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Showing content from https://en.cppreference.com/w/cpp/language/../algorithm/../numeric/complex/atanh.html below:

std::atanh(std::complex) - cppreference.com

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 [â1; +1] along the real axis.

[edit] Parameters [edit] 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 [âiÏ/2; +iÏ/2] along the imaginary axis.

[edit] 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 (+â,+0) and FE_DIVBYZERO is raised
If z is (x,+â) (for any finite positive x), the result is (+0,Ï/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 (+â,y) (for any finite positive y), the result is (+0,Ï/2)
If z is (+â,+â), the result is (+0,Ï/2)
If z is (+â,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,+â), the result is (Â±0,Ï/2) (the sign of the real part is unspecified)
If z is (NaN,NaN), the result is (NaN,NaN)

[edit] 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 (-â,-1] and [+1,+â) of the real axis.

The mathematical definition of the principal value of the inverse hyperbolic tangent is

atanh z =

For any

z

atanh(z) =

[edit] Example

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)

[edit] See also computes area hyperbolic sine of a complex number (\({\small\operatorname{arsinh}{z}}\)arsinh(z))
(function template) [edit] computes area hyperbolic cosine of a complex number (\({\small\operatorname{arcosh}{z}}\)arcosh(z))
(function template) [edit] computes hyperbolic tangent of a complex number (\({\small\tanh{z}}\)tanh(z))
(function template) [edit] computes the inverse hyperbolic tangent (\({\small\operatorname{artanh}{x}}\)artanh(x))
(function) [edit]

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