix; jtheta1 evaluates the first Jacobi theta function and ljtheta1 evaluates its logarithm.

Examples

jtheta1(1 + 1i, q = exp(-pi/2))
jtheta1(1 + 1i, q = exp(-pi/2))

Jacobi theta function two

Description

Evaluates the second Jacobi theta function.

Usage

jtheta2(z, tau = NULL, q = NULL)

ljtheta2(z, tau = NULL, q = NULL)
jtheta2(z, tau = NULL, q = NULL)

ljtheta2(z, tau = NULL, q = NULL)

Arguments

`z`	complex number, vector, or matrix
`tau`	lattice parameter, a complex number with strictly positive imaginary part; the two complex numbers `tau` and `q` are related by `q = exp(1ipitau)`, and only one of them must be supplied
`q`	the nome, a complex number whose modulus is strictly less than one, but not zero

Value

A complex number, vector or matrix; jtheta2 evaluates the second Jacobi theta function and ljtheta2 evaluates its logarithm.

Examples

jtheta2(1 + 1i, q = exp(-pi/2))
jtheta2(1 + 1i, q = exp(-pi/2))

Jacobi theta function three

Description

Evaluates the third Jacobi theta function.

Usage

jtheta3(z, tau = NULL, q = NULL)

ljtheta3(z, tau = NULL, q = NULL)
jtheta3(z, tau = NULL, q = NULL)

ljtheta3(z, tau = NULL, q = NULL)

Arguments

`z`	complex number, vector, or matrix
`tau`	lattice parameter, a complex number with strictly positive imaginary part; the two complex numbers `tau` and `q` are related by `q = exp(1ipitau)`, and only one of them must be supplied
`q`	the nome, a complex number whose modulus is strictly less than one, but not zero

Value

A complex number, vector or matrix; jtheta3 evaluates the third Jacobi theta function and ljtheta3 evaluates its logarithm.

Examples

jtheta3(1 + 1i, q = exp(-pi/2))
jtheta3(1 + 1i, q = exp(-pi/2))

Jacobi theta function four

Description

Evaluates the fourth Jacobi theta function.

Usage

jtheta4(z, tau = NULL, q = NULL)

ljtheta4(z, tau = NULL, q = NULL)
jtheta4(z, tau = NULL, q = NULL)

ljtheta4(z, tau = NULL, q = NULL)

Arguments

`z`	complex number, vector, or matrix
`tau`	lattice parameter, a complex number with strictly positive imaginary part; the two complex numbers `tau` and `q` are related by `q = exp(1ipitau)`, and only one of them must be supplied
`q`	the nome, a complex number whose modulus is strictly less than one, but not zero

Value

A complex number, vector or matrix; jtheta4 evaluates the fourth Jacobi theta function and ljtheta4 evaluates its logarithm.

Examples

jtheta4(1 + 1i, q = exp(-pi/2))
jtheta4(1 + 1i, q = exp(-pi/2))

Klein j-function and its inverse

Description

Evaluation of the Klein j-invariant function and its inverse.

Usage

kleinj(tau, transfo = FALSE)

kleinjinv(j)
kleinj(tau, transfo = FALSE)

kleinjinv(j)

Arguments

`tau`	a complex number with strictly positive imaginary part, or a vector or matrix of such complex numbers; missing values allowed
`transfo`	Boolean, whether to use a transformation of the values of `tau` close to the real line; using this option can fix some failures of the computation (at the cost of speed), e.g. when the algorithm reaches the maximal number of iterations
`j`	a complex number

Value

A complex number, vector or matrix.

Note

The Klein-j function is the one with the factor 1728.

Examples

( j <- kleinj(2i) )
66^3
kleinjinv(j)
( j <- kleinj(2i) )
66^3
kleinjinv(j)

Lambda modular function

Description

Evaluation of the lambda modular function.

