lemniscate function

Lemniscate functions

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

sl(z)

cl(z)

asl(z)

acl(z)

slh(z)

clh(z)

Arguments

• z: a real number or a complex number

Returns

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)