NAME¶

std::acosh(std::complex) - std::acosh(std::complex)

Synopsis¶

Defined in header <complex>
template< class T > (since C++11)
complex<T> acosh( const complex<T>& z );

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

Parameters¶

z - complex value

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
[−iπ; +iπ] along the imaginary axis.

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 (±0,+0), the result is (+0,π/2)
* If z is (x,+∞) (for any finite x), the result is (+∞,π/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 (-∞,y) (for any positive finite y), the result is (+∞,π)
* If z is (+∞,y) (for any positive finite y), the result is (+∞,+0)
* If z is (-∞,+∞), the result is (+∞,3π/4)
* If z is (±∞,NaN), the result is (+∞,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 (+∞,NaN)
* If z is (NaN,NaN), the result is (NaN,NaN)

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

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 (-∞,+1)
of the real axis.

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

For any z, acosh(z) =

√
z-1
√
1-z

acos(z), or simply i acos(z) in the upper half of the complex plane.

Example¶

// Run this code

#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 half-plane, 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)

See also¶

acos(std::complex) computes arc cosine of a complex number (\({\small\arccos{z}
(C++11) }\)arccos(z))
(function template)
asinh(std::complex) computes area hyperbolic sine of a complex number
(C++11) (\({\small\operatorname{arsinh}{z} }\)arsinh(z))
(function template)
atanh(std::complex) computes area hyperbolic tangent of a complex number
(C++11) (\({\small\operatorname{artanh}{z} }\)artanh(z))
(function template)
computes hyperbolic cosine of a complex number
cosh(std::complex) (\({\small\cosh{z} }\)cosh(z))
(function template)
acosh
acoshf computes the inverse hyperbolic cosine
acoshl (\({\small\operatorname{arcosh}{x} }\)arcosh(x))
(C++11) (function)
(C++11)
(C++11)