# 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 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(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) (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 atanhf atanhl (C++11)(C++11)(C++11) | computes the inverse hyperbolic tangent (artanh(x)) (function)

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