exphp: Power Hyperbolas and Power Hyperbolic Functions

In FatTailsR: Kiener Distributions and Fat Tails in Finance

Power Hyperbolas and Power Hyperbolic Functions

Description

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.

Usage

exphp(x, k = 1)

coshp(x, k = 1)

sinhp(x, k = 1)

tanhp(x, k = 1)

sechp(x, k = 1)

cosechp(x, k = 1)

cotanhp(x, k = 1)

Arguments

x	a numeric value, vector or matrix.
k	a numeric value, preferably strictly positive.

Details

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.

See Also

The nonlinear transformation kashp, the inverse power hyperbolas and the inverse power hyperbolic functions loghp.

Examples

### 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

FatTailsR documentation built on May 29, 2024, 1:33 a.m.