std::acos(std::complex)

template< class T > 
complex<T> acos( const complex<T>& z );

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

Parameters

Return value

If no errors occur, complex arc cosine of z is returned, in the range of a strip unbounded along the imaginary axis and in the interval [0, +] along the real 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 (/2,-0)
• If z is (0,NaN), the result is (/2,NaN)
• If z is (x,+) (for any finite x), the result is (/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 (,-)
• If z is (+,y) (for any positive finite y), the result is (+0,-)
• If z is (-,+), the result is (3/4,-)
• If z is (+,+), the result is (/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 (-,-1) and (1,) of the real axis.

The mathematical definition of the principal value of arc cosine is

.

For any z, acos(z) = - acos(-z).

Example

#include <cmath>
#include <complex>
#include <iostream>

int main()
{
    std::cout << std::fixed;
    std::complex<double> z1(-2.0, 0.0);
    std::cout << "acos" << z1 << " = " << std::acos(z1) << '\n';

    std::complex<double> z2(-2.0, -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';
}

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)

See also