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Related

assoc_legendre_q-imagpart-legendre_q-plot2d-sqrt
plot2d(imagpart(f(y)),[x,0.8,1.0]);

Alpha:1;

Fr:sqrt(2);

y:-%i*((4.0*(Alpha+1)...

Calculate

assoc_legendre_q-diff-expand-legendre_q

f(x):= x+1;

f(1);

g(x):= ((x^2-1)^(-1/2...

Calculate

assoc_legendre_q-imagpart-legendre_q-numer-sqrt

Alpha:1;

Fr:sqrt(2);

y:-%i*((4.0*(Alpha+1)...

Calculate

assoc_legendre_q-diff-float-imagpart-legendre_q-numer-plot2d-realpart
plot2d(cre*realpart(F1(x0*y))+ F2(x0*y),[y,0.0,0.99]);

x0:-%i*0.577;

F1(x):=(1-x*x)^(-0.5)...

DiffF1(x):= ''(diff(F...

Calculate

assoc_legendre_q-diff-expand-legendre_q

f(x):= x+1;

expand(g(1));

g(x):= ((x^2-1)^(-1/2...

Calculate

assoc_legendre_q-legendre_q-plot2d-realpart-sqrt
plot2d(realpart(f(y)),[x,0.2,0.5]);

Alpha:1;

Fr:sqrt(2);

y:-%i*((4.0*(Alpha+1)...

Calculate

assoc_legendre_q-legendre_q-plot2d-realpart-sqrt
plot2d(realpart(f(y)),[x,0.1,1]);

Alpha:1;

Fr:sqrt(2);

y:-%i*((4.0*(Alpha+1)...

