std::acos(std::complex) - cppreference.com

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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.

### [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

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)

### [edit] See also

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

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