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

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

template< class T > complex<T> acosh( const complex<T>& z );

(since C++11)

Computes complex arc hyperbolic cosine of a complex value z with branch cut at values less than 1 along the real axis.

[edit] Parameters [edit] Return value

If no errors occur, the complex arc hyperbolic cosine of z is returned, in the range of a half-strip of nonnegative values along the real axis and in the interval [âiÏ; +iÏ] 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::acosh(std::conj(z)) == std::conj(std::acosh(z)).
If z is (Â±0,+0), the result is (+0,Ï/2).
If z is (x,+â) (for any finite x), the result is (+â,Ï/2).
If z is (x,NaN) (for any^[1] 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,Ï).
If z is (+â,y) (for any positive finite y), the result is (+â,+0).
If z is (-â,+â), the result is (+â,3Ï/4).
If z is (Â±â,NaN), the result is (+â,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,NaN).
If z is (NaN,NaN), the result is (NaN,NaN).

â per C11 DR471, this holds for non-zero x only. If z is (0,NaN), the result should be (NaN,Ï/2).

[edit] Notes

Although the C++ standard names this function "complex arc hyperbolic cosine", 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 cosine", and, less common, "complex area hyperbolic cosine".

Inverse hyperbolic cosine is a multivalued function and requires a branch cut on the complex plane. The branch cut is conventionally placed at the line segment (-â,+1) of the real axis.

The mathematical definition of the principal value of the inverse hyperbolic cosine is acosh z = ln(z + âz+1 âz-1).

For any

z

acosh(z) = acos(z)

, or simply

i acos(z)

in the upper half of the complex plane.

[edit] Example

Output:

acosh(0.500000,0.000000) = (0.000000,-1.047198)
acosh(0.500000,-0.000000) (the other side of the cut) = (0.000000,1.047198)
acosh(1.000000,1.000000) = (1.061275,0.904557)
i*acos(1.000000,1.000000) = (1.061275,0.904557)

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

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