Calculate

assoc_legendre_q

Run Example
(%i1)a:-%i/(3^(0.5));
(%o1)                        - 0.57735026918963 %i
(%i2) a:%ig(x):= ((x^2-1)^(-1/2))*assoc_legendre_q(8,-1,x);
                                 - 1
                                 ---
                           2      2
(%o2)          %ig(x) := (x  - 1)    assoc_legendre_q(8, - 1, x)
(%i3) g(x):= legendre_q(5,x);
(%o3)                      g(x) := legendre_q(5, x)
(%i4) expand(g(a));
                             - 1
                             ---
                       2      2     - 1    5
(%o4) (63 (%ig(x) := (x  - 1)    "Q"   (x))
                                    8
                            - 1
                            ---
                      2      2     - 1
          %ig(x) := (x  - 1)    "Q"   (x)
                                   8
 log(- -------------------------------------
                          - 1
                          ---
                    2      2     - 1
       (%ig(x) := (x  - 1)    "Q"   (x)) - 1
                                 8
                     1
 - -------------------------------------))/16
                      - 1
                      ---
                2      2     - 1
   (%ig(x) := (x  - 1)    "Q"   (x)) - 1
                             8
                          - 1
                          ---
                    2      2     - 1    3
 - (35 (%ig(x) := (x  - 1)    "Q"   (x))
                                 8
                            - 1
                            ---
                      2      2     - 1
          %ig(x) := (x  - 1)    "Q"   (x)
                                   8
 log(- -------------------------------------
                          - 1
                          ---
                    2      2     - 1
       (%ig(x) := (x  - 1)    "Q"   (x)) - 1
                                 8
                     1
 - -------------------------------------))/8
                      - 1
                      ---
                2      2     - 1
   (%ig(x) := (x  - 1)    "Q"   (x)) - 1
                             8
                          - 1
                          ---
                    2      2     - 1
 + (15 (%ig(x) := (x  - 1)    "Q"   (x))
                                 8
                            - 1
                            ---
                      2      2     - 1
          %ig(x) := (x  - 1)    "Q"   (x)
                                   8
 log(- -------------------------------------
                          - 1
                          ---
                    2      2     - 1
       (%ig(x) := (x  - 1)    "Q"   (x)) - 1
                                 8
                     1
 - -------------------------------------))/16
                      - 1
                      ---
                2      2     - 1
   (%ig(x) := (x  - 1)    "Q"   (x)) - 1
                             8
                         - 1
                         ---
                   2      2     - 1    4
   63 (%ig(x) := (x  - 1)    "Q"   (x))
                                8
 - -------------------------------------
                     8
                         - 1
                         ---
                   2      2     - 1    2
   49 (%ig(x) := (x  - 1)    "Q"   (x))
                                8          8
 + ------------------------------------- - --
                     8                     15
(%i5) g1(x):=''(diff(g(x),x));
                             x + 1   6        5             x + 1   4        3
(%o5)/R/ g1(x) := (315 log(- -----) x  - 630 x  - 525 log(- -----) x  + 840 x
                             x - 1                          x - 1
                              x + 1   2                    x + 1        2
                  + 225 log(- -----) x  - 226 x - 15 log(- -----))/(16 x  - 16)
                              x - 1                        x - 1
(%i6) expand(g1(a));
                              - 1
                              ---
                        2      2     - 1    6
(%o6) (315 (%ig(x) := (x  - 1)    "Q"   (x))
                                     8
                            - 1
                            ---
                      2      2     - 1
          %ig(x) := (x  - 1)    "Q"   (x)
                                   8
 log(- -------------------------------------
                          - 1
                          ---
                    2      2     - 1
       (%ig(x) := (x  - 1)    "Q"   (x)) - 1
                                 8
                     1
 - -------------------------------------))
                      - 1
                      ---
                2      2     - 1
   (%ig(x) := (x  - 1)    "Q"   (x)) - 1
                             8
                        - 1
                        ---
                  2      2     - 1    2
/(16 (%ig(x) := (x  - 1)    "Q"   (x))  - 16)
                               8
                           - 1
                           ---
                     2      2     - 1    4
 - (525 (%ig(x) := (x  - 1)    "Q"   (x))
                                  8
                            - 1
                            ---
                      2      2     - 1
          %ig(x) := (x  - 1)    "Q"   (x)
                                   8
 log(- -------------------------------------
                          - 1
                          ---
                    2      2     - 1
       (%ig(x) := (x  - 1)    "Q"   (x)) - 1
                                 8
                     1
 - -------------------------------------))
                      - 1
                      ---
                2      2     - 1
   (%ig(x) := (x  - 1)    "Q"   (x)) - 1
                             8
                        - 1
                        ---
                  2      2     - 1    2
/(16 (%ig(x) := (x  - 1)    "Q"   (x))  - 16)
                               8
                           - 1
                           ---
                     2      2     - 1    2
 + (225 (%ig(x) := (x  - 1)    "Q"   (x))
                                  8
                            - 1
                            ---
                      2      2     - 1
          %ig(x) := (x  - 1)    "Q"   (x)
                                   8
 log(- -------------------------------------
                          - 1
                          ---
                    2      2     - 1
       (%ig(x) := (x  - 1)    "Q"   (x)) - 1
                                 8
                     1
 - -------------------------------------))
                      - 1
                      ---
                2      2     - 1
   (%ig(x) := (x  - 1)    "Q"   (x)) - 1
                             8
                        - 1
                        ---
                  2      2     - 1    2
/(16 (%ig(x) := (x  - 1)    "Q"   (x))  - 16)
                               8
                                  - 1
                                  ---
                            2      2     - 1
                %ig(x) := (x  - 1)    "Q"   (x)
                                         8
 - (15 log(- -------------------------------------
                                - 1
                                ---
                          2      2     - 1
             (%ig(x) := (x  - 1)    "Q"   (x)) - 1
                                       8
                     1
 - -------------------------------------))
                      - 1
                      ---
                2      2     - 1
   (%ig(x) := (x  - 1)    "Q"   (x)) - 1
                             8
                        - 1
                        ---
                  2      2     - 1    2
/(16 (%ig(x) := (x  - 1)    "Q"   (x))  - 16)
                               8
                            - 1
                            ---
                      2      2     - 1    5
     630 (%ig(x) := (x  - 1)    "Q"   (x))
                                   8
 - ------------------------------------------
                         - 1
                         ---
                   2      2     - 1    2
   16 (%ig(x) := (x  - 1)    "Q"   (x))  - 16
                                8
                            - 1
                            ---
                      2      2     - 1    3
     840 (%ig(x) := (x  - 1)    "Q"   (x))
                                   8
 + ------------------------------------------
                         - 1
                         ---
                   2      2     - 1    2
   16 (%ig(x) := (x  - 1)    "Q"   (x))  - 16
                                8
                            - 1
                            ---
                      2      2     - 1
      226 %ig(x) := (x  - 1)    "Q"   (x)
                                   8
 - ------------------------------------------
                         - 1
                         ---
                   2      2     - 1    2
   16 (%ig(x) := (x  - 1)    "Q"   (x))  - 16
                                8
(%i7) A1:(4.0)/(7.0);
(%o7)                          0.57142857142857
(%i8) B:(125*25)/(56*21);
                                     3125
(%o8)                                ----
                                     1176
(%i9) E:(125)/(1512);
                                     125
(%o9)                                ----
                                     1512
(%i10) F:(-25)/(196);
                                       25
(%o10)                               - ---
                                       196
(%i11) Exp1:B+a*A1*E*2.0*a/((1-a*a)^2)-F-E/(1-a*a);
                              125
(%o11) - ---------------------------------------------
                                      - 1
                                      ---
                                2      2     - 1    2
         1512 (1 - (%ig(x) := (x  - 1)    "Q"   (x)) )
                                             8
                                                         - 1
                                                         ---
                                                   2      2     - 1    2
                     0.09448223733938 (%ig(x) := (x  - 1)    "Q"   (x))
                                                                8          3275
                   + --------------------------------------------------- + ----
                                                  - 1                      1176
                                                  ---
                                            2      2     - 1    2 2
                          (1 - (%ig(x) := (x  - 1)    "Q"   (x)) )
                                                         8
(%i12) Exp1:9/200+27*3/50;
                                      333
(%o12)                                ---
                                      200
(%i13) Exp2: g(a)-a*A1*g1(a);

