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

#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 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)
C documentation for
casin