std::asinh(std::complex)

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.

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 $(- i \infty,- i)$ and $(i, i \infty)$ of the imaginary axis.

The mathematical definition of the principal value of the inverse hyperbolic sine is $asinh z = ln(z + \sqrt 1+z 2)$

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)

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