std::acos(std::complex) (3) - Linux Manuals

std::acos(std::complex): std::acos(std::complex)

NAME

std::acos(std::complex) - std::acos(std::complex)

Synopsis

Defined in header <complex>
template< class T > (since C++11)
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

z - complex value

Return value

If no errors occur, complex arc cosine of z is returned, in the range [0 ; ∞) along the real axis and in the range [−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::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 (-∞,-1) and (1,∞) of the real axis.
The mathematical definition of the principal value of arc cosine is acos z =

1
2

π + iln(iz +
√
1-z2
)
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)