rat: replaced 0.57142857142857 by 4/7 = 0.57142857142857
                                            - 1
                                            ---
                                      2      2     - 1
                         (%ig(x) := (x  - 1)    "Q"   (x)) + 1
                                                   8
(%o13)/R/ - (12285 log(- -------------------------------------)
                                            - 1
                                            ---
                                      2      2     - 1
                         (%ig(x) := (x  - 1)    "Q"   (x)) - 1
                                                   8
                    - 1                                        - 1
                    ---                                        ---
              2      2     - 1    7                      2      2     - 1    6
 (%ig(x) := (x  - 1)    "Q"   (x))  - 24570 (%ig(x) := (x  - 1)    "Q"   (x))
                           8                                          8
                                  - 1
                                  ---
                            2      2     - 1
               (%ig(x) := (x  - 1)    "Q"   (x)) + 1
                                         8
 - 17535 log(- -------------------------------------)
                                  - 1
                                  ---
                            2      2     - 1
               (%ig(x) := (x  - 1)    "Q"   (x)) - 1
                                         8
                    - 1                                        - 1
                    ---                                        ---
              2      2     - 1    5                      2      2     - 1    4
 (%ig(x) := (x  - 1)    "Q"   (x))  + 26880 (%ig(x) := (x  - 1)    "Q"   (x))
                           8                                          8
                                 - 1
                                 ---
                           2      2     - 1
              (%ig(x) := (x  - 1)    "Q"   (x)) + 1
                                        8
 + 4575 log(- -------------------------------------)
                                 - 1
                                 ---
                           2      2     - 1
              (%ig(x) := (x  - 1)    "Q"   (x)) - 1
                                        8
                    - 1                                       - 1
                    ---                                       ---
              2      2     - 1    3                     2      2     - 1    2
 (%ig(x) := (x  - 1)    "Q"   (x))  - 2374 (%ig(x) := (x  - 1)    "Q"   (x))
                           8                                         8
                                - 1
                                ---
                          2      2     - 1
             (%ig(x) := (x  - 1)    "Q"   (x)) + 1
                                       8
 + 675 log(- -------------------------------------)
                                - 1
                                ---
                          2      2     - 1
             (%ig(x) := (x  - 1)    "Q"   (x)) - 1
                                       8
                    - 1
                    ---
              2      2     - 1
 (%ig(x) := (x  - 1)    "Q"   (x)) - 896)
                           8
                          - 1
                          ---
                    2      2     - 1    2
/(1680 (%ig(x) := (x  - 1)    "Q"   (x))  - 1680)
                                 8
(%i14) Exp2: g(a)+ a*(12/30)*g1(a);
                                         - 1
                                         ---
                                   2      2     - 1
                      (%ig(x) := (x  - 1)    "Q"   (x)) + 1
                                                8
(%o14)/R/ (2835 log(- -------------------------------------)
                                         - 1
                                         ---
                                   2      2     - 1
                      (%ig(x) := (x  - 1)    "Q"   (x)) - 1
                                                8
                    - 1                                       - 1
                    ---                                       ---
              2      2     - 1    7                     2      2     - 1    6
 (%ig(x) := (x  - 1)    "Q"   (x))  - 5670 (%ig(x) := (x  - 1)    "Q"   (x))
                           8                                         8
                                 - 1
                                 ---
                           2      2     - 1
              (%ig(x) := (x  - 1)    "Q"   (x)) + 1
                                        8
 - 5145 log(- -------------------------------------)
                                 - 1
                                 ---
                           2      2     - 1
              (%ig(x) := (x  - 1)    "Q"   (x)) - 1
                                        8
                    - 1                                       - 1
                    ---                                       ---
              2      2     - 1    5                     2      2     - 1    4
 (%ig(x) := (x  - 1)    "Q"   (x))  + 8400 (%ig(x) := (x  - 1)    "Q"   (x))
                           8                                         8
                                 - 1
                                 ---
                           2      2     - 1
              (%ig(x) := (x  - 1)    "Q"   (x)) + 1
                                        8
 + 2625 log(- -------------------------------------)
                                 - 1
                                 ---
                           2      2     - 1
              (%ig(x) := (x  - 1)    "Q"   (x)) - 1
                                        8
                    - 1                                       - 1
                    ---                                       ---
              2      2     - 1    3                     2      2     - 1    2
 (%ig(x) := (x  - 1)    "Q"   (x))  - 2954 (%ig(x) := (x  - 1)    "Q"   (x))
                           8                                         8
                                - 1
