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// (C) Copyright John Maddock 2005.
// Use, modification and distribution are subject to the
// Boost Software License, Version 1.0. (See accompanying file
// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
#ifndef BOOST_MATH_COMPLEX_ATANH_INCLUDED
#define BOOST_MATH_COMPLEX_ATANH_INCLUDED
#ifndef BOOST_MATH_COMPLEX_DETAILS_INCLUDED
# include <boost/math/complex/details.hpp>
#endif
#ifndef BOOST_MATH_LOG1P_INCLUDED
# include <boost/math/special_functions/log1p.hpp>
#endif
#include <boost/assert.hpp>
#ifdef BOOST_NO_STDC_NAMESPACE
namespace std{ using ::sqrt; using ::fabs; using ::acos; using ::asin; using ::atan; using ::atan2; }
#endif
namespace boost{ namespace math{
template<class T>
std::complex<T> atanh(const std::complex<T>& z)
{
//
// References:
//
// Eric W. Weisstein. "Inverse Hyperbolic Tangent."
// From MathWorld--A Wolfram Web Resource.
// http://mathworld.wolfram.com/InverseHyperbolicTangent.html
//
// Also: The Wolfram Functions Site,
// http://functions.wolfram.com/ElementaryFunctions/ArcTanh/
//
// Also "Abramowitz and Stegun. Handbook of Mathematical Functions."
// at : http://jove.prohosting.com/~skripty/toc.htm
//
// See also: https://svn.boost.org/trac/boost/ticket/7291
//
static const T pi = boost::math::constants::pi<T>();
static const T half_pi = pi / 2;
static const T one = static_cast<T>(1.0L);
static const T two = static_cast<T>(2.0L);
static const T four = static_cast<T>(4.0L);
static const T zero = static_cast<T>(0);
static const T log_two = boost::math::constants::ln_two<T>();
#ifdef BOOST_MSVC
#pragma warning(push)
#pragma warning(disable:4127)
#endif
T x = std::fabs(z.real());
T y = std::fabs(z.imag());
T real, imag; // our results
T safe_upper = detail::safe_max(two);
T safe_lower = detail::safe_min(static_cast<T>(2));
//
// Begin by handling the special cases specified in C99:
//
if((boost::math::isnan)(x))
{
if((boost::math::isnan)(y))
return std::complex<T>(x, x);
else if((boost::math::isinf)(y))
return std::complex<T>(0, ((boost::math::signbit)(z.imag()) ? -half_pi : half_pi));
else
return std::complex<T>(x, x);
}
else if((boost::math::isnan)(y))
{
if(x == 0)
return std::complex<T>(x, y);
if((boost::math::isinf)(x))
return std::complex<T>(0, y);
else
return std::complex<T>(y, y);
}
else if((x > safe_lower) && (x < safe_upper) && (y > safe_lower) && (y < safe_upper))
{
T yy = y*y;
T mxm1 = one - x;
///
// The real part is given by:
//
// real(atanh(z)) == log1p(4*x / ((x-1)*(x-1) + y^2))
//
real = boost::math::log1p(four * x / (mxm1*mxm1 + yy));
real /= four;
if((boost::math::signbit)(z.real()))
real = (boost::math::changesign)(real);
imag = std::atan2((y * two), (mxm1*(one+x) - yy));
imag /= two;
if(z.imag() < 0)
imag = (boost::math::changesign)(imag);
}
else
{
//
// This section handles exception cases that would normally cause
// underflow or overflow in the main formulas.
//
// Begin by working out the real part, we need to approximate
// real = boost::math::log1p(4x / ((x-1)^2 + y^2))
// without either overflow or underflow in the squared terms.
//
T mxm1 = one - x;
if(x >= safe_upper)
{
// x-1 = x to machine precision:
if((boost::math::isinf)(x) || (boost::math::isinf)(y))
{
real = 0;
}
else if(y >= safe_upper)
{
// Big x and y: divide through by x*y:
real = boost::math::log1p((four/y) / (x/y + y/x));
}
else if(y > one)
{
// Big x: divide through by x:
real = boost::math::log1p(four / (x + y*y/x));
}
else
{
// Big x small y, as above but neglect y^2/x:
real = boost::math::log1p(four/x);
}
}
else if(y >= safe_upper)
{
if(x > one)
{
// Big y, medium x, divide through by y:
real = boost::math::log1p((four*x/y) / (y + mxm1*mxm1/y));
}
else
{
// Small or medium x, large y:
real = four*x/y/y;
}
}
else if (x != one)
{
// y is small, calculate divisor carefully:
T div = mxm1*mxm1;
if(y > safe_lower)
div += y*y;
real = boost::math::log1p(four*x/div);
}
else
real = boost::math::changesign(two * (std::log(y) - log_two));
real /= four;
if((boost::math::signbit)(z.real()))
real = (boost::math::changesign)(real);
//
// Now handle imaginary part, this is much easier,
// if x or y are large, then the formula:
// atan2(2y, (1-x)*(1+x) - y^2)
// evaluates to +-(PI - theta) where theta is negligible compared to PI.
//
if((x >= safe_upper) || (y >= safe_upper))
{
imag = pi;
}
else if(x <= safe_lower)
{
//
// If both x and y are small then atan(2y),
// otherwise just x^2 is negligible in the divisor:
//
if(y <= safe_lower)
imag = std::atan2(two*y, one);
else
{
if((y == zero) && (x == zero))
imag = 0;
else
imag = std::atan2(two*y, one - y*y);
}
}
else
{
//
// y^2 is negligible:
//
if((y == zero) && (x == one))
imag = 0;
else
imag = std::atan2(two*y, mxm1*(one+x));
}
imag /= two;
if((boost::math::signbit)(z.imag()))
imag = (boost::math::changesign)(imag);
}
return std::complex<T>(real, imag);
#ifdef BOOST_MSVC
#pragma warning(pop)
#endif
}
} } // namespaces
#endif // BOOST_MATH_COMPLEX_ATANH_INCLUDED