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std::asinh(std::complex)

From cppreference.com
< cpp‎ | numeric‎ | complex
Defined in header <complex>
template< class T >
complex<T> asinh( const complex<T>& z );
(since C++11)

Computes complex arc hyperbolic sine of a complex value z with branch cuts outside the interval [−i; +i] along the imaginary axis.

Contents

[edit] Parameters

z - complex value

[edit] Return value

If no errors occur, the complex arc hyperbolic sine of z is returned, in the range of a strip mathematically unmounded along the real axis and in the interval [−iπ/2; +iπ/2] 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::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)

[edit] 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

[edit] Example

#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) = sinh(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)

[edit] See also

computes area hyperbolic cosine of a complex number
(function template) [edit]
computes area hyperbolic tangent of a complex number
(function template) [edit]
computes hyperbolic sine of a complex number (sh(z))
(function template) [edit]
(C++11)
computes the inverse hyperbolic sine (arsinh(x))
(function) [edit]
C documentation for casinh