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)