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 floatingpoint 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)(C++11)(C++11)  computes the inverse hyperbolic sine (arsinh(x)) (function) 
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