std::acos(std::complex)(3)

💖 std::acos(std::complex)(3) 💖

okay bestie, this is from the C++ Standard Libary manual page — yes, that exact spelling, we are respecting the lore. This one is all about complex arc cosine, and it is lowkey iconic.

🌸 NAME 🌸

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

The NAME section is literally repeating itself. The way that it ate and left no crumbs. 🍓

✨ 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 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 + i ln(i z + sqrt(1 - z^2)).

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

okay that’s the whole page, bestie — mwah! 💋 source/version: http://cppreference.com · date: 2024.06.10