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legendre_q

legendre_q (5, x);

Calculate

legendre_q

legendre_q (5, x);

Calculate

legendre_q-sqrt

legendre_q(8,-i/sqrt(...

Calculate

legendre_q-sqrt

legendre_q(8,-i/sqrt(...

Calculate

legendre_q

legendre_q (5, i);

Calculate

legendre_q

legendre_q(8,0.5);

Calculate

legendre_q

legendre_q (5, x);

Calculate

legendre_q-sqrt