/C++

# std::atanh(std::complex)

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.

### 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)`

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 sine is atanh z =

 ln(1+z)-ln(1-z) 2
.

For any z, atanh(z) =

 atan(iz) i

### Example

```#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)```

 asinh(std::complex) (C++11) computes area hyperbolic sine of a complex number (function template) acosh(std::complex) (C++11) computes area hyperbolic cosine of a complex number (function template) tanh(std::complex) computes hyperbolic tangent of a complex number (function template) atanh (C++11) computes the inverse hyperbolic tangent (artanh(x)) (function) C documentation for `catanh`