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// (C) Copyright John Maddock 2006.
// 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_SPECIAL_SPHERICAL_HARMONIC_HPP
#define BOOST_MATH_SPECIAL_SPHERICAL_HARMONIC_HPP
#ifdef _MSC_VER
#pragma once
#endif
#include <boost/math/special_functions/legendre.hpp>
#include <boost/math/tools/workaround.hpp>
#include <complex>
namespace boost{
namespace math{
namespace detail{
//
// Calculates the prefix term that's common to the real
// and imaginary parts. Does *not* fix up the sign of the result
// though.
//
template <class T, class Policy>
inline T spherical_harmonic_prefix(unsigned n, unsigned m, T theta, const Policy& pol)
{
BOOST_MATH_STD_USING
if(m > n)
return 0;
T sin_theta = sin(theta);
T x = cos(theta);
T leg = detail::legendre_p_imp(n, m, x, static_cast<T>(pow(fabs(sin_theta), T(m))), pol);
T prefix = boost::math::tgamma_delta_ratio(static_cast<T>(n - m + 1), static_cast<T>(2 * m), pol);
prefix *= (2 * n + 1) / (4 * constants::pi<T>());
prefix = sqrt(prefix);
return prefix * leg;
}
//
// Real Part:
//
template <class T, class Policy>
T spherical_harmonic_r(unsigned n, int m, T theta, T phi, const Policy& pol)
{
BOOST_MATH_STD_USING // ADL of std functions
bool sign = false;
if(m < 0)
{
// Reflect and adjust sign if m < 0:
sign = m&1;
m = abs(m);
}
if(m&1)
{
// Check phase if theta is outside [0, PI]:
T mod = boost::math::tools::fmod_workaround(theta, T(2 * constants::pi<T>()));
if(mod < 0)
mod += 2 * constants::pi<T>();
if(mod > constants::pi<T>())
sign = !sign;
}
// Get the value and adjust sign as required:
T prefix = spherical_harmonic_prefix(n, m, theta, pol);
prefix *= cos(m * phi);
return sign ? T(-prefix) : prefix;
}
template <class T, class Policy>
T spherical_harmonic_i(unsigned n, int m, T theta, T phi, const Policy& pol)
{
BOOST_MATH_STD_USING // ADL of std functions
bool sign = false;
if(m < 0)
{
// Reflect and adjust sign if m < 0:
sign = !(m&1);
m = abs(m);
}
if(m&1)
{
// Check phase if theta is outside [0, PI]:
T mod = boost::math::tools::fmod_workaround(theta, T(2 * constants::pi<T>()));
if(mod < 0)
mod += 2 * constants::pi<T>();
if(mod > constants::pi<T>())
sign = !sign;
}
// Get the value and adjust sign as required:
T prefix = spherical_harmonic_prefix(n, m, theta, pol);
prefix *= sin(m * phi);
return sign ? T(-prefix) : prefix;
}
template <class T, class U, class Policy>
std::complex<T> spherical_harmonic(unsigned n, int m, U theta, U phi, const Policy& pol)
{
BOOST_MATH_STD_USING
//
// Sort out the signs:
//
bool r_sign = false;
bool i_sign = false;
if(m < 0)
{
// Reflect and adjust sign if m < 0:
r_sign = m&1;
i_sign = !(m&1);
m = abs(m);
}
if(m&1)
{
// Check phase if theta is outside [0, PI]:
U mod = boost::math::tools::fmod_workaround(theta, 2 * constants::pi<U>());
if(mod < 0)
mod += 2 * constants::pi<U>();
if(mod > constants::pi<U>())
{
r_sign = !r_sign;
i_sign = !i_sign;
}
}
//
// Calculate the value:
//
U prefix = spherical_harmonic_prefix(n, m, theta, pol);
U r = prefix * cos(m * phi);
U i = prefix * sin(m * phi);
//
// Add in the signs:
//
if(r_sign)
r = -r;
if(i_sign)
i = -i;
static const char* function = "boost::math::spherical_harmonic<%1%>(int, int, %1%, %1%)";
return std::complex<T>(policies::checked_narrowing_cast<T, Policy>(r, function), policies::checked_narrowing_cast<T, Policy>(i, function));
}
} // namespace detail
template <class T1, class T2, class Policy>
inline std::complex<typename tools::promote_args<T1, T2>::type>
spherical_harmonic(unsigned n, int m, T1 theta, T2 phi, const Policy& pol)
{
typedef typename tools::promote_args<T1, T2>::type result_type;
typedef typename policies::evaluation<result_type, Policy>::type value_type;
return detail::spherical_harmonic<result_type, value_type>(n, m, static_cast<value_type>(theta), static_cast<value_type>(phi), pol);
}
template <class T1, class T2>
inline std::complex<typename tools::promote_args<T1, T2>::type>
spherical_harmonic(unsigned n, int m, T1 theta, T2 phi)
{
return boost::math::spherical_harmonic(n, m, theta, phi, policies::policy<>());
}
template <class T1, class T2, class Policy>
inline typename tools::promote_args<T1, T2>::type
spherical_harmonic_r(unsigned n, int m, T1 theta, T2 phi, const Policy& pol)
{
typedef typename tools::promote_args<T1, T2>::type result_type;
typedef typename policies::evaluation<result_type, Policy>::type value_type;
return policies::checked_narrowing_cast<result_type, Policy>(detail::spherical_harmonic_r(n, m, static_cast<value_type>(theta), static_cast<value_type>(phi), pol), "bost::math::spherical_harmonic_r<%1%>(unsigned, int, %1%, %1%)");
}
template <class T1, class T2>
inline typename tools::promote_args<T1, T2>::type
spherical_harmonic_r(unsigned n, int m, T1 theta, T2 phi)
{
return boost::math::spherical_harmonic_r(n, m, theta, phi, policies::policy<>());
}
template <class T1, class T2, class Policy>
inline typename tools::promote_args<T1, T2>::type
spherical_harmonic_i(unsigned n, int m, T1 theta, T2 phi, const Policy& pol)
{
typedef typename tools::promote_args<T1, T2>::type result_type;
typedef typename policies::evaluation<result_type, Policy>::type value_type;
return policies::checked_narrowing_cast<result_type, Policy>(detail::spherical_harmonic_i(n, m, static_cast<value_type>(theta), static_cast<value_type>(phi), pol), "boost::math::spherical_harmonic_i<%1%>(unsigned, int, %1%, %1%)");
}
template <class T1, class T2>
inline typename tools::promote_args<T1, T2>::type
spherical_harmonic_i(unsigned n, int m, T1 theta, T2 phi)
{
return boost::math::spherical_harmonic_i(n, m, theta, phi, policies::policy<>());
}
} // namespace math
} // namespace boost
#endif // BOOST_MATH_SPECIAL_SPHERICAL_HARMONIC_HPP
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