reduce-historical/r36/xlog/FPS.LOG

REDUCE 3.6, 15-Jul-95, patched to 6 Mar 96 ...


%       Examples for the algorithmic calculation of formal 
%       Puiseux, Laurent and power series,
%
%       Wolfram Koepf, Freie Universitaet Berlin, Germany
%       (taken from the original paper and adapted to REDUCE 
%        form by Winfried Neun, ZIB Berlin)

% Formal Laurent series

on errcont;



fps(E^x,x);


              k
             x
infsum(--------------,k,0,infinity)
        factorial(k)


fps(E^x/(x^3),x);


               k
              x
infsum(-----------------,k,0,infinity)
                      3
        factorial(k)*x


fps(x * e^(x^4),x);


            4*k
           x   *x
infsum(--------------,k,0,infinity)
        factorial(k)


fps(sin (x + y),x);


         2*k       k
        x   *( - 1) *cos(y)*x
infsum(-----------------------,k,0,infinity)
         factorial(2*k + 1)

            2*k       k
           x   *( - 1) *sin(y)
 + infsum(---------------------,k,0,infinity)
             factorial(2*k)


simplede (sin x,x);


df(y,x,2) + y
 %find a DE for sin

simplede (sin (x)^2,x,w);


df(w,x,3) + 4*df(w,x)
 % DE in w and x   

fps(asin x,x);


            2*k
           x   *factorial(2*k)*x
infsum(------------------------------,k,0,infinity)
         2*k             2
        2   *factorial(k) *(2*k + 1)


fps((asin x)^2,x);


         2*k  2*k             2  2
        x   *2   *factorial(k) *x
infsum(----------------------------,k,0,infinity)
        factorial(2*k + 1)*(k + 1)


fps(e^(asin x),x);


         2*k  k         2
        x   *2 *prod(2*j  - 2*j + 1,j,1,k)*x
infsum(--------------------------------------,k,0,infinity)
                 factorial(2*k + 1)

            2*k         2
           x   *prod(4*j  - 8*j + 5,j,1,k)
 + infsum(---------------------------------,k,0,infinity)
                   factorial(2*k)


fps(e^(asinh x),x);


            2*k       k
         - x   *( - 1) *factorial(2*k)
infsum(--------------------------------,k,0,infinity) + x
           k             2
          4 *factorial(k) *(2*k - 1)


fps((x + sqrt(1+x^2))^A,x);


         2*k       k  2*k              - a                 a
        x   *( - 1) *2   *pochhammer(------,k)*pochhammer(---,k)
                                       2                   2
infsum(----------------------------------------------------------,k,0,infinity) 
                             factorial(2*k)

           2*k       k  2*k              - a + 1                 a + 1
          x   *( - 1) *2   *pochhammer(----------,k)*pochhammer(-------,k)*a*x
                                           2                       2
+ infsum(----------------------------------------------------------------------,
                                   factorial(2*k + 1)

         k,0,infinity)


fps(e^(x^2)*erf x,x);


           2*k           2*k
        2*x   *sqrt(pi)*2   *factorial(k)*x
infsum(-------------------------------------,k,0,infinity)
               factorial(2*k + 1)*pi


fps(e^x - 2 e^(-x/2) * cos(sqrt(3) * x/2 -pi/3),x);


           3*k  2
        9*x   *x *(k + 1)
infsum(--------------------,k,0,infinity)
        factorial(3*k + 3)


% fps(int(e^(-a^2*t^2) * cos(2*x*t),t,0,infinity),x)  % not yet

% fps(4/x * int(e^(t^2)*erf(t),t,0,sqrt(x)/2),x);

fps(sin x * e^x,x);


         k  k/2      k*pi
        x *2   *sin(------)
                      4
infsum(---------------------,k,0,infinity)
           factorial(k)


fps(cos x * e^(2*x),x);


         k  k/2           1
        x *5   *cos(atan(---)*k)
                          2
infsum(--------------------------,k,0,infinity)
              factorial(k)


fps(1/(x-x^3),x);


          k       k    k
         x *( - 1)  - x
 infsum(-----------------,k,0,infinity)*x + 1
                    k
            2*( - 1)
----------------------------------------------
                      x


fps(1/(x^2 + 3 x + 2),x);


           k       k    k       k
        2*x *( - 2)  - x *( - 1)
infsum(---------------------------,k,0,infinity)
                    k       k
            2*( - 1) *( - 2)


fps(x/(1-x-x^2),x);


          x
fps(--------------,x,0)
                2
     (1 - x) - x


% Logarithmic singularities and Puisieux series

fps(sin sqrt x,x);


