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std::atanh(std::complex)

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

Computes the complex arc hyperbolic tangent of z with branch cuts outside the interval [−1; +1] along the real axis.

Contents

[edit] Parameters

z - complex value

[edit] Return value

If no errors occur, the complex arc hyperbolic tangent of z is returned, in the range of a half-strip mathematically unbounded 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::atanh(std::conj(z)) == std::conj(std::atanh(z))
  • std::atanh(-z) == -std::atanh(z)
  • If z is (+0,+0), the result is (+0,+0)
  • If z is (+0,NaN), the result is (+0,NaN)
  • If z is (+1,+0), the result is (+∞,+0) and FE_DIVBYZERO is raised
  • If z is (x,+∞) (for any finite positive x), the result is (+0,π/2)
  • If z is (x,NaN) (for any finite nonzero x), the result is (NaN,NaN) and FE_INVALID may be raised
  • If z is (+∞,y) (for any finite positive y), the result is (+0,π/2)
  • If z is (+∞,+∞), the result is (+0,π/2)
  • If z is (+∞,NaN), the result is (+0,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 (±0,π/2) (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 tangent", 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 tangent", and, less common, "complex area hyperbolic tangent".

Inverse hyperbolic tangent is a multivalued function and requires a branch cut on the complex plane. The branch cut is conventionally placed at the line segmentd (-∞,-1] and [+1,+∞) of the real axis.

The mathematical definition of the principal value of the inverse hyperbolic sine is atanh z =
ln(1+z)-ln(z-1)
2
.


For any z, atanh(z) =
atan(iz)
i

[edit] Example

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

Output:

atanh(2.000000,0.000000) = (0.549306,1.570796)
atanh(2.000000,-0.000000) (the other side of the cut) = (0.549306,-1.570796)
atanh(1.000000,2.000000) = (0.173287,1.178097)
atan(-2.000000,1.000000)/i = (0.173287,1.178097)

[edit] See also

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