FDp52_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 = 5/2, 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 = FDp52_M(x)
#
#  FILE NAMES 
#
#    FDp52_M.mpl    -- MacLeod's piecewise Chebyshev series approx 
#    Clenshaw_M.mpl -- Clenshaw's evaluation of Chebyshev sum
#
#  DATE
#
#    October 17, 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,FDp52_Cheb;
#
   Digits := 19;
#
#  Initialize constants 2*e and 1/GAMMA(9/2) = 16/(105*sqrt(Pi)) 
#
   twoe  := 5.4365636569180904e0;
   const := 8.5971746064420006e-2;
 
#   
#  Left tail region (-inf,-1) Chebyshev coefficients
#
   c1 := array(0..11 , [ 1.9694416685896693e0 ,
                        -0.149691794643492e-1 ,
                         0.3006955816627e-3   ,
                        -0.89462485950e-5     ,
                         0.3298072025e-6      ,
                        -0.139239298e-7       ,
                         0.6455885e-9         ,
                        -0.320623e-10         ,
                         0.16783e-11          ,
                        -0.916e-13            ,
                         0.52e-14             ,
                        -0.3e-15              ] );
#
#  Middle region I [-1,2) Chebyshev coefficients
#
   c2 := array(0..21 , [ 4.2642838398655301e0 ,
                         2.3437426884912867e0 ,
                         0.6727119780052076e0 ,
                         0.1148826327965569e0 , 
                         0.109363968046758e-1 ,
                         0.2567173957015e-3   ,
                        -0.505889983911e-4    , 
                        -0.7376215774e-6      , 
                         0.7352998758e-6      , 
                        -0.166421736e-7       , 
                        -0.140920499e-7       , 
                         0.9949192e-9         ,
                         0.2991457e-9         ,
                        -0.401332e-10         ,
                        -0.63546e-11          ,
                         0.14793e-11          ,
                         0.1181e-12           ,
                        -0.524e-13            ,
                        -0.11e-14             ,
                         0.18e-14             ,
                        -0.1e-15              ,
                        -0.1e-15              ] );
#
#  Right tail region [2,infinity) Chebyshev coefficients
#
   c3 := array(0..61 , [ 30.2895676859802579e0 ,
                          1.1678976642060562e0 ,
                          0.6420591800821472e0 ,
                          0.3461723868407417e0 ,
                          0.1840816790781889e0 ,
                          0.973092435354509e-1 ,
                          0.513973292675393e-1 ,
                          0.271709801041757e-1 ,
                          0.143833271401165e-1 ,
                          0.76264863952155e-2  ,
                          0.40503695767202e-2  , 
                          0.21543961464149e-2  ,
                          0.11475689901777e-2  ,
                          0.6120622369282e-3   ,
                          0.3268340337859e-3   ,
                          0.1747145522742e-3   ,
                          0.934878457860e-4    ,
                          0.500692212553e-4    , 
                          0.268373821846e-4    , 
                          0.143957191251e-4    ,
                          0.77272440700e-5     ,
                          0.41503820336e-5     ,
                          0.22305118261e-5     ,
                          0.11993697093e-5     ,
                          0.6452344369e-6      , 
                          0.3472822881e-6      ,
                          0.1869964215e-6      ,
                          0.1007300272e-6      ,
                          0.542807561e-7       ,
                          0.292607829e-7       , 
                          0.157785918e-7       ,
                          0.85110768e-8        ,
                          0.45922760e-8        ,
                          0.24785001e-8        ,
                          0.13380255e-8        , 
                          0.7225103e-9         , 
                          0.3902350e-9         , 
                          0.2108157e-9         ,
                          0.1139122e-9         ,
                          0.615638e-10         ,
                          0.332781e-10         ,
                          0.179919e-10         , 
                          0.97288e-11          ,
                          0.52617e-11          ,
                          0.28461e-11          , 
                          0.15397e-11          ,
                          0.8331e-12           ,
                          0.4508e-12           ,
                          0.2440e-12           ,
                          0.1321e-12           ,
                          0.715e-13            ,
                          0.387e-13            , 
                          0.210e-13            ,
                          0.114e-13            ,
                          0.61e-14             , 
                          0.33e-14             , 
                          0.18e-14             , 
                          0.11e-14             ,
                          0.5e-15              ,
                          0.3e-15              , 
                          0.2e-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(c1,11,t);
     FDp52_Cheb := ex*ChebSum;
#
   elif x < 2 then
#
     t := (2*x-1)/3;
     ChebSum := Clenshaw_M(c2,21,t);
     FDp52_Cheb := ChebSum;
#
   else
#
     xsq := x*x;
     FDp52_Cheb := const*x*xsq*evalf(sqrt(x));
     t := (50 - xsq)/(42 + xsq);
     ChebSum := Clenshaw_M(c3,61,t);
     FDp52_Cheb := FDp52_Cheb*(1 + ChebSum/xsq);
   fi;
#
   end: