Description Usage Arguments Details See Also Examples
These functions define the power hyperbola exphp
and the associated
power hyperbolic cosine, sine, tangent, secant, cosecant, cotangent.
They are similar to the traditional hyperbolic functions with term
x
receiving a nonlinear transformation via the function
kashp
.
1 2 3 4 5 6 7 8 9 10 11 12 13 |
x |
a numeric value, vector or matrix. |
k |
a numeric value, preferably strictly positive. |
exphp
function is defined for x in (-Inf, +Inf) by:
exphp(x, k) = exp( kashp(x, k) ) = exp( k * asinh(x / 2 / k) )
coshp
function is defined for x in (-Inf, +Inf) by:
coshp(x, k) = cosh( kashp(x, k) )
sinhp
function is defined for x in (-Inf, +Inf) by:
sinhp(x, k) = sinh( kashp(x, k) )
tanhp
function is defined for x in (-Inf, +Inf) by:
tanhp(x, k) = tanh( kashp(x, k) )
sechp
function is defined for x in (-Inf, +Inf) by:
sechp(x, k) = 1 / coshp(x, k)
cosechp
function is defined for x in (-Inf, 0) U (0, +Inf) by:
cosechp(x, k) = 1 / sinhp(x, k)
cotanhp
function is defined for x in (-Inf, 0) U (0, +Inf) by:
cotanhp(x, k) = 1 / tanhp(x, k)
The undesired case k = 0 returns 0 for sinhp and tanhp, 1 for exphp, coshp and sechp, Inf for cosechp and cotanhp.
If k is a vector of length > 1, then the use of the function
outer
is recommanded.
The nonlinear transformation kashp
, the inverse power
hyperbolas and the inverse power hyperbolic functions loghp
.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 | ### Example 1
x <- (-3:3)*3
exphp(x, k = 4)
coshp(x, k = 4)
sinhp(x, k = 4)
tanhp(x, k = 4)
### Example 2 outer + plot(exphp, coshp, sinhp, tanhp)
xmin <- -10
xd <- 0.5
x <- seq(xmin, -xmin, xd) ; names(x) <- x
k <- c(0.6, 1, 1.5, 2, 3.2, 10) ; names(k) <- k
olty <- c(2, 1, 2, 1, 2, 1, 1)
olwd <- c(1, 1, 2, 2, 3, 4, 2)
ocol <- c(2, 2, 4, 4, 3, 3, 1)
op <- par(mfrow = c(2,2), mgp = c(1.5,0.8,0), mar = c(3,3,2,1))
## exphp(x, k)
Texphp <- ts(cbind(outer(-x, k, exphp), "exp(-x/2)" = exp(-x/2)),
start = xmin, deltat = xd)
plot(Texphp, plot.type = "single", ylim = c(0,20),
lty = olty, lwd = olwd, col = ocol, xaxs = "i", yaxs = "i", xlab = "",
ylab = "", main = "exphp(-x, k)" )
legend("topright", title = expression(kappa), legend = colnames(Texphp),
inset = 0.02, lty = olty, lwd = olwd, col = ocol, cex = 0.7 )
## coshp(x, k)
Tcoshp <- ts(cbind(outer(x, k, coshp), "cosh(x/2)" = cosh(x/2)),
start = xmin, deltat = xd)
plot(Tcoshp, plot.type = "single", ylim = c(0,20),
lty = olty, lwd = olwd, col = ocol, xaxs = "i", yaxs = "i",
xlab = "", ylab = "", main = "coshp(x, k)" )
legend("top", title = expression(kappa), legend = colnames(Tcoshp),
inset = 0.02, lty = olty, lwd = olwd, col = ocol, cex = 0.7 )
## sinhp(x, k)
Tsinhp <- ts(cbind(outer(x, k, sinhp), "sinh(x/2)" = sinh(x/2)),
start = xmin, deltat=xd)
plot(Tsinhp, plot.type = "single", ylim = c(-10,10),
lty = olty, lwd = olwd, col = ocol, xaxs = "i", yaxs = "i",
xlab = "", ylab = "", main = "sinhp(x, k)" )
legend("topleft", title = expression(kappa), legend = colnames(Tsinhp),
inset = 0.02, lty = olty, lwd= olwd, col = ocol, cex = 0.7 )
## tanhp(x, k)
Ttanhp <- ts(cbind(outer(x, k, tanhp), "tanh(x/2)" = tanh(x/2)),
start = xmin, deltat = xd)
plot(Ttanhp, plot.type = "single", ylim = c(-1,1),
lty = olty, lwd = olwd, col = ocol, xaxs = "i", yaxs = "i", xlab = "",
ylab = "", main = "tanhp(x, k)" )
legend("topleft", title = expression(kappa), legend = colnames(Ttanhp),
inset = 0.02, lty = olty, lwd = olwd, col = ocol, cex = 0.7 )
### End Example 3
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