Usage

lambda(tau, transfo = FALSE)
lambda(tau, transfo = FALSE)

Arguments

`tau`	a complex number with strictly positive imaginary part, or a vector or matrix of such complex numbers; missing values allowed
`transfo`	Boolean, whether to use a transformation of the values of `tau` close to the real line; using this option can fix some failures of the computation (at the cost of speed), e.g. when the algorithm reaches the maximal number of iterations

Value

A complex number, vector or matrix.

Note

The lambda function is the square of the elliptic modulus.

Examples

x <- 2
lambda(1i*sqrt(x)) + lambda(1i*sqrt(1/x)) # should be one
x <- 2
lambda(1i*sqrt(x)) + lambda(1i*sqrt(1/x)) # should be one

Lemniscate functions

Description

Lemniscate sine, cosine, arcsine, arccosine, hyperbolic sine, and hyperbolic cosine functions.

Usage

sl(z)

cl(z)

asl(z)

acl(z)

slh(z)

clh(z)
sl(z)

cl(z)

asl(z)

acl(z)

slh(z)

clh(z)

Arguments

`z`	a real number or a complex number

Value

A complex number.

Examples

sl(1+1i) * cl(1+1i) # should be 1
## | the lemniscate ####
# lemniscate parameterization
p <- Vectorize(function(s) {
  a <- Re(cl(s))
  b <- Re(sl(s))
  c(a, a * b) / sqrt(1 + b*b)
})
# lemnniscate constant
ombar <- 2.622 # gamma(1/4)^2 / (2 * sqrt(2*pi))
# plot
s_ <- seq(0, ombar, length.out = 100)
lemniscate <- t(p(s_))
plot(lemniscate, type = "l", col = "blue", lwd = 3)
lines(cbind(lemniscate[, 1L], -lemniscate[, 2L]), col="red", lwd = 3)
sl(1+1i) * cl(1+1i) # should be 1
## | the lemniscate ####
# lemniscate parameterization
p <- Vectorize(function(s) {
  a <- Re(cl(s))
  b <- Re(sl(s))
  c(a, a * b) / sqrt(1 + b*b)
})
# lemnniscate constant
ombar <- 2.622 # gamma(1/4)^2 / (2 * sqrt(2*pi))
# plot
s_ <- seq(0, ombar, length.out = 100)
lemniscate <- t(p(s_))
plot(lemniscate, type = "l", col = "blue", lwd = 3)
lines(cbind(lemniscate[, 1L], -lemniscate[, 2L]), col="red", lwd = 3)

Nome

Description

The nome in function of the parameter $m$ .

Usage

nome(m)
nome(m)

Arguments

`m`	the parameter, square of elliptic modulus, real or complex number

Value

A complex number.

Examples

nome(-2)
nome(-2)

Rogers-Ramanujan continued fraction

Description

Evaluates the Rogers-Ramanujan continued fraction.

Usage

RR(q)
RR(q)

Arguments

`q`	the nome, a complex number whose modulus is strictly less than one, and which is not zero

Value

A complex number

Note

This function is sometimes denoted by $R$ .

Alternating Rogers-Ramanujan continued fraction

Description

Evaluates the alternating Rogers-Ramanujan continued fraction.

Usage

RRa(q)
RRa(q)

Arguments

`q`	the nome, a complex number whose modulus is strictly less than one, and which is not zero

Value

A complex number

Note

This function is sometimes denoted by $S$ .

Square to disk

Description

Conformal map from the unit square to the unit disk. The function is vectorized.

Usage

square2disk(z)
square2disk(z)

Arguments

`z`	a complex number in the unit square $[0,1] \times [0,1]$

Value

A complex number in the unit disk.

Examples

x <- y <- seq(0, 1, length.out = 25L)
Grid <- transform(
  expand.grid(X = x, Y = y), 
  Z = complex(real = X, imaginary = Y)
)
u <- square2disk(Grid$Z)
plot(u, pch = 19, asp = 1)
x <- y <- seq(0, 1, length.out = 25L)
Grid <- transform(
  expand.grid(X = x, Y = y), 
  Z = complex(real = X, imaginary = Y)
)
u <- square2disk(Grid$Z)
plot(u, pch = 19, asp = 1)

Square to upper half-plane

Description

Conformal map from the unit square to the upper half-plane. The function is vectorized.

