Costa surface

Description

Computes a mesh of the Costa surface.

Usage

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

Arguments

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")

Dixon elliptic functions

Description

The Dixon elliptic functions.

Usage

sm(z)

cm(z)

Arguments

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)

Eisenstein series

Description

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

Usage

EisensteinE(n, q)

Arguments

Value

A complex number, the value of the Eisenstein series.

Rogers-Ramanujan continued fraction

Description

Evaluates the Rogers-Ramanujan continued fraction.

Usage

RR(q)

Arguments

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)

Arguments

Value

A complex number

Note

This function is sometimes denoted by S.

Arithmetic-geometric mean

Description

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

Usage

agm(x, y)

Arguments

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

Amplitude function

Description

Evaluation of the amplitude function.

Usage

am(u, m)

Arguments

Value

A complex number.

Examples

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

Disk to upper half-plane

Description

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

Usage

disk2H(z)

Arguments

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)

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)

Arguments

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)
}

Elliptic alpha function

Description

Evaluates the elliptic alpha function.

Usage

ellipticAlpha(z)

Arguments

Value

A complex number.

References

Weisstein, Eric W. "Elliptic Alpha Function".

Elliptic invariants

Description

Elliptic invariants from half-periods.

Usage

ellipticInvariants(omega1omega2)

Arguments

Value

The elliptic invariants, a vector of two complex numbers.

Dedekind eta function

Description

Evaluation of the Dedekind eta function.

Usage

eta(tau)

Arguments

Value

A vector of complex numbers.

Examples

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

Half-periods

Description

Half-periods from elliptic invariants.

Usage

halfPeriods(g2g3)

Arguments

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)

Arguments

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

Jacobi theta function one

Description

Evaluates the first Jacobi theta function.

Usage

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

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

Arguments

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))

Jacobi theta function two

Description

Evaluates the second Jacobi theta function.

Usage

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

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

Arguments

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))

Jacobi theta function three

Description

Evaluates the third Jacobi theta function.

Usage

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

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

Arguments

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))

Jacobi theta function four

Description

Evaluates the fourth Jacobi theta function.

Usage

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

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

Arguments

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))

Jacobi theta function with characteristics

Description

Evaluates the Jacobi theta function with characteristics.

Usage

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

Arguments

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))

Klein j-function and its inverse

Description

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

Usage

kleinj(tau, transfo = FALSE)

kleinjinv(j)

Arguments

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)

Lambda modular function

Description

Evaluation of the lambda modular function.

Usage

lambda(tau, transfo = FALSE)

Arguments

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

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)

Arguments

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)

Nome

Description

The nome in function of the parameter m.

Usage

nome(m)

Arguments

Value

A complex number.

Examples

nome(-2)

Square to upper half-plane

Description

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

Usage

square2H(z)

Arguments

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)

Square to disk

Description

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

Usage

square2disk(z)

Arguments

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)

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)

Arguments

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)

Arguments

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

Inverse of Weierstrass elliptic function

Description

Evaluation of the inverse of the Weierstrass elliptic function.

Usage

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

Arguments

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

Weierstrass sigma function

Description

Evaluation of the Weierstrass sigma function.

Usage

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

Arguments

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))

Weierstrass zeta function

Description

Evaluation of the Weierstrass zeta function.

Usage

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

Arguments

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))