Std::atanh(std::complex) (3) Linux Manual Page

std::atanh(std::complex) – std::atanh(std::complex)

Synopsis

Defined in header<complex>

    template <class T>
    (since C++ 11)

        complex<T> atanh(const complex<T> &z);

Computes the complex arc hyperbolic tangent of z with branch cuts outside the interval [−1; +1] along the real axis.

Parameters

z – complex value

Return value

If no errors occur, the complex arc hyperbolic tangent of z is returned, in the range of a half-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::atanh(std::conj(z)) == std::conj(std::atanh(z))
* std::atanh(-z) == -std::atanh(z)
* If z is (+0,+0), the result is (+0,+0)
* If z is (+0,NaN), the result is (+0,NaN)
* If z is (+1,+0), the result is (+∞,+0) and FE_DIVBYZERO is raised
* If z is (x,+∞) (for any finite positive x), the result is (+0,π/2)
* If z is (x,NaN) (for any finite nonzero x), the result is (NaN,NaN) and FE_INVALID may be raised
* If z is (+∞,y) (for any finite positive y), the result is (+0,π/2)
* If z is (+∞,+∞), the result is (+0,π/2)
* If z is (+∞,NaN), the result is (+0,NaN)
* If z is (NaN,y) (for any finite y), the result is (NaN,NaN) and FE_INVALID may be raised
* If z is (NaN,+∞), the result is (±0,π/2) (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 tangent", 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 tangent", and, less common, "complex area hyperbolic tangent".
Inverse hyperbolic tangent is a multivalued function and requires a branch cut on the complex plane. The branch cut is conventionally placed at the line segmentd (-∞,-1] and [+1,+∞) of the real axis.
The mathematical definition of the principal value of the inverse hyperbolic tangent is atanh z =
ln(1+z)-ln(1-z)
2
.
For any z, atanh(z) =
atan(iz)
i

Example

// Run this code

#include <iostream>

#include <complex>
int main()

{
    std::cout << std::fixed;
    std::complex<double> z1(2, 0);
    std::cout << "atanh" << z1 << " = " << std::atanh(z1) << '\n';
    std::complex<double> z2(2, -0.0);
    std::cout << "atanh" << z2 << " (the other side of the cut) = "
              << std::atanh(z2) << '\n';
    // for any z, atanh(z) = atanh(iz)/i
    std::complex<double> z3(1, 2);
    std::complex<double> i(0, 1);
    std::cout << "atanh" << z3 << " = " << std::atanh(z3) << '\n'
              << "atan" << z3 * i << "/i = " << std::atan(z3 * i) / i << '\n';
}

Output:

  atanh(2.000000,0.000000) = (0.549306,1.570796)
  atanh(2.000000,-0.000000) (the other side of the cut) = (0.549306,-1.570796)
  atanh(1.000000,2.000000) = (0.173287,1.178097)
  atan(-2.000000,1.000000)/i = (0.173287,1.178097)

std::atanh(std::complex) (3) Linux Manual Page