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

| | | | | --------------------------------------------------------------- | | ------------- | | 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.

[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 [−iπ/2; +iπ/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 (+∞,+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)

[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 (-∞,-1] and [+1,+∞) 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] | | |