

   HHyyppeerrbboolliicc FFuunnccttiioonnss

        cosh(x)
        sinh(x)
        tanh(x)
        acosh(x)
        asinh(x)
        atanh(x)

   DDeessccrriippttiioonn::

        These functions give the obvious hyperbolic functions.
        They respectively compute the hyperbolic cosine, sine,
        tangent, arc-cosine, arc-sine, arc-tangent.

   SSeeee AAllssoo::

        `cos', `sin', `tan', `acos', `asin', `atan'.

   EExxaammpplleess::

        Ceps <- .Machine$double.eps # ``Computer epsilon''
        x <- rnorm(500)
        all(abs(cosh(x) - (exp(x) + exp(-x))/2) < 10*Ceps)
        all(abs(sinh(x) - (exp(x) - exp(-x))/2) < 10*Ceps)
        all(Mod(cosh(x) - cos(1i*x)) < 10*Ceps)
        all(Mod(sinh(x) - sin(1i*x)/1i) < 10*Ceps)
        all(abs(tanh(x)*cosh(x) - sinh(x)) < 10*Ceps)

        ## Inverse:
        all(abs(asinh(sinh(x)) - x) < 10*Ceps)
        x[abs(acosh(cosh(x)) - abs(x)) > 100*Ceps] #- imprecise for small x
        all(abs(atanh(tanh(x)) - x) < 100*Ceps)

        all(abs(asinh(x) - log(x + sqrt(x^2 + 1))) < 10*Ceps)
        cx <- cosh(x)
        all(abs(acosh(cx) - log(cx + sqrt(cx^2 - 1))) < 1000*Ceps)

