std::asinh(std::complex)

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)

[edit] 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

.

[edit] Example

Run this code

#include <complex>
#include <iostream>
 
int main()
{
 std::cout << std::fixed ;
 std::complex <double> z1(0.0, -2.0);
 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(1.0, 2.0);
 std::complex <double> i(0.0, 1.0);
 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)