                                ---
                          2      2     - 1
             (%ig(x) := (x  - 1)    "Q"   (x)) + 1
                                       8
 - 315 log(- -------------------------------------)
                                - 1
                                ---
                          2      2     - 1
             (%ig(x) := (x  - 1)    "Q"   (x)) - 1
                                       8
                    - 1
                    ---
              2      2     - 1
 (%ig(x) := (x  - 1)    "Q"   (x)) + 128)
                           8
                         - 1
                         ---
                   2      2     - 1    2
/(240 (%ig(x) := (x  - 1)    "Q"   (x))  - 240)
                                8
(%i15) C2:expand(Exp1/Exp2);
                                - 1
                                ---
                          2      2     - 1    2
(%o15) (1998 (%ig(x) := (x  - 1)    "Q"   (x)) )
                                       8
                           - 1
                           ---
                     2      2     - 1    7
/(14175 (%ig(x) := (x  - 1)    "Q"   (x))
                                  8
                            - 1
                            ---
                      2      2     - 1
          %ig(x) := (x  - 1)    "Q"   (x)
                                   8
 log(- -------------------------------------
                          - 1
                          ---
                    2      2     - 1
       (%ig(x) := (x  - 1)    "Q"   (x)) - 1
                                 8
                     1
 - -------------------------------------)
                      - 1
                      ---
                2      2     - 1
   (%ig(x) := (x  - 1)    "Q"   (x)) - 1
                             8
                            - 1
                            ---
                      2      2     - 1    5
 - 25725 (%ig(x) := (x  - 1)    "Q"   (x))
                                   8
                            - 1
                            ---
                      2      2     - 1
          %ig(x) := (x  - 1)    "Q"   (x)
                                   8
 log(- -------------------------------------
                          - 1
                          ---
                    2      2     - 1
       (%ig(x) := (x  - 1)    "Q"   (x)) - 1
                                 8
                     1
 - -------------------------------------)
                      - 1
                      ---
                2      2     - 1
   (%ig(x) := (x  - 1)    "Q"   (x)) - 1
                             8
                            - 1
                            ---
                      2      2     - 1    3
 + 13125 (%ig(x) := (x  - 1)    "Q"   (x))
                                   8
                            - 1
                            ---
                      2      2     - 1
          %ig(x) := (x  - 1)    "Q"   (x)
                                   8
 log(- -------------------------------------
                          - 1
                          ---
                    2      2     - 1
       (%ig(x) := (x  - 1)    "Q"   (x)) - 1
                                 8
                     1
 - -------------------------------------)
                      - 1
                      ---
                2      2     - 1
   (%ig(x) := (x  - 1)    "Q"   (x)) - 1
                             8
                           - 1
                           ---
                     2      2     - 1
 - 1575 (%ig(x) := (x  - 1)    "Q"   (x))
                                  8
                            - 1
                            ---
                      2      2     - 1
          %ig(x) := (x  - 1)    "Q"   (x)
                                   8
 log(- -------------------------------------
                          - 1
                          ---
                    2      2     - 1
       (%ig(x) := (x  - 1)    "Q"   (x)) - 1
                                 8
                     1
 - -------------------------------------)
                      - 1
                      ---
                2      2     - 1
   (%ig(x) := (x  - 1)    "Q"   (x)) - 1
                             8
                            - 1
                            ---
                      2      2     - 1    6
 - 28350 (%ig(x) := (x  - 1)    "Q"   (x))
                                   8
                            - 1
                            ---
                      2      2     - 1    4
 + 42000 (%ig(x) := (x  - 1)    "Q"   (x))
                                   8
                            - 1
                            ---
                      2      2     - 1    2
 - 14770 (%ig(x) := (x  - 1)    "Q"   (x))  + 640)
                                   8
                                  - 1
                                  ---
                            2      2     - 1    7
 - 1998/(14175 (%ig(x) := (x  - 1)    "Q"   (x))
                                         8
                            - 1
                            ---
                      2      2     - 1
          %ig(x) := (x  - 1)    "Q"   (x)
                                   8
 log(- -------------------------------------
                          - 1
                          ---
                    2      2     - 1
       (%ig(x) := (x  - 1)    "Q"   (x)) - 1
                                 8
                     1
 - -------------------------------------)
                      - 1
                      ---
                2      2     - 1
   (%ig(x) := (x  - 1)    "Q"   (x)) - 1
                             8
                            - 1
                            ---
                      2      2     - 1    5
 - 25725 (%ig(x) := (x  - 1)    "Q"   (x))
                                   8
                            - 1
                            ---
                      2      2     - 1
          %ig(x) := (x  - 1)    "Q"   (x)
                                   8
 log(- -------------------------------------
                          - 1
                          ---
                    2      2     - 1
