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

| | | | | --------------------------------------------------------------- | | ------------- | | template< class T > complex<T> asinh( const complex<T>& z ); | | (since C++11) |

Computes complex arc hyperbolic sine of a complex value z with branch cuts outside the interval [−i; +i] along the imaginary axis.

[edit] Parameters

[edit] Return value

If no errors occur, the complex arc hyperbolic sine of z is returned, in the range of a 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::asinh(std::conj(z)) == std::conj(std::asinh(z))
std::asinh(-z) == -std::asinh(z)
If z is (+0,+0), the result is (+0,+0)
If z is (x,+∞) (for any positive finite x), the result is (+∞,π/2)
If z is (x,NaN) (for any finite x), the result is (NaN,NaN) and FE_INVALID may be raised
If z is (+∞,y) (for any positive finite y), the result is (+∞,+0)
If z is (+∞,+∞), the result is (+∞,π/4)
If z is (+∞,NaN), the result is (+∞,NaN)
If z is (NaN,+0), the result is (NaN,+0)
If z is (NaN,y) (for any finite nonzero y), the result is (NaN,NaN) and FE_INVALID may be raised
If z is (NaN,+∞), the result is (±∞,NaN) (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 sine", 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 sine", and, less common, "complex area hyperbolic sine".

Inverse hyperbolic sine is a multivalued function and requires a branch cut on the complex plane. The branch cut is conventionally placed at the line segments (-_i_∞,-i) and (i,_i_∞) of the imaginary axis.

The mathematical definition of the principal value of the inverse hyperbolic sine is asinh z = ln(z + √1+z2
).

For any z, asinh(z) = .

[edit] Example

Output:

asinh(0.000000,-2.000000) = (1.316958,-1.570796) asinh(-0.000000,-2.000000) (the other side of the cut) = (-1.316958,-1.570796) asinh(1.000000,2.000000) = (1.469352,1.063440) asin(-2.000000,1.000000) / i = (1.469352,1.063440)

[edit] See also

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