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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 #endif #ifndef BOOST_MATH_LOG1P_INCLUDED # include #endif #include #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 [[deprecated("Replaced by C++11")]] std::complex atanh(const std::complex& 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(); static const T half_pi = pi / 2; static const T one = static_cast(1.0L); static const T two = static_cast(2.0L); static const T four = static_cast(4.0L); static const T zero = static_cast(0); static const T log_two = boost::math::constants::ln_two(); #ifdef _MSC_VER #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(2)); // // Begin by handling the special cases specified in C99: // if((boost::math::isnan)(x)) { if((boost::math::isnan)(y)) return std::complex(x, x); else if((boost::math::isinf)(y)) return std::complex(0, ((boost::math::signbit)(z.imag()) ? -half_pi : half_pi)); else return std::complex(x, x); } else if((boost::math::isnan)(y)) { if(x == 0) return std::complex(x, y); if((boost::math::isinf)(x)) return std::complex(0, y); else return std::complex(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(real, imag); #ifdef _MSC_VER #pragma warning(pop) #endif } } } // namespaces #endif // BOOST_MATH_COMPLEX_ATANH_INCLUDED