| /* specfunc/gsl_sf_dilog.h |
| * |
| * Copyright (C) 1996, 1997, 1998, 1999, 2000, 2004 Gerard Jungman |
| * |
| * This program is free software; you can redistribute it and/or modify |
| * it under the terms of the GNU General Public License as published by |
| * the Free Software Foundation; either version 2 of the License, or (at |
| * your option) any later version. |
| * |
| * This program is distributed in the hope that it will be useful, but |
| * WITHOUT ANY WARRANTY; without even the implied warranty of |
| * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU |
| * General Public License for more details. |
| * |
| * You should have received a copy of the GNU General Public License |
| * along with this program; if not, write to the Free Software |
| * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA. |
| */ |
| |
| /* Author: G. Jungman */ |
| |
| #ifndef __GSL_SF_DILOG_H__ |
| #define __GSL_SF_DILOG_H__ |
| |
| #include <gsl/gsl_sf_result.h> |
| |
| #undef __BEGIN_DECLS |
| #undef __END_DECLS |
| #ifdef __cplusplus |
| # define __BEGIN_DECLS extern "C" { |
| # define __END_DECLS } |
| #else |
| # define __BEGIN_DECLS /* empty */ |
| # define __END_DECLS /* empty */ |
| #endif |
| |
| __BEGIN_DECLS |
| |
| |
| /* Real part of DiLogarithm(x), for real argument. |
| * In Lewin's notation, this is Li_2(x). |
| * |
| * Li_2(x) = - Re[ Integrate[ Log[1-s] / s, {s, 0, x}] ] |
| * |
| * The function in the complex plane has a branch point |
| * at z = 1; we place the cut in the conventional way, |
| * on [1, +infty). This means that the value for real x > 1 |
| * is a matter of definition; however, this choice does not |
| * affect the real part and so is not relevant to the |
| * interpretation of this implemented function. |
| */ |
| int gsl_sf_dilog_e(const double x, gsl_sf_result * result); |
| double gsl_sf_dilog(const double x); |
| |
| |
| /* DiLogarithm(z), for complex argument z = x + i y. |
| * Computes the principal branch. |
| * |
| * Recall that the branch cut is on the real axis with x > 1. |
| * The imaginary part of the computed value on the cut is given |
| * by -Pi*log(x), which is the limiting value taken approaching |
| * from y < 0. This is a conventional choice, though there is no |
| * true standardized choice. |
| * |
| * Note that there is no canonical way to lift the defining |
| * contour to the full Riemann surface because of the appearance |
| * of a "hidden branch point" at z = 0 on non-principal sheets. |
| * Experts will know the simple algebraic prescription for |
| * obtaining the sheet they want; non-experts will not want |
| * to know anything about it. This is why GSL chooses to compute |
| * only on the principal branch. |
| */ |
| int |
| gsl_sf_complex_dilog_xy_e( |
| const double x, |
| const double y, |
| gsl_sf_result * result_re, |
| gsl_sf_result * result_im |
| ); |
| |
| |
| |
| /* DiLogarithm(z), for complex argument z = r Exp[i theta]. |
| * Computes the principal branch, thereby assuming an |
| * implicit reduction of theta to the range (-2 pi, 2 pi). |
| * |
| * If theta is identically zero, the imaginary part is computed |
| * as if approaching from y > 0. For other values of theta no |
| * special consideration is given, since it is assumed that |
| * no other machine representations of multiples of pi will |
| * produce y = 0 precisely. This assumption depends on some |
| * subtle properties of the machine arithmetic, such as |
| * correct rounding and monotonicity of the underlying |
| * implementation of sin() and cos(). |
| * |
| * This function is ok, but the interface is confusing since |
| * it makes it appear that the branch structure is resolved. |
| * Furthermore the handling of values close to the branch |
| * cut is subtle. Perhap this interface should be deprecated. |
| */ |
| int |
| gsl_sf_complex_dilog_e( |
| const double r, |
| const double theta, |
| gsl_sf_result * result_re, |
| gsl_sf_result * result_im |
| ); |
| |
| |
| |
| /* Spence integral; spence(s) := Li_2(1-s) |
| * |
| * This function has a branch point at 0; we place the |
| * cut on (-infty,0). Because of our choice for the value |
| * of Li_2(z) on the cut, spence(s) is continuous as |
| * s approaches the cut from above. In other words, |
| * we define spence(x) = spence(x + i 0+). |
| */ |
| int |
| gsl_sf_complex_spence_xy_e( |
| const double x, |
| const double y, |
| gsl_sf_result * real_sp, |
| gsl_sf_result * imag_sp |
| ); |
| |
| |
| __END_DECLS |
| |
| #endif /* __GSL_SF_DILOG_H__ */ |
