std::acos(std::complex)

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

Computes complex arc cosine of a complex value z. Branch cuts exist outside the interval $[-1 ; +1]$ along the real axis.

Parameters

z

-

complex value

Return value

If no errors occur, complex arc cosine of z is returned, in the range $[0 ; \infty)$ along the real axis and in the range $[- i π ; i π]$ 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::acos(std::conj(z)) == std::conj(std::acos(z))
If z is (±0,+0), the result is (π/2,-0)
If z is (±0,NaN), the result is (π/2,NaN)
If z is (x,+∞) (for any finite x), the result is (π/2,-∞)
If z is (x,NaN) (for any nonzero 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 (+∞,+∞), the result is (π/4,-∞)
If z is (±∞,NaN), the result is (NaN,±∞) (the sign of the imaginary part is unspecified)
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)

Notes

Inverse cosine (or arc cosine) is a multivalued function and requires a branch cut on the complex plane. The branch cut is conventionally placed at the line segments $(-\infty,-1)$ and $(1,\infty)$ of the real axis.

The mathematical definition of the principal value of arc cosine is

acos z = 12 π + i ln(i z + \sqrt 1-z 2)

For any z, $acos(z) = π - acos(-z)$

Example

Run this code

#include <iostream>
#include <cmath>
#include <complex>
 
int main()
{
    std::cout << std::fixed;
    std::complex<double> z1(-2, 0);
    std::cout << "acos" << z1 << " = " << std::acos(z1) << '\n';
 
    std::complex<double> z2(-2, -0.0);
    std::cout << "acos" << z2 << " (the other side of the cut) = "
              << std::acos(z2) << '\n';
 
    // for any z, acos(z) = pi - acos(-z)
    const double pi = std::acos(-1);
    std::complex<double> z3 = pi - std::acos(z2);
    std::cout << "cos(pi - acos" << z2 << ") = " << std::cos(z3) << '\n';
}

Output:

acos(-2.000000,0.000000) = (3.141593,-1.316958)
acos(-2.000000,-0.000000) (the other side of the cut) = (3.141593,1.316958)
cos(pi - acos(-2.000000,-0.000000)) = (2.000000,0.000000)

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