# std::atanh(std::complex)

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< cpp‎ | numeric‎ | 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)`

### 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 <complex>
    #include <iostream>

    int main()
    {
        std::cout << std::fixed;
        std::complex<double> z1(2.0, 0.0);
        std::cout << "atanh" << z1 << " = " << std::atanh(z1) << '\n';

        std::complex<double> z2(2.0, -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(1.0, 2.0);
        std::complex<double> i(0.0, 1.0);
        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) (C++11) | computes area hyperbolic sine of a complex number (\({\small\operatorname{arsinh}{z} }\)arsinh(z)) (function template)
acosh(std::complex) (C++11) | computes area hyperbolic cosine of a complex number (\({\small\operatorname{arcosh}{z} }\)arcosh(z)) (function template)
tanh(std::complex) | computes hyperbolic tangent of a complex number (\({\small\tanh{z} }\)tanh(z)) (function template)
atanh atanhf atanhl (C++11)(C++11)(C++11) | computes the inverse hyperbolic tangent (\({\small\operatorname{artanh}{x} }\)artanh(x)) (function)

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