std::asin(std::complex)

template< class T >
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

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) -

Example

#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';
}

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