std::asin(std::complex)(3)

okay bestie, you're on the std::asin(std::complex)(3) page from the C++ Standard Libary — yes, that's the manual name right there, and yes the header really says “Libary”, not me 😭. This is the complex-arc-sine lore. 💅

💖 NAME

std::asin(std::complex) — std::asin(std::complex).

✨ SYNOPSIS ✨

Defined in header <complex>:

template< class T >                                (since C++11)
std::complex<T> asin( const std::complex<T>& z );

Computes complex arc sine of a complex value z. Branch cut exists outside the interval [−1, +1] along the real axis.

🫶 PARAMETERS

• z — complex value.

🌈 RETURN VALUE 🌈

If no errors occur, complex arc sine of z is returned, in the range of a strip unbounded along the imaginary axis and in the interval [−π/2, +π/2] along the real axis.

Errors and special cases are handled as if the operation is implemented by -i * std::asinh(i * z), where i is the imaginary unit.

🧚 NOTES 🧚

Inverse sine (or arc sine) 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 sine is:

\(\small \arcsin z = -{\rm i}\ln({\rm i}z+\sqrt{1-z^2})\)arcsin z = -iln(iz + √ 1-z2 ).

For any z:

\(\small{ \arcsin(z) = \arccos(-z) - \frac{\pi}{2} }\)asin(z) = acos(-z) - π/2 .

💅 EXAMPLE 💅

// Run this code — go off, bestie:

#include <cmath>
#include <complex>
#include <iostream>

int main()
{
    std::cout << std::fixed;
    std::complex<double> z1(-2.0, 0.0);
    std::cout << "asin" << z1 << " = " << std::asin(z1) << '\n';

    std::complex<double> z2(-2.0, -0.0);
    std::cout << "asin" << z2 << " (the other side of the cut) = "
              << std::asin(z2) << '\n';

    // for any z, asin(z) = acos(−z) − pi / 2
    const double pi = std::acos(-1);
    std::complex<double> z3 = std::acos(z2) - pi / 2;
    std::cout << "sin(acos" << z2 << " - pi / 2) = " << std::sin(z3) <<
       '\n';
}

🍓 OUTPUT 🍓

asin(-2.000000,0.000000) = (-1.570796,1.316958)
asin(-2.000000,-0.000000) (the other side of the cut) =
(-1.570796,-1.316958)
sin(acos(-2.000000,-0.000000) - pi / 2) = (2.000000,0.000000)

👑 SEE ALSO 👑

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

That's the whole page, bestie. Source: http://cppreference.com; version/date: 2024.06.10; page: std::asin(std::complex)(3). 🫶💖✨