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# std::acosh(std::complex)

From cppreference.com
< cpp‎ | numeric‎ | complex

Defined in header `<complex>` |  |
template< class T > complex<T> acosh( const complex<T>& z ); |  | (since C++11)

Computes complex arc hyperbolic cosine of a complex value `z` with branch cut at values less than 1 along the real axis.

### [edit] Parameters

z | - | complex value

### [edit] Return value

If no errors occur, the complex arc hyperbolic cosine of `z` is returned, in the range of a half-strip of nonnegative values along the real axis and in the interval [−iπ; +iπ] along the imaginary axis.

### [edit] Error handling and special values

Errors are reported consistent with math_errhandling

If the implementation supports IEEE floating-point arithmetic,

- std::acosh(std::conj(z)) == std::conj(std::acosh(z))
- If `z` is `(±0,+0)`, the result is `(+0,π/2)`
- If `z` is `(x,+∞)` (for any finite x), the result is `(+∞,π/2)`
- If `z` is `(x,NaN)` (for any[1] finite x), the result is `(NaN,NaN)` and FE_INVALID may be raised.
- If `z` is `(-∞,y)` (for any positive finite y), the result is `(+∞,π)`
- If `z` is `(+∞,y)` (for any positive finite y), the result is `(+∞,+0)`
- If `z` is `(-∞,+∞)`, the result is `(+∞,3π/4)`
- If `z` is `(±∞,NaN)`, the result is `(+∞,NaN)`
- If `z` is `(NaN,y)` (for any finite y), the result is `(NaN,NaN)` and FE_INVALID may be raised.
- If `z` is `(NaN,+∞)`, the result is `(+∞,NaN)`
- If `z` is `(NaN,NaN)`, the result is `(NaN,NaN)`

1. ↑ per C11 DR471, this holds for non-zero x only. If `z` is `(0,NaN)`, the result should be `(NaN,π/2)`

### [edit] Notes

Although the C++ standard names this function "complex arc hyperbolic cosine", the inverse functions of the hyperbolic functions are the area functions. Their argument is the area of a hyperbolic sector, not an arc. The correct name is "complex inverse hyperbolic cosine", and, less common, "complex area hyperbolic cosine".

Inverse hyperbolic cosine is a multivalued function and requires a branch cut on the complex plane. The branch cut is conventionally placed at the line segment (-∞,+1) of the real axis.

The mathematical definition of the principal value of the inverse hyperbolic sine is acosh z = ln(z + √z+1 √z-1)

For any z, acosh(z) =

√z-1
√1-z

acos(z), or simply i acos(z) in the upper half of the complex plane.

### [edit] Example

Run this code

    #include <iostream>
    #include <complex>

    int main()
    {
        std::cout << std::fixed;
        std::complex<double> z1(0.5, 0);
        std::cout << "acosh" << z1 << " = " << std::acosh(z1) << '\n';

        std::complex<double> z2(0.5, -0.0);
        std::cout << "acosh" << z2 << " (the other side of the cut) = "
                  << std::acosh(z2) << '\n';

        // in upper half-plane, acosh = i acos
        std::complex<double> z3(1, 1), i(0, 1);
        std::cout << "acosh" << z3 << " = " << std::acosh(z3) << '\n'
                  << "i*acos" << z3 << " = " << i*std::acos(z3) << '\n';
    }

Output:

    acosh(0.500000,0.000000) = (0.000000,-1.047198)
    acosh(0.500000,-0.000000) (the other side of the cut) = (0.000000,1.047198)
    acosh(1.000000,1.000000) = (1.061275,0.904557)
    i*acos(1.000000,1.000000) = (1.061275,0.904557)

### [edit] See also

acos(std::complex) (C++11) | computes arc cosine of a complex number (arccos(z)) (function template) [edit]
asinh(std::complex) (C++11) | computes area hyperbolic sine of a complex number (function template) [edit]
atanh(std::complex) (C++11) | computes area hyperbolic tangent of a complex number (function template) [edit]
cosh(std::complex) | computes hyperbolic cosine of a complex number (ch(z)) (function template) [edit]
acosh (C++11) | computes the inverse hyperbolic cosine (arcosh(x)) (function) [edit]
C documentation for cacosh

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