## Copyright (C) 2006 Sylvain Pelissier <sylvain.pelissier@gmail.com>
##
## 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, see <http://www.gnu.org/licenses/>.

## -*- texinfo -*-
## @deftypefn {Function File} {@var{y} =} expint_Ei (@var{x})
## Compute the exponential integral,
## @verbatim
##                      infinity
##                     /
##    expint_Ei(x) = - | exp(t)/t dt
##                     /
##                     -x
## @end verbatim
## @seealso{expint, expint_E1}
## @end deftypefn

function y = expint_Ei(x)
  if (nargin != 1)
    print_usage;
  endif
  y = zeros(size(x));
  F = @(x) exp(-x)./x;
  s = prod(size(x));
  for t = 1:s;
    if(x(t)<0 && imag(x(t)) == 0)
      y(t) = -quad(F,-x(t),Inf);
    else
      if(abs(x(t)) > 2 && imag(x(t)) == 0)
        y(t) = expint_Ei(2) - quad(F,-x(t),-2);
      else
        if(abs(x(t)) >= 10)
          if(imag(x(t)) <= 0)
            a1 = 4.03640;
            a2 = 1.15198;
            b1 = 5.03637;
            b2 = 4.19160;
            y(t) = -(x(t).^2 - a1.*x(t) + a2)./((x(t).^2-b1.*x(t)+b2).*(-x(t)).*exp(-x(t)))-i.*pi;
          else
            y(t) = conj(expint_Ei(conj(x(t))));
          endif;
        ## Serie Expansion
        else
          for k = 1:100;
            y(t) = y(t) + x(t).^k./(k.*factorial(k));
          endfor
          y(t) = 0.577215664901532860606512090082402431 + log(x(t)) + y(t);
        endif
      endif
    endif
  endfor
endfunction
