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

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

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 [âi; +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 [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::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,+â) (for any positive finite x), the result is (+â,Ï/2)
If z is (x,NaN) (for any finite x), the result is (NaN,NaN) and FE_INVALID may be raised
If z is (+â,y) (for any positive finite y), the result is (+â,+0)
If z is (+â,+â), the result is (+â,Ï/4)
If z is (+â,NaN), the result is (+â,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,+â), the result is (Â±â,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 (-iâ,-i) and (i,iâ) of the imaginary axis.

The mathematical definition of the principal value of the inverse hyperbolic sine is asinh z = ln(z + â1+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]

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