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std::acos(std::complex)

From cppreference.com
< cpp‎ | numeric‎ | 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.

Contents

[edit] Parameters

z - complex value

[edit] 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.

[edit] 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)

[edit] 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)

[edit] Example

#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)

[edit] See also

computes arc sine of a complex number (arcsin(z))
(function template) [edit]
computes arc tangent of a complex number (arctan(z))
(function template) [edit]
computes cosine of a complex number (cos(z))
(function template) [edit]
computes arc cosine (arccos(x))
(function) [edit]
applies the function std::acos to each element of valarray
(function template) [edit]
C documentation for cacos