std::atanh(std::complex)(3)

NAME
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 segments
   (-∞,-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)

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