         (2*k + 1)/2       k
        x           *( - 1)
infsum(----------------------,k,0,infinity)
         factorial(2*k + 1)


fps(((1 + sqrt x)/x)^(1/3),x);


         (6*k + 1)/6             2
        x           *pochhammer(---,2*k)
                                 3
infsum(----------------------------------,k,0,infinity)
              3*factorial(2*k + 1)

            k              - 1
           x *pochhammer(------,2*k)
                           3
 + infsum(---------------------------,k,0,infinity)
               1/3
              x   *factorial(2*k)


fps(asech x,x);


***** Problem using limit operator 


% some more (Wolfram Koepf, priv. comm.)

fps((1+x)^alpha,x);


         k       k
        x *( - 1) *pochhammer( - alpha,k)
infsum(-----------------------------------,k,0,infinity)
                  factorial(k)


fps((1+sqrt(1+x))^beta,x);


          k       k  beta
         x *( - 1) *2    *pochhammer( - beta,2*k)
infsum(---------------------------------------------,k,0,infinity)
         2*k
        2   *factorial(k)*pochhammer( - beta + 1,k)


fps(sin(x)^2+cos(x)^2,x);


1


fps(sin(x)^2*cos(x)^2,x);


            2*k       k  2*k  2
           x   *( - 4) *2   *x
infsum(----------------------------,k,0,infinity)
        factorial(2*k + 1)*(k + 1)


fps(sin(x)*cos(x^2),x);


                2
fps(sin(x)*cos(x ),x,0)


fps((x-1)^(-1),x);


           k
infsum( - x ,k,0,infinity)


fps(atan(x+y),x);


fps(atan(x + y),x,0)


fps((1-x^5)^6,x);


 30      25       20       15       10      5
x   - 6*x   + 15*x   - 20*x   + 15*x   - 6*x  + 1


fps(asec x,x);


***** Problem using limit operator 


fps(besseli(0,x),x);


         2*k
        x   *besseli(0,0)
infsum(--------------------,k,0,infinity)
         2*k             2
        2   *factorial(k)

fps(besseli(1,x),x);


                 2*k
                x   *besseli(0,0)*x
infsum(--------------------------------------,k,0,infinity)
           2*k
        2*2   *factorial(k + 1)*factorial(k)


fps(exp(x^(1/3)),x);


            (3*k + 1)/3
           x
infsum(--------------------,k,0,infinity)
        factorial(3*k + 1)

                  k
                 x
 + infsum(----------------,k,0,infinity)
           factorial(3*k)

              (3*k + 2)/3
           3*x           *(k + 1)
 + infsum(------------------------,k,0,infinity)
             factorial(3*k + 3)

fps(log(1-x),x);


            k
         - x *x
infsum(---------,k,0,infinity)
         k + 1

fps(exp x*sinh x,x);


             k  k
            x *2 *x
infsum(------------------,k,0,infinity)
        factorial(k + 1)

fps(atan x,x);


         2*k       k
        x   *( - 1) *x
infsum(----------------,k,0,infinity)
           2*k + 1

fps(sin x+sinh x,x);


                4*k
             2*x   *x
infsum(--------------------,k,0,infinity)
        factorial(4*k + 1)

fps(sin x*sinh x,x);


             4*k       k  2*k  2
            x   *( - 1) *2   *x
infsum(------------------------------,k,0,infinity)
        factorial(4*k + 1)*(2*k + 1)

fps(int(erf(x),x),x);