Usage

square2H(z)
square2H(z)

Arguments

`z`	a complex number in the unit square $[0,1] \times [0,1]$

Value

A complex number in the upper half-plane.

Examples

n <- 1024L
x <- y <- seq(0.0001, 0.9999, length.out = n)
Grid <- transform(
  expand.grid(X = x, Y = y), 
  Z = complex(real = X, imaginary = Y)
)
K <- kleinj(square2H(Grid$Z))
dim(K) <- c(n, n)
# plot
if(require("RcppColors")) {
  img <- colorMap5(K)
} else {
  img <- as.raster((Arg(K) + pi)/(2*pi))
}
opar <- par(mar = c(0, 0, 0, 0))
plot(NULL, xlim = c(0, 1), ylim = c(0, 1), asp = 1, 
     axes = FALSE, xaxs = "i", yaxs = "i", xlab = NA, ylab = NA)
rasterImage(img, 0, 0, 1, 1)
par(opar)
n <- 1024L
x <- y <- seq(0.0001, 0.9999, length.out = n)
Grid <- transform(
  expand.grid(X = x, Y = y), 
  Z = complex(real = X, imaginary = Y)
)
K <- kleinj(square2H(Grid$Z))
dim(K) <- c(n, n)
# plot
if(require("RcppColors")) {
  img <- colorMap5(K)
} else {
  img <- as.raster((Arg(K) + pi)/(2*pi))
}
opar <- par(mar = c(0, 0, 0, 0))
plot(NULL, xlim = c(0, 1), ylim = c(0, 1), asp = 1, 
     axes = FALSE, xaxs = "i", yaxs = "i", xlab = NA, ylab = NA)
rasterImage(img, 0, 0, 1, 1)
par(opar)

Neville theta functions

Description

Evaluation of the Neville theta functions.

Usage

theta.s(z, tau = NULL, m = NULL)

theta.c(z, tau = NULL, m = NULL)

theta.n(z, tau = NULL, m = NULL)

theta.d(z, tau = NULL, m = NULL)
theta.s(z, tau = NULL, m = NULL)

theta.c(z, tau = NULL, m = NULL)

theta.n(z, tau = NULL, m = NULL)

theta.d(z, tau = NULL, m = NULL)

Arguments

`z`	a complex number, vector, or matrix
`tau`	complex number with strictly positive imaginary part; it is related to `m` and only one of them must be supplied
`m`	the "parameter", square of the elliptic modulus; it is related to `tau` and only one of them must be supplied

Value

A complex number, vector or matrix.

Weierstrass elliptic function

Description

Evaluation of the Weierstrass elliptic function and its derivatives.

Usage

wp(z, g = NULL, omega = NULL, tau = NULL, derivative = 0L)
wp(z, g = NULL, omega = NULL, tau = NULL, derivative = 0L)

Arguments

`z`	complex number, vector or matrix
`g`	the elliptic invariants, a vector of two complex numbers; only one of `g`, `omega` and `tau` must be given
`omega`	the half-periods, a vector of two complex numbers; only one of `g`, `omega` and `tau` must be given
`tau`	the half-periods ratio; supplying `tau` is equivalent to supply `omega = c(1/2, tau/2)`
`derivative`	differentiation order, an integer between 0 and 3

Value

A complex number, vector or matrix.

Examples

omega1 <- 1.4 - 1i
omega2 <- 1.6 + 0.5i
omega <- c(omega1, omega2)
e1 <- wp(omega1, omega = omega)
e2 <- wp(omega2, omega = omega)
e3 <- wp(-omega1-omega2, omega = omega)
e1 + e2 + e3 # should be 0
omega1 <- 1.4 - 1i
omega2 <- 1.6 + 0.5i
omega <- c(omega1, omega2)
e1 <- wp(omega1, omega = omega)
e2 <- wp(omega2, omega = omega)
e3 <- wp(-omega1-omega2, omega = omega)
e1 + e2 + e3 # should be 0

Inverse of Weierstrass elliptic function

Description

Evaluation of the inverse of the Weierstrass elliptic function.

