NAME¶

std::asinh(std::complex) - std::asinh(std::complex)

Synopsis¶

Defined in header <complex>
template< class T > (since C++11)
complex<T> asinh( const complex<T>& z );

Computes complex arc hyperbolic sine of a complex value z with branch cuts outside
the interval [−i; +i] along the imaginary axis.

Parameters¶

z - complex value

Return 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.

Error handling and special values

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)
* If z is (+0,+0), the result is (+0,+0)
* If z is (x,+∞) (for any positive finite x), the result is (+∞,π/2)
* If z is (x,NaN) (for any finite x), the result is (NaN,NaN) and FE_INVALID may
be raised
* If z is (+∞,y) (for any positive finite y), the result is (+∞,+0)
* If z is (+∞,+∞), the result is (+∞,π/4)
* If z is (+∞,NaN), the result is (+∞,NaN)
* If z is (NaN,+0), the result is (NaN,+0)
* If z is (NaN,y) (for any finite nonzero y), the result is (NaN,NaN) and
FE_INVALID may be raised
* If z is (NaN,+∞), the result is (±∞,NaN) (the sign of the real part is
unspecified)
* If z is (NaN,NaN), the result is (NaN,NaN)

Notes¶

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

Example¶

// Run this code

#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)

See also¶

acosh(std::complex) computes area hyperbolic cosine of a complex number
(C++11) (\({\small\operatorname{arcosh}{z} }\)arcosh(z))
(function template)
atanh(std::complex) computes area hyperbolic tangent of a complex number
(C++11) (\({\small\operatorname{artanh}{z} }\)artanh(z))
(function template)
computes hyperbolic sine of a complex number (\({\small\sinh{z}
sinh(std::complex) }\)sinh(z))
(function template)
asinh
asinhf computes the inverse hyperbolic sine
asinhl (\({\small\operatorname{arsinh}{x} }\)arsinh(x))
(C++11) (function)
(C++11)
(C++11)