text: /* 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__ */
  