/C++

# std::asin(std::complex)

Defined in header <complex>
template< class T >
complex<T> asin( const complex<T>& z );
(since C++11)

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.

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

#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)

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