       (%ig(x) := (x  - 1)    "Q"   (x)) - 1
                                 8
                     1
 - -------------------------------------)
                      - 1
                      ---
                2      2     - 1
   (%ig(x) := (x  - 1)    "Q"   (x)) - 1
                             8
                            - 1
                            ---
                      2      2     - 1    3
 + 13125 (%ig(x) := (x  - 1)    "Q"   (x))
                                   8
                            - 1
                            ---
                      2      2     - 1
          %ig(x) := (x  - 1)    "Q"   (x)
                                   8
 log(- -------------------------------------
                          - 1
                          ---
                    2      2     - 1
       (%ig(x) := (x  - 1)    "Q"   (x)) - 1
                                 8
                     1
 - -------------------------------------)
                      - 1
                      ---
                2      2     - 1
   (%ig(x) := (x  - 1)    "Q"   (x)) - 1
                             8
                           - 1
                           ---
                     2      2     - 1
 - 1575 (%ig(x) := (x  - 1)    "Q"   (x))
                                  8
                            - 1
                            ---
                      2      2     - 1
          %ig(x) := (x  - 1)    "Q"   (x)
                                   8
 log(- -------------------------------------
                          - 1
                          ---
                    2      2     - 1
       (%ig(x) := (x  - 1)    "Q"   (x)) - 1
                                 8
                     1
 - -------------------------------------)
                      - 1
                      ---
                2      2     - 1
   (%ig(x) := (x  - 1)    "Q"   (x)) - 1
                             8
                            - 1
                            ---
                      2      2     - 1    6
 - 28350 (%ig(x) := (x  - 1)    "Q"   (x))
                                   8
                            - 1
                            ---
                      2      2     - 1    4
 + 42000 (%ig(x) := (x  - 1)    "Q"   (x))
                                   8
                            - 1
                            ---
                      2      2     - 1    2
 - 14770 (%ig(x) := (x  - 1)    "Q"   (x))  + 640)
                                   8
(%i16) 
Run Example
f(x):= x+1;
(%o1)                            f(x) := x + 1
(%i2) g(1);
(%o2)                                g(1)
(%i3) g(x):= ((x^2-1)^(-1/2))*assoc_legendre_q(8,-1,x);
                                - 1
                                ---
                          2      2
(%o3)           g(x) := (x  - 1)    assoc_legendre_q(8, - 1, x)
(%i4) expand(g(x1));
                 9            2          x1       1
        225225 x1  sqrt(1 - x1 ) log(- ------ - ------)
                                       x1 - 1   x1 - 1
(%o4) - -----------------------------------------------
                 2        2                     2
         80640 x1  sqrt(x1  - 1) - 80640 sqrt(x1  - 1)
            7            2          x1       1
   540540 x1  sqrt(1 - x1 ) log(- ------ - ------)
                                  x1 - 1   x1 - 1
 + -----------------------------------------------
            2        2                     2
    80640 x1  sqrt(x1  - 1) - 80640 sqrt(x1  - 1)
            5            2          x1       1
   436590 x1  sqrt(1 - x1 ) log(- ------ - ------)
                                  x1 - 1   x1 - 1
 - -----------------------------------------------
            2        2                     2
    80640 x1  sqrt(x1  - 1) - 80640 sqrt(x1  - 1)
            3            2          x1       1
   132300 x1  sqrt(1 - x1 ) log(- ------ - ------)
                                  x1 - 1   x1 - 1
 + -----------------------------------------------
            2        2                     2
    80640 x1  sqrt(x1  - 1) - 80640 sqrt(x1  - 1)
                       2          x1       1
   11025 x1 sqrt(1 - x1 ) log(- ------ - ------)
                                x1 - 1   x1 - 1
 - ---------------------------------------------
           2        2                     2
   80640 x1  sqrt(x1  - 1) - 80640 sqrt(x1  - 1)
                      8            2
             450450 x1  sqrt(1 - x1 )
 + ---------------------------------------------
           2        2                     2
   80640 x1  sqrt(x1  - 1) - 80640 sqrt(x1  - 1)
                      6            2
             930930 x1  sqrt(1 - x1 )
 - ---------------------------------------------
           2        2                     2
   80640 x1  sqrt(x1  - 1) - 80640 sqrt(x1  - 1)
                      4            2
             602910 x1  sqrt(1 - x1 )
 + ---------------------------------------------
           2        2                     2
   80640 x1  sqrt(x1  - 1) - 80640 sqrt(x1  - 1)
                      2            2
             125406 x1  sqrt(1 - x1 )
 - ---------------------------------------------
           2        2                     2
   80640 x1  sqrt(x1  - 1) - 80640 sqrt(x1  - 1)
                                2
                4096 sqrt(1 - x1 )
 + ---------------------------------------------
           2        2                     2
   80640 x1  sqrt(x1  - 1) - 80640 sqrt(x1  - 1)
(%i5) g1(x):=''(diff(g(x),x));
                                 x + 1            2 2
(%o5)/R/ g1(x) := ((225225 log(- -----) sqrt(1 - x )
                                 x - 1
                2     2       x + 1    12
 + 225225 sqrt(x  - 1)  log(- -----)) x
                              x - 1
                       2 2                2     2   11
 + (- 450450 sqrt(1 - x )  - 450450 sqrt(x  - 1) ) x
                     2     2                 x + 1            2 2
 + ((- 1576575 sqrt(x  - 1)  - 765765) log(- -----) sqrt(1 - x )
                                             x - 1
                2     2       x + 1    10
 - 765765 sqrt(x  - 1)  log(- -----)) x
                              x - 1
