FDm12_M := proc(x)
#
#  ---------------------------------------------------------------
#  
#  PURPOSE
#
#    This Maple 6 procedure Computes the Fermi-Dirac integral
#
#       Fj(x) = 1/GAMMMA(j+1)*int(t^j/(exp(t-x)+1),t=0..infinity)
#
#    of order j = -1/2, correct to at least 15 significant decimal
#    digits of accuracy using MacLeod's piecewise Chebyshev series
#    approximation scheme. (See reference below.)
#
#    The magnitude of the relative error is at most 0.4x10^(-15)
#    on the x domain (-infinity,infinity). 
#
#  USAGE
#
#    y = FDm12_M(x)
#
#  FILE NAMES 
#
#    FDm12_M.mpl    -- MacLeod's piecewise Chebyshev series approx 
#    Clenshaw_M.mpl -- Clenshaw's evaluation of Chebyshev sum
#
#  DATE
#
#    October 12, 2000
#
#  PROGRAMMER
#
#    Dr. F.G. Lether
#    Dept. Math. 
#    Univ. Georgia
#    Athens, GA. 30602 USA
#
#    email : fglether@math.uga.edu
#
#  REFERENCE
#
#    MacLeod, Allan.J.(1998), Algorithm 778: Fermi-Dirac Functions
#    of Order -1/2,1/2,3/2,5/2, Trans. Math. Soft. (TOMS) 24,1-12
#
#  ---------------------------------------------------------------
#  
   local twoe,const,c1,c2,c3,ex,t,xsq,ChebSum,FDm12_Cheb;
#
   Digits := 19;
#
#  Initialize constants 2*e and 1/GAMMA(3/2) = 2/sqrt(Pi) 
#
   twoe  := 5.4365636569180904;
   const := 1.1283791670955126;
#   
#  Left tail region (-inf,-1) Chebyshev coefficients
#
   c1 := array(0..14 , [ 1.7863596385102264e0 ,
                        -0.999372007632333e-1 ,
                         0.64144652216054e-2  ,
                        -0.4356415371345e-3   ,
                         0.305216700310e-4    ,
                        -0.21810648110e-5     , 
                         0.1580050781e-6      ,
                        -0.115620570e-7       ,
                         0.8525860e-9         ,
                        -0.632529e-10         ,
                         0.47159e-11          ,
                        -0.3530e-12           ,
                         0.265e-13            ,
                        -0.20e-14             ,
                         0.2e-15              ] );
#
#  Middle region I [-1,2) Chebyshev coefficients
#
   c2 := array(0..23 , [ 1.6877111526052352e0 ,
                         0.5978360226336983e0 ,
                         0.357226004541669e-1 ,
                        -0.132144786506426e-1 ,
                        -0.4040134207447e-3   ,
                         0.5330011846887e-3   ,
                        -0.148923504863e-4    ,
                        -0.218863822916e-4    , 
                         0.19652084277e-5     ,
                         0.8565830466e-6      ,
                        -0.1407723133e-6      ,
                        -0.305175803e-7       ,
                         0.83524532e-8        , 
                         0.9025750e-9         ,
                        -0.4455471e-9         ,
                        -0.148342e-10         , 
                         0.219266e-10         ,
                        -0.6579e-12           , 
                        -0.10009e-11          ,
                         0.936e-13            , 
                         0.420e-13            ,
                        -0.71e-14             ,
                        -0.16e-14             , 
                         0.4e-15              ] );
#
#  Right tail region [2,infinity) Chebyshev coefficients
#
   c3 := array(0..58 , [ 0.8707195029590563e0 ,
                         0.59833110231733e-2  ,
                        -0.432670470895746e-1 , 
                        -0.393083681608590e-1 , 
                        -0.191482688045932e-1 , 
                        -0.65582880980158e-2  ,
                        -0.22276691516312e-2  , 
                        -0.8466786936178e-3   , 
                        -0.2807459489219e-3   ,
                        -0.955575024348e-4     , 
                        -0.362367662803e-4     ,
                        -0.109158468869e-4     , 
                        -0.39356701000e-5      ,
                        -0.13108192725e-5      ,
                        -0.2468816388e-6       ,
                        -0.1048380311e-6       ,
                         0.236181487e-7        ,  
                         0.227145359e-7        ,
                         0.145775174e-7        , 
                         0.153926767e-7        ,
                         0.56924772e-8         ,
                         0.50623068e-8         , 
                         0.23426075e-8         ,
                         0.12652275e-8         ,
                         0.8927773e-9          ,
                         0.2994501e-9          ,
                         0.2822785e-9          ,
                         0.910685e-10          , 
                         0.696285e-10          ,
                         0.366225e-10          , 
                         0.124351e-10          , 
                         0.145019e-10          ,
                         0.16645e-11           , 
                         0.45856e-11           , 
                         0.6092e-12            ,
                         0.9331e-12            ,
                         0.5238e-12            , 
                        -0.56e-14              ,
                         0.3170e-12            ,
                        -0.926e-13             ,
                         0.1265e-12            , 
                        -0.327e-13             ,
                         0.276e-13             , 
                         0.33e-14              ,
                        -0.42e-14              ,
                         0.101e-13             ,
                        -0.73e-14              ,
                         0.64e-14              , 
                        -0.37e-14              ,
                         0.23e-14              ,
                        -0.9e-15               ,
                         0.2e-15               , 
                         0.2e-15               , 
                        -0.3e-15               , 
                         0.4e-15               ,
                        -0.3e-15               ,
                         0.2e-15               ,
                        -0.1e-15               , 
                         0.1e-15               ] ); 
#
#  Compute MacLeod approximation depending on x in
#  (-infinity,-1), [1,2) or [2,infinity)
#   
   if x < -1 then
#
     ex := evalf(exp(x)); 
     t :=  twoe*ex - 1;
     ChebSum := Clenshaw_M(c1,14,t);
     FDm12_Cheb := ex*ChebSum;
#
   elif x < 2 then
#
     t := (2*x-1)/3;
     ChebSum := Clenshaw_M(c2,23,t);
     FDm12_Cheb := ChebSum;
#
   else
#
     FDm12_Cheb := const*evalf(sqrt(x));
     xsq := x*x;
     t := (50 - xsq)/(42 + xsq);
     ChebSum := Clenshaw_M(c3,58,t);
     FDm12_Cheb := FDm12_Cheb*(1 - ChebSum/xsq);
   fi;
#
   end: