Description

Eq. (15.16) in Eichner (2017) as a result of Cardano's formula.

Usage

 `1` ```J1(u, cc = sqrt(5/3)) ```

Arguments

 `u` Numeric vector. `cc` Numeric constant, defaults to √(5/3).

Details

Using, for brevity's sake, J_{1a}(u, c) := -q_c(u) and J_{1b}(u, c) := J_{1a}(u, c)^2 + p_c^3, the definition of J_1 reads:

J_1(u, c) := [J_{1a}(u, c) + √(J_{1b}(u, c))]^{1/3} + [J_{1a}(u, c) - √(J_{1b}(u, c))]^{1/3}.

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).

Value

Vector of same length and mode as `u`.

Note

Eq. (15.16) in Eichner (2017), and hence J_1(u, c), requires c to be in [√(5/3), 3). If `cc` does not satisfy this requirement a warning (only) is issued.

`J_admissible`.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13``` ```u <- seq(0, 1, by = 0.01) c0 <- expression(sqrt(5/3)) c1 <- expression(sqrt(3) - 0.01) cgrid <- c(1.35, seq(1.4, 1.7, by = 0.1)) cvals <- c(eval(c0), cgrid, eval(c1)) Y <- sapply(cvals, function(cc, u) J1(u, cc = cc), u = u) cols <- rainbow(ncol(Y), end = 9/12) matplot(u, Y, type = "l", lty = "solid", col = cols, ylab = expression(J[1](u, c))) abline(h = 0) legend("topleft", title = "c", legend = c(c0, cgrid, c1), lty = 1, col = cols, cex = 0.8) ```