                   2     2                      2 2                 2     2   9
 + ((3153150 sqrt(x  - 1)  + 1381380) sqrt(1 - x )  + 1381380 sqrt(x  - 1) ) x
                   2     2                 x + 1            2 2
 + ((4729725 sqrt(x  - 1)  + 977130) log(- -----) sqrt(1 - x )
                                           x - 1
                2     2       x + 1    8
 + 977130 sqrt(x  - 1)  log(- -----)) x
                              x - 1
                     2     2                      2 2                 2     2
 + ((- 8408400 sqrt(x  - 1)  - 1533840) sqrt(1 - x )  - 1533840 sqrt(x  - 1) )
  7                     2     2                 x + 1            2 2
 x  + ((- 5093550 sqrt(x  - 1)  - 568890) log(- -----) sqrt(1 - x )
                                                x - 1
                2     2       x + 1    6
 - 568890 sqrt(x  - 1)  log(- -----)) x
                              x - 1
                   2     2                     2 2                2     2   5
 + ((7664580 sqrt(x  - 1)  + 728316) sqrt(1 - x )  + 728316 sqrt(x  - 1) ) x
                   2     2                 x + 1            2 2
 + ((2315250 sqrt(x  - 1)  + 143325) log(- -----) sqrt(1 - x )
                                           x - 1
                2     2       x + 1    4
 + 143325 sqrt(x  - 1)  log(- -----)) x
                              x - 1
                     2     2                     2 2                2     2   3
 + ((- 2676240 sqrt(x  - 1)  - 129502) sqrt(1 - x )  - 129502 sqrt(x  - 1) ) x
                    2     2                x + 1            2 2
 + ((- 385875 sqrt(x  - 1)  - 11025) log(- -----) sqrt(1 - x )
                                           x - 1
               2     2       x + 1    2
 - 11025 sqrt(x  - 1)  log(- -----)) x
                             x - 1
                  2     2                   2 2              2     2
 + ((264670 sqrt(x  - 1)  + 4096) sqrt(1 - x )  + 4096 sqrt(x  - 1) ) x
               2     2       x + 1            2 2
 + 11025 sqrt(x  - 1)  log(- -----) sqrt(1 - x ) )
                             x - 1
              2     3           2   4                2     3           2   2
/(80640 sqrt(x  - 1)  sqrt(1 - x ) x  - 161280 sqrt(x  - 1)  sqrt(1 - x ) x
               2     3           2
 + 80640 sqrt(x  - 1)  sqrt(1 - x ))
(%i6) expand(g1(-%i));
                                   %i         1
                     175 %i log(-------- - --------)
                                - %i - 1   - %i - 1    1237
(%o6)              - ------------------------------- - ----
                                    4                   18
(%i7) 
Run Example
Fr:sqrt(2);
(%o1)                               sqrt(2)
(%i2) Alpha:1;
(%o2)                                  1
(%i3) eta:1;
(%o3)                                  1
(%i4) y:-%i*((4.0*(Alpha+1)/(Fr*Fr))-1)^(-0.5);
(%o4)                        - 0.57735026918963 %i
(%i5) f(x):= (x^2-1)^(-Alpha/2)*assoc_legendre_q(8,-1,x);
                              - Alpha
                              -------
                        2        2
(%o5)         f(x) := (x  - 1)        assoc_legendre_q(8, - 1, x)
(%i6) f1(x):=''(diff(f(x),x));
                                 x + 1            2 2
(%o6)/R/ f1(x) := ((225225 log(- -----) sqrt(1 - x )
                                 x - 1
                2     2       x + 1    12
 + 225225 sqrt(x  - 1)  log(- -----)) x
                              x - 1
                       2 2                2     2   11
 + (- 450450 sqrt(1 - x )  - 450450 sqrt(x  - 1) ) x
                     2     2                 x + 1            2 2
 + ((- 1576575 sqrt(x  - 1)  - 765765) log(- -----) sqrt(1 - x )
                                             x - 1
                2     2       x + 1    10
 - 765765 sqrt(x  - 1)  log(- -----)) x
                              x - 1
                   2     2                      2 2                 2     2   9
 + ((3153150 sqrt(x  - 1)  + 1381380) sqrt(1 - x )  + 1381380 sqrt(x  - 1) ) x
                   2     2                 x + 1            2 2
 + ((4729725 sqrt(x  - 1)  + 977130) log(- -----) sqrt(1 - x )
                                           x - 1
                2     2       x + 1    8
 + 977130 sqrt(x  - 1)  log(- -----)) x
                              x - 1
                     2     2                      2 2                 2     2
 + ((- 8408400 sqrt(x  - 1)  - 1533840) sqrt(1 - x )  - 1533840 sqrt(x  - 1) )
  7                     2     2                 x + 1            2 2
 x  + ((- 5093550 sqrt(x  - 1)  - 568890) log(- -----) sqrt(1 - x )
                                                x - 1
                2     2       x + 1    6
 - 568890 sqrt(x  - 1)  log(- -----)) x
                              x - 1
                   2     2                     2 2                2     2   5
 + ((7664580 sqrt(x  - 1)  + 728316) sqrt(1 - x )  + 728316 sqrt(x  - 1) ) x
                   2     2                 x + 1            2 2
 + ((2315250 sqrt(x  - 1)  + 143325) log(- -----) sqrt(1 - x )
                                           x - 1
                2     2       x + 1    4
 + 143325 sqrt(x  - 1)  log(- -----)) x
                              x - 1
                     2     2                     2 2                2     2   3
 + ((- 2676240 sqrt(x  - 1)  - 129502) sqrt(1 - x )  - 129502 sqrt(x  - 1) ) x
                    2     2                x + 1            2 2
 + ((- 385875 sqrt(x  - 1)  - 11025) log(- -----) sqrt(1 - x )
                                           x - 1
               2     2       x + 1    2
 - 11025 sqrt(x  - 1)  log(- -----)) x
                             x - 1
                  2     2                   2 2              2     2
 + ((264670 sqrt(x  - 1)  + 4096) sqrt(1 - x )  + 4096 sqrt(x  - 1) ) x
               2     2       x + 1            2 2
 + 11025 sqrt(x  - 1)  log(- -----) sqrt(1 - x ) )
                             x - 1
              2     3           2   4                2     3           2   2
/(80640 sqrt(x  - 1)  sqrt(1 - x ) x  - 161280 sqrt(x  - 1)  sqrt(1 - x ) x
               2     3           2
 + 80640 sqrt(x  - 1)  sqrt(1 - x ))
(%i7) f(y);

