kader (version 0.0.8)

J2: J2

Description

Eq. (20) in Eichner (2017) (based on "Bronstein's formula for k = 3")

Usage

J2(u, cc = sqrt(5))

Arguments

u

Numeric vector.

cc

Numeric constant, defaults to \(\sqrt 5\).

Value

Vector of same length and mode as u.

Details

\(J_2(u, c) = 2\sqrt(-p_c) * sin(1/3 * arcsin(q_c(u) / (-p_c)^{3/2}))\)

For implementation details of \(q_c(u)\) and \(p_c\) see qc and pc, respectively.

For further mathematical details see Eichner (2017) and/or Eichner & Stute (2013).

See Also

J_admissible.

Examples

# NOT RUN {
u <- seq(0, 1, by = 0.01)
c0 <- expression(sqrt(3) + 0.01)
c1 <- expression(sqrt(5))
cgrid <- seq(1.85, 2.15, by = 0.1)
cvals <- c(eval(c0), cgrid, eval(c1))

Y <- sapply(cvals, function(cc, u) J2(u, cc = cc), u = u)
cols <- rainbow(ncol(Y), end = 9/12)
matplot(u, Y, type = "l", lty = "solid", col = cols,
  ylab = expression(J[2](u, c)))
abline(h = 0)
legend("topleft", title = "c", legend = c(c0, cgrid, c1),
  lty = 1, col = cols, cex = 0.8)
# }
# NOT RUN {
# }