Defined in header <complex>  

template< class T > complex<T> atanh( const complex<T>& z );  (since C++11) 
Computes the complex arc hyperbolic tangent of z
with branch cuts outside the interval [−1; +1] along the real axis.
z    complex value 
If no errors occur, the complex arc hyperbolic tangent of z
is returned, in the range of a halfstrip mathematically unbounded along the real axis and in the interval [−iπ/2; +iπ/2] along the imaginary axis.
Errors are reported consistent with math_errhandling
.
If the implementation supports IEEE floatingpoint arithmetic,
std::atanh(std::conj(z)) == std::conj(std::atanh(z))
std::atanh(z) == std::atanh(z)
z
is (+0,+0)
, the result is (+0,+0)
z
is (+0,NaN)
, the result is (+0,NaN)
z
is (+1,+0)
, the result is (+∞,+0)
and FE_DIVBYZERO
is raised z
is (x,+∞)
(for any finite positive x), the result is (+0,π/2)
z
is (x,NaN)
(for any finite nonzero x), the result is (NaN,NaN)
and FE_INVALID
may be raised z
is (+∞,y)
(for any finite positive y), the result is (+0,π/2)
z
is (+∞,+∞)
, the result is (+0,π/2)
z
is (+∞,NaN)
, the result is (+0,NaN)
z
is (NaN,y)
(for any finite y), the result is (NaN,NaN)
and FE_INVALID
may be raised z
is (NaN,+∞)
, the result is (±0,π/2)
(the sign of the real part is unspecified) z
is (NaN,NaN)
, the result is (NaN,NaN)
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(1z) 
2 
For any z, atanh(z) =
atan(iz) 
i 
#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)
(C++11)  computes area hyperbolic sine of a complex number (function template) 
(C++11)  computes area hyperbolic cosine of a complex number (function template) 
computes hyperbolic tangent of a complex number (function template) 

(C++11)(C++11)(C++11)  computes the inverse hyperbolic tangent (artanh(x)) (function) 
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