rat: replaced -0.5773502691896 by -2911/5042 = -0.5773502578342

rat: replaced -0.8660254037844 by -2521/2911 = -0.8660254208176
                                                        2911 %i - 5042
(%o7)/R/ - (3380620685652680218917209899760877975 log(- --------------)
                                                        2911 %i + 5042
                  10
 sqrt(33895685) %i   + 11710813807668027250965697227478080900 sqrt(33895685)
   9                                                2911 %i - 5042
 %i  - 24340469894165108924508022846146815760 log(- --------------)
                                                    2911 %i + 5042
                  8                                                           7
 sqrt(33895685) %i  - 72607048463639735392924405261621428240 sqrt(33895685) %i
                                                2911 %i - 5042
 + 58978833217413474998881283623595108640 log(- --------------) sqrt(33895685)
                                                2911 %i + 5042
   6                                                            5
 %i  + 141070275889217638962006568334802793920 sqrt(33895685) %i
                                                2911 %i - 5042
 - 53617123215842699667191033285360467200 log(- --------------) sqrt(33895685)
                                                2911 %i + 5042
   4                                                           3
 %i  - 88028361019382779306595208019746555648 sqrt(33895685) %i
                                                2911 %i - 5042
 + 13404281331236489893149906549523910400 log(- --------------) sqrt(33895685)
                                                2911 %i + 5042
   2
 %i  + 8625524638023996517736179394844557312 sqrt(33895685) %i)
                                               2
/(329554328690930458480124574262598065520640 %i
 - 988663024963209245860744467349204936949760)
(%i8) f1(y);

rat: replaced 3.9236886160714256E-6 by 9/2293760 = 3.9236886160714282E-6

rat: replaced 2420991.336787477 by 7262974/3 = 2420991.333333334

rat: replaced -1156319.34156379 by -3468958/3 = -1156319.33333333
                             1 - 0.57735026918963 %i                2
           5203437 %i log(- -------------------------) - 10894461 %i
                            - 0.57735026918963 %i - 1
(%o8)/R/ - ----------------------------------------------------------
                                    1146880
(%i9) a1:realpart(f(y));

rat: replaced -0.5773502691896 by -2911/5042 = -0.5773502578342

rat: replaced -0.8660254037844 by -2521/2911 = -0.8660254208176
(%o9)                                  0
(%i10) a1,numer;
(%o10)                                 0
(%i11) a2:imagpart(f(y));

rat: replaced -0.5773502691896 by -2911/5042 = -0.5773502578342

rat: replaced -0.8660254037844 by -2521/2911 = -0.8660254208176
                                                3/2      2911
(%o11) (9070259435341722918574883865270 33895685    atan(----)
                                                         5042
 + 322042023817932177430228058238493416020 sqrt(33895685))
/1318217353654139704340869041611803002470400
(%i12) a2,numer;
(%o12)                         2.133285863830809
(%i13) b1:imagpart(f1(y));

