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.
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.
Examples
### Example 1x <-(-3:3)*3exphp(x, k =4)coshp(x, k =4)sinhp(x, k =4)tanhp(x, k =4)### Example 2 outer + plot(exphp, coshp, sinhp, tanhp)xmin <--10xd <-0.5x <- 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
See Also
The nonlinear transformation kashp, the inverse power hyperbolas and the inverse power hyperbolic functions loghp.