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)

See also¶

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