*** ci already defined as operator 

*** si already defined as operator 

            2*k                k
         - x   *sqrt(pi)*( - 1)
infsum(---------------------------,k,0,infinity)
        factorial(k)*pi*(2*k - 1)

fps(sqrt(2-x),x);


            k
         - x *sqrt(2)*factorial(2*k)
infsum(------------------------------,k,0,infinity)
          k             2
         8 *factorial(k) *(2*k - 1)

fps(sqrt(1+x)+sqrt(1-x),x);


                 2*k
            - 2*x   *factorial(4*k)
infsum(--------------------------------,k,0,infinity)
         2*k               2
        4   *factorial(2*k) *(4*k - 1)

fps(exp(a+b*x)*exp(c+d*x),x);


         k  a + c        k
        x *e     *(b + d)
infsum(--------------------,k,0,infinity)
           factorial(k)

fps(1/cos(asin x),x);


         2*k
        x   *factorial(2*k)
infsum(---------------------,k,0,infinity)
         2*k             2
        2   *factorial(k)

fps(sqrt(1-x^2)+x*asin x,x);


                2*k
               x   *factorial(2*k)
infsum(-----------------------------------,k,0,infinity)
         k             2     2
        4 *factorial(k) *(4*k  - 4*k + 1)

fps(sqrt(1-sqrt(x)),x);


                 (2*k + 1)/2
              - x           *factorial(4*k)
infsum(------------------------------------------,k,0,infinity)
           4*k
        2*2   *factorial(2*k + 1)*factorial(2*k)

                    k
                 - x *factorial(4*k)
 + infsum(--------------------------------,k,0,infinity)
            2*k               2
           4   *factorial(2*k) *(4*k - 1)

fps(cos(n*acos x),x);


         2*k  2*k      n*pi                - n                 n
        x   *2   *cos(------)*pochhammer(------,k)*pochhammer(---,k)
                        2                  2                   2
infsum(--------------------------------------------------------------,k,0,
                               factorial(2*k)

       infinity) + infsum(

     2*k  2*k              - n + 1                 n + 1          n*pi
    x   *2   *pochhammer(----------,k)*pochhammer(-------,k)*sin(------)*n*x
                             2                       2             2
   --------------------------------------------------------------------------,k,
                               factorial(2*k + 1)

   0,infinity)

fps(cos x+I*sin x,x);


           k       k
          x *( - 1)
infsum(-----------------,k,0,infinity)
         k
        i *factorial(k)

fps(cos(3*asinh x),x);


         2*k       k         2
        x   *( - 1) *prod(4*j  - 8*j + 13,j,1,k)
infsum(------------------------------------------,k,0,infinity)
                     factorial(2*k)

fps(cos(n*asinh x),x);


         2*k       k  2*k              - i*n                 i*n
        x   *( - 1) *2   *pochhammer(--------,k)*pochhammer(-----,k)
                                        2                     2
infsum(--------------------------------------------------------------,k,0,
                               factorial(2*k)

       infinity)

fps(sin(n*log(x+sqrt(1+x^2))),x);


         2*k       k  2*k              - i*n + 1                 i*n + 1
infsum((x   *( - 1) *2   *pochhammer(------------,k)*pochhammer(---------,k)*n*x
                                          2                         2

        )/factorial(2*k + 1),k,0,infinity)

fps(sqrt(1+x^2)*asinh x-x,x);


           2*k       k  2*k                                3
        2*x   *( - 1) *2   *factorial(k + 1)*factorial(k)*x
infsum(------------------------------------------------------,k,0,infinity)
                         factorial(2*k + 3)


fps(int(erf(x)/x,x),x);


              2*k                k
           2*x   *sqrt(pi)*( - 1) *x
infsum(----------------------------------,k,0,infinity)
                            2
        factorial(k)*pi*(4*k  + 4*k + 1)

                erf(x)
 + sub(x=0,int(--------,x))
                  x

fps(asin(x)^2/x^4,x);


            2*k  2*k             2
           x   *2   *factorial(k)
infsum(-------------------------------,k,0,infinity)
                            2
        factorial(2*k + 1)*x *(k + 1)



% we had problems here:

fps(cos(asin x),x);


              2*k
           - x   *factorial(2*k)
infsum(----------------------------,k,0,infinity)
         k             2
        4 *factorial(k) *(2*k - 1)

fps(sinh(log x),x);


fps(sinh(log(x)),x,0)

fps(atan(cot x),x);


