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zeta.m

# 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 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, see <http://www.gnu.org/licenses/>.

## ZETA compute the Riemann's Zeta function.


function z = zeta(t)
      if (nargin != 1)
                  usage ("zeta(x)");
      endif
      z = zeros(size(t));
      for j = 1:prod(size(t))
            if(real(t(j)) >= 0)
                  if(imag(t(j)) == 0 && real(t(j)) > 1)
                        F= @(x) 1./(gamma(t(j))).*x.^(t(j)-1)./(exp(x)-1);
                        z(j) = quad(F,0,Inf);
                  elseif(t(j) == 0)
                        z(j) = -0.5;
                  elseif(t(j) == 1)
                        z(j) = Inf;
                  else
                        for k = 1:100
                              z(j) += (-1).^(k-1)./(k.^t(j));
                        endfor
                        z(j) = 1./(1-2.^(1-t(j))).*z(j);
                  endif
            else
                  z(j) = 2.^t(j).*pi.^(t(j)-1).*sin(pi.*t(j)./2).*gamma(1-t(j)).*zeta(1-t(j));
            endif
      endfor
endfunction;

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