123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131 |
- /* 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 3 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_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__ */
|