Could not find the limit of: atan(cot(x)),x,0

***** problem using limit operator 

 
% we can cure this one by defining the limit:

let limit(atan(cot ~x),x,0) => pi/2;


fps(atan(cot x),x);


 pi - 2*x
----------
    2


fps(exp(nnn*x)*cos(mmm*x),x);


         k              2                                          2
infsum((x *((impart(mmm)  + 2*impart(mmm)*repart(nnn) + impart(nnn)

                                                       2              2    k
              - 2*impart(nnn)*repart(mmm) + repart(mmm)  + repart(nnn) )**---
                                                                           2

                       impart(nnn) - repart(mmm)                    2
            *cos(atan(---------------------------)*k) + (impart(mmm)
                       impart(mmm) + repart(nnn)

                                                         2
                - 2*impart(mmm)*repart(nnn) + impart(nnn)

                                                         2              2    k
                + 2*impart(nnn)*repart(mmm) + repart(mmm)  + repart(nnn) )**---
                                                                             2

                       impart(nnn) + repart(mmm)                    2
            *cos(atan(---------------------------)*k) - (impart(mmm)
                       impart(mmm) - repart(nnn)

                                                         2
                - 2*impart(mmm)*repart(nnn) + impart(nnn)

                                                         2              2    k
                + 2*impart(nnn)*repart(mmm) + repart(mmm)  + repart(nnn) )**---
                                                                             2

                       impart(nnn) + repart(mmm)                      2
            *sin(atan(---------------------------)*k)*i + (impart(mmm)
                       impart(mmm) - repart(nnn)

                                                         2
                + 2*impart(mmm)*repart(nnn) + impart(nnn)

                                                         2              2    k
                - 2*impart(nnn)*repart(mmm) + repart(mmm)  + repart(nnn) )**---
                                                                             2

                       impart(nnn) - repart(mmm)
            *sin(atan(---------------------------)*k)*i))/(2*factorial(k)),k,0,
                       impart(mmm) + repart(nnn)

       infinity)

fps(sqrt(2-x^2),x);


            2*k
         - x   *sqrt(2)*factorial(2*k)
infsum(--------------------------------,k,0,infinity)
           k             2
          8 *factorial(k) *(2*k - 1)

fps(ci x,x);


                   2*k       k
                  x   *( - 1) *infinity*x
ci(0) + infsum(------------------------------,k,0,infinity)
                factorial(2*k + 1)*(2*k + 1)

fps(log(1-2*x*y+x^2),x);


                     2
fps(log(1 - 2*x*y + x ),x,0)


FPS(sin x,x,pi);


                   2*k       k
        ( - pi + x)   *( - 1) *( - pi + x)
infsum(------------------------------------,k,0,infinity)
                factorial(2*k + 1)


% This one takes ages :

%fps(acos(cos(x)),x);

fps_search_depth := 7;


fps_search_depth := 7
 % does not find aa DE with the default
fps(sin(x^(1/3)),x);


            2*k       k    k                         k
infsum(( - x   *( - 1) *108 *factorial(k)*x)/(6*46656 *factorial(3*k + 1)

                                          7                 5
          *factorial(2*k + 1)*pochhammer(---,k)*pochhammer(---,k)),k,0,infinity)
                                          6                 6

            (6*k + 2)/3       k  2*k  3*k                              k
 + infsum((x           *( - 1) *2   *3   *factorial(k + 1)*x)/(20*46656

                                                                11
             *factorial(3*k + 3)*factorial(2*k + 1)*pochhammer(----,k)
                                                                6

                          7
             *pochhammer(---,k)),k,0,infinity) + infsum(
                          6

                   (6*k + 1)/3       k  2*k  3*k
                  x           *( - 1) *2   *3   *factorial(k)
   --------------------------------------------------------------------------,k,
         k                                           7                 5
    46656 *factorial(3*k)*factorial(2*k)*pochhammer(---,k)*pochhammer(---,k)
                                                     6                 6

   0,infinity)


end;
(TIME:  fps 165079 183049)