Usage

wpinv(w, g = NULL, omega = NULL, tau = NULL)
wpinv(w, g = NULL, omega = NULL, tau = NULL)

Arguments

`w`	complex number
`g`	the elliptic invariants, a vector of two complex numbers; only one of `g`, `omega` and `tau` must be given
`omega`	the half-periods, a vector of two complex numbers; only one of `g`, `omega` and `tau` must be given
`tau`	the half-periods ratio; supplying `tau` is equivalent to supply `omega = c(1/2, tau/2)`

Value

A complex number.

Examples

library(jacobi)
omega <- c(1.4 - 1i, 1.6 + 0.5i)
w <- 1 + 1i
z <- wpinv(w, omega = omega)
wp(z, omega = omega) # should be w
library(jacobi)
omega <- c(1.4 - 1i, 1.6 + 0.5i)
w <- 1 + 1i
z <- wpinv(w, omega = omega)
wp(z, omega = omega) # should be w

Weierstrass sigma function

Description

Evaluation of the Weierstrass sigma function.

Usage

wsigma(z, g = NULL, omega = NULL, tau = NULL)
wsigma(z, g = NULL, omega = NULL, tau = NULL)

Arguments

`z`	a complex number, vector or matrix
`g`	the elliptic invariants, a vector of two complex numbers; only one of `g`, `omega` and `tau` must be given
`omega`	the half-periods, a vector of two complex numbers; only one of `g`, `omega` and `tau` must be given
`tau`	the half-periods ratio; supplying `tau` is equivalent to supply `omega = c(1/2, tau/2)`

Value

A complex number, vector or matrix.

Examples

wsigma(1, g = c(12, -8))
# should be equal to:
sin(1i*sqrt(3))/(1i*sqrt(3)) / sqrt(exp(1))
wsigma(1, g = c(12, -8))
# should be equal to:
sin(1i*sqrt(3))/(1i*sqrt(3)) / sqrt(exp(1))

Weierstrass zeta function

Description

Evaluation of the Weierstrass zeta function.

Usage

wzeta(z, g = NULL, omega = NULL, tau = NULL)
wzeta(z, g = NULL, omega = NULL, tau = NULL)

Arguments

`z`	complex number, vector or matrix
`g`	the elliptic invariants, a vector of two complex numbers; only one of `g`, `omega` and `tau` must be given
`omega`	the half-periods, a vector of two complex numbers; only one of `g`, `omega` and `tau` must be given
`tau`	the half-periods ratio; supplying `tau` is equivalent to supply `omega = c(1/2, tau/2)`

Value

A complex number, vector or matrix.

Examples

# Mirror symmetry property:
z <- 1 + 1i
g <- c(1i, 1+2i)
wzeta(Conj(z), Conj(g))
Conj(wzeta(z, g))
# Mirror symmetry property:
z <- 1 + 1i
g <- c(1i, 1+2i)
wzeta(Conj(z), Conj(g))
Conj(wzeta(z, g))

Package 'jacobi'

Help Index

Arithmetic-geometric mean

Description

Usage

Arguments

Value

Examples

Amplitude function

Description

Usage

Arguments

Value

Examples

Costa surface

Description

Usage

Arguments

Value

Examples

Disk to upper half-plane

Description

Usage

Arguments

Value

Examples

Disk to square

Description

Usage

Arguments

Value

Examples

Dixon elliptic functions

Description

Usage

Arguments

Value

Examples

Eisenstein series

Description

Usage

Arguments

Value

Elliptic alpha function

Description

Usage

Arguments

Value

References

Elliptic invariants

Description

Usage

Arguments

Value

Dedekind eta function

Description

Usage

Arguments

Value

Examples

Half-periods

Description

Usage

Arguments

Value

Jacobi elliptic functions

Description

Usage

Arguments

Value

Examples

Jacobi theta function with characteristics

Description

Usage

Arguments

Details

Value

Note

References

Examples