std::acos(std::complex)

 

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std::complex

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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 of a strip unbounded along the imaginary axis and in the interval [0, +π] 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 (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 <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';
}

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)

See also

asin(std::complex) (C++11)	computes arc sine of a complex number (\({\small\arcsin{z}}\)arcsin(z)) (function template)
atan(std::complex) (C++11)	computes arc tangent of a complex number (\({\small\arctan{z}}\)arctan(z)) (function template)
cos(std::complex)	computes cosine of a complex number (\({\small\cos{z}}\)cos(z)) (function template)
acosacosfacosl (C++11)(C++11)	computes arc cosine (\({\small\arccos{x}}\)arccos(x)) (function)
acos(std::valarray)	applies the function std::acos to each element of valarray (function template)
C documentation for cacos