std::asinh(std::complex)(3)

NAME
std::asinh(std::complex) - std::asinh(std::complex)

Synopsis
   Defined in header <complex>
   template< class T > (since C++11)
   complex<T> asinh( const complex<T>& z );
   Computes complex arc hyperbolic sine of a complex value z with branch cuts outside
   the interval [−i; +i] along the imaginary axis.

Parameters
   z - complex value

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
   [−iπ/2; +iπ/2] 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::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,+∞) (for any positive finite x), the result is (+∞,π/2)
     * If z is (x,NaN) (for any 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 (+∞,+0)
     * If z is (+∞,+∞), the result is (+∞,π/4)
     * If z is (+∞,NaN), the result is (+∞,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,+∞), the result is (±∞,NaN) (the sign of the real part is
       unspecified)
     * If z is (NaN,NaN), the result is (NaN,NaN)

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
   (-i∞,-i) and (i,i∞) of the imaginary axis.
   The mathematical definition of the principal value of the inverse hyperbolic sine is
   asinh z = ln(z +
   √
   1+z2
   )
   For any z, asinh(z) =
   asin(iz)
   i

Example
// Run this code
 #include <iostream>
 #include <complex>
 int main()
 {
     std::cout << std::fixed;
     std::complex<double> z1(0, -2);
     std::cout << "asinh" << z1 << " = " << std::asinh(z1) << '\n';
     std::complex<double> z2(-0.0, -2);
     std::cout << "asinh" << z2 << " (the other side of the cut) = "
               << std::asinh(z2) << '\n';
     // for any z, asinh(z) = asin(iz)/i
     std::complex<double> z3(1,2);
     std::complex<double> i(0,1);
     std::cout << "asinh" << z3 << " = " << std::asinh(z3) << '\n'
               << "asin" << z3*i << "/i = " << std::asin(z3*i)/i << '\n';
 }

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)

See also
   acosh(std::complex) computes area hyperbolic cosine of a complex number
   (C++11) (\({\small\operatorname{arcosh}{z} }\)arcosh(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 sine of a complex number (\({\small\sinh{z}
   sinh(std::complex) }\)sinh(z))
                       (function template)
   asinh
   asinhf computes the inverse hyperbolic sine
   asinhl (\({\small\operatorname{arsinh}{x} }\)arsinh(x))
   (C++11) (function)
   (C++11)
   (C++11)