acosh hyperbolic cosine inverse

## Syntax

    t = acosh(x)

## Arguments

x, t

  each is a real or complex vector or matrix. `t` has the sizes of `x`.

## Description

`acosh(x)` returns `t` such that `cosh(t)==x` and `real(t)>=0`. `-t` are always other possible answers.

For real input numbers `-1 < x < 1`, `real(t)==0` and `imag(t)` belongs to `]0, %pi[`.

For complex numbers `x`, `imag(t)` belongs to `[-pi, pi]` and any `t + k*%pi*%i` with integer k are other possible answers.

## Sample

subplot(2,2,1) x = -20:0.2:-1; t = acosh(x); plot(x, real(acosh(x))) xtitle("acosh(x) = a + pi.i", "x <= -1", "a > 0") xstring(-18, 0.5, "-a + pi.i other solution") subplot(2,1,2) x = -1:0.01:1; plot(x, imag(acosh(x))) title("acosh(x) = b.i is pure imaginary") xlabel("-1 < x < 1") ylabel("0 < b < pi") subplot(2,2,2) x = 1:0.2:20; plot(x, acosh(x)) xlabel("x >= 1") ylabel("acosh(x)")

## Examples

    x = [ -2 -1.001 -1 -0.999 -0.5  0
           2  1.001  1  0.999  0.5  0]'
    t = acosh(x)
    cosh(t) - x
    cosh(-t) - x

    // With complex numbers:
    acosh([-0.01*%i  0.01*%i
           -0.1*%i   0.1*%i
           -%i       %i
           -10*%i    10*%i
           -1 - %i,  1 + %i
           -2 - 2*%i, 2 + 2*%i
           ])

## See also

- cosh
- sinh
- tanh