std::asin(std::complex) (3) - Linux Manuals

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

NAME

std::asin(std::complex) - std::asin(std::complex)

Synopsis

Defined in header <complex>
template< class T > (since C++11)
complex<T> asin( const 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 asin z = -iln(iz +
√
1-z2
)
For any z, asin(z) = acos(-z) -

π
2

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:

  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)