std::acosh(std::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.

Parameters

z

-

complex value

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.

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)

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

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 $(-\infty,+1)$ of the real axis.

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

For any z,

acosh(z) = \sqrt z-1 \sqrt 1-z acos(z)

, or simply

i acos(z)

in the upper half of the complex plane.

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)

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