Defined in header <complex>  

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.
z    complex value 
If no errors occur, the complex arc hyperbolic cosine of z
is returned, in the range of a halfstrip of nonnegative values along the real axis and in the interval [−iπ; +iπ] along the imaginary axis.
Errors are reported consistent with math_errhandling
.
If the implementation supports IEEE floatingpoint arithmetic,
std::acosh(std::conj(z)) == std::conj(std::acosh(z))
z
is (±0,+0)
, the result is (+0,π/2)
z
is (x,+∞)
(for any finite x), the result is (+∞,π/2)
z
is (x,NaN)
(for any^{[1]} finite x), the result is (NaN,NaN)
and FE_INVALID
may be raised. z
is (∞,y)
(for any positive finite y), the result is (+∞,π)
z
is (+∞,y)
(for any positive finite y), the result is (+∞,+0)
z
is (∞,+∞)
, the result is (+∞,3π/4)
z
is (±∞,NaN)
, the result is (+∞,NaN)
z
is (NaN,y)
(for any finite y), the result is (NaN,NaN)
and FE_INVALID
may be raised. z
is (NaN,+∞)
, the result is (+∞,NaN)
z
is (NaN,NaN)
, the result is (NaN,NaN)
z
is (0,NaN)
, the result should be (NaN,π/2)
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 (∞,+1) of the real axis.
The mathematical definition of the principal value of the inverse hyperbolic cosine is acosh z = ln(z + √z+1 √z1) For any z, acosh(z) =
√z1 
√1z 
#include <iostream> #include <complex> int main() { std::cout << std::fixed; std::complex<double> z1(0.5, 0); std::cout << "acosh" << z1 << " = " << std::acosh(z1) << '\n'; std::complex<double> z2(0.5, 0.0); std::cout << "acosh" << z2 << " (the other side of the cut) = " << std::acosh(z2) << '\n'; // in upper halfplane, acosh = i acos std::complex<double> z3(1, 1), i(0, 1); std::cout << "acosh" << z3 << " = " << std::acosh(z3) << '\n' << "i*acos" << z3 << " = " << i*std::acos(z3) << '\n'; }
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)
(C++11)  computes arc cosine of a complex number (arccos(z)) (function template) 
(C++11)  computes area hyperbolic sine of a complex number (function template) 
(C++11)  computes area hyperbolic tangent of a complex number (function template) 
computes hyperbolic cosine of a complex number (ch(z)) (function template) 

(C++11)(C++11)(C++11)  computes the inverse hyperbolic cosine (arcosh(x)) (function) 
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