Defined in header
<complex> | ||
---|---|---|
template< class T > complex<T> asinh( const complex<T>& z ); | (since C++11) |
Computes complex arc hyperbolic sine of a complex value z
with branch cuts outside the interval [−i; +i] along the imaginary axis.
z | - | complex value |
If no errors occur, the complex arc hyperbolic sine of z
is returned, in the range of a strip mathematically unbounded along the real axis and in the interval [−iπ/2; +iπ/2] along the imaginary axis.
Errors are reported consistent with math_errhandling
.
If the implementation supports IEEE floating-point arithmetic,
std::asinh(std::conj(z)) == std::conj(std::asinh(z))
std::asinh(-z) == -std::asinh(z)
z
is (+0,+0)
, the result is (+0,+0)
z
is (x,+∞)
(for any positive finite x), the result is (+∞,π/2)
z
is (x,NaN)
(for any finite x), the result is (NaN,NaN)
and FE_INVALID
may be raised z
is (+∞,y)
(for any positive finite y), the result is (+∞,+0)
z
is (+∞,+∞)
, the result is (+∞,π/4)
z
is (+∞,NaN)
, the result is (+∞,NaN)
z
is (NaN,+0)
, the result is (NaN,+0)
z
is (NaN,y)
(for any finite nonzero y), the result is (NaN,NaN)
and FE_INVALID
may be raised z
is (NaN,+∞)
, the result is (±∞,NaN)
(the sign of the real part is unspecified) z
is (NaN,NaN)
, the result is (NaN,NaN)
Although the C++ standard names this function "complex arc hyperbolic sine", the inverse functions of the hyperbolic functions are the area functions. Their argument is the area of a hyperbolic sector, not an arc. The correct name is "complex inverse hyperbolic sine", and, less common, "complex area hyperbolic sine".
Inverse hyperbolic sine is a multivalued function and requires a branch cut on the complex plane. The branch cut is conventionally placed at the line segments (-i∞,-i) and (i,i∞) of the imaginary axis.
The mathematical definition of the principal value of the inverse hyperbolic sine is asinh z = ln(z + √1+z2
) For any z, asinh(z) =
asin(iz) |
i |
#include <iostream> #include <complex> int main() { std::cout << std::fixed; std::complex<double> z1(0, -2); std::cout << "asinh" << z1 << " = " << std::asinh(z1) << '\n'; std::complex<double> z2(-0.0, -2); std::cout << "asinh" << z2 << " (the other side of the cut) = " << std::asinh(z2) << '\n'; // for any z, asinh(z) = asin(iz)/i std::complex<double> z3(1,2); std::complex<double> i(0,1); std::cout << "asinh" << z3 << " = " << std::asinh(z3) << '\n' << "asin" << z3*i << "/i = " << std::asin(z3*i)/i << '\n'; }
Output:
asinh(0.000000,-2.000000) = (1.316958,-1.570796) asinh(-0.000000,-2.000000) (the other side of the cut) = (-1.316958,-1.570796) asinh(1.000000,2.000000) = (1.469352,1.063440) asin(-2.000000,1.000000)/i = (1.469352,1.063440)
(C++11)
| computes area hyperbolic cosine of a complex number (function template) |
(C++11)
| computes area hyperbolic tangent of a complex number (function template) |
computes hyperbolic sine of a complex number (sh(z)) (function template) |
|
(C++11)
| computes the inverse hyperbolic sine (arsinh(x)) (function) |
C documentation for casinh |
© cppreference.com
Licensed under the Creative Commons Attribution-ShareAlike Unported License v3.0.
http://en.cppreference.com/w/cpp/numeric/complex/asinh