std::acosh(std::complex)(3)

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 <complex>
 #include <iostream>
 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
   (C++11) (\({\small\arccos{z}}\)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)
   C documentation for
   cacosh