std::asin(std::complex)
<complex>
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.
Contents
[edit] Parameters
[edit] 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.
[edit] 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
).
[edit] 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'; }
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)
[edit] See also
(function template) [edit]
(function template) [edit]