rat: replaced 3.9236886160714256E-6 by 9/2293760 = 3.9236886160714282E-6

rat: replaced 2420991.336787477 by 7262974/3 = 2420991.333333334

rat: replaced -1156319.34156379 by -3468958/3 = -1156319.33333333
(%o13)                                0.0
(%i14) b1,numer;
(%o14)                                0.0
(%i15) b2:realpart(f1(y));

rat: replaced 3.9236886160714256E-6 by 9/2293760 = 3.9236886160714282E-6

rat: replaced 2420991.336787477 by 7262974/3 = 2420991.333333334

rat: replaced -1156319.34156379 by -3468958/3 = -1156319.33333333
               5203437 (- %pi - 2.094395102393195) + 1.0894461E+7
(%o15)       - --------------------------------------------------
                                    1146880
(%i16) b2,numer;
(%o16)                         14.25665407107008
(%i17) A:(14+8*Alpha-Fr^2)*(Alpha+1)/((5+2*Alpha)*(3+2*Alpha)*Fr^2);
                                       4
(%o17)                                 -
                                       7
(%i18) B:((3+2*Alpha)^3)/(4*(5+2*Alpha)*(1+Alpha))* (1+0.5*(14+8*Alpha-Fr^2)*(Alpha+1)/( (3+2*Alpha)*(5+2*Alpha)*(Alpha+2))  );
(%o18)                         2.657312925170068
(%i19) Hp(x):=125/1512 * 1.0/(1-x^2) - 25/196;
                                     125
                                     ---- 1.0
                                     1512       25
(%o19)                      Hp(x) := -------- - ---
                                           2    196
                                      1 - x
(%i20) Hp1(x):=''(diff(Hp(x),x));
                                   0.16534391534392 x
(%o20)                   Hp1(x) := ------------------
                                             2 2
                                       (1 - x )
(%i21) c: (B-Hp(y)-A*Hp1(y)*y)/(f(y)+A*f1(y)*y);

rat: replaced -0.5773502691896 by -2911/5042 = -0.5773502578342

rat: replaced -0.8660254037844 by -2521/2911 = -0.8660254208176

rat: replaced 3.9236886160714256E-6 by 9/2293760 = 3.9236886160714282E-6

rat: replaced 2420991.336787477 by 7262974/3 = 2420991.333333334

rat: replaced -1156319.34156379 by -3468958/3 = -1156319.33333333

rat: replaced -0.5773502691896 by -2911/5042 = -0.5773502578342

rat: replaced 2.740575396825397 by 5525/2016 = 2.740575396825397
                                                           2
(%o21)/R/ - (6322179395893717996884334280558521916672000 %i
 - 18966538933756010706182684660084573877248000)
                                               2911 %i - 5042
/(23664344799568761532420469298326145825 log(- --------------) sqrt(33895685)
                                               2911 %i + 5042
   10                                                           9
 %i   + 81975696653676190756759880592346566300 sqrt(33895685) %i
                                                 2911 %i - 5042
 - 170383289259155762471556159923027710320 log(- --------------) sqrt(33895685)
                                                 2911 %i + 5042
   8                                                            7
 %i  - 508249339245478147750470836831349997680 sqrt(33895685) %i
                                                 2911 %i - 5042
 + 412851832521894324992168985365165760480 log(- --------------) sqrt(33895685)
                                                 2911 %i + 5042
   6
 %i  + (987491931224523472734045978343619557440 sqrt(33895685)
                                                  5
 + 7229597933149398193967400886360229467519956) %i
                                                    2911 %i - 5042
 + (- 375319862510898897670337232997523270400 log(- --------------)
                                                    2911 %i + 5042
 sqrt(33895685) - 3453016847779170083790574913795149052328852
        1 - 0.57735026918963 %i      4
 log(- -------------------------)) %i
       - 0.57735026918963 %i - 1
 + (- 616198527135679455146166456138225889536 sqrt(33895685)
                                                   3
 - 21688794652606718616926625705672801588442704) %i
                                                 2911 %i - 5042
 + (93829969318655429252049345846667372800 log(- --------------) sqrt(33895685)
                                                 2911 %i + 5042
 + 10359050950825006037462966775689863470892368
        1 - 0.57735026918963 %i      2
 log(- -------------------------)) %i  + 60378672466167975624153255763911901184
       - 0.57735026918963 %i - 1
 sqrt(33895685) %i)
(%i22) c,numer;
(%o22) 25288718329649728703067018940643095793920000
                                 2911 %i - 5042
/(- 6.2647591077563691E+42 log(- --------------)
                                 2911 %i + 5042
 - 13812067798604176121253541689485012523221220
        1 - 0.57735026918963 %i
 log(- -------------------------) + 4.2042893418033475E+43 %i)
       - 0.57735026918963 %i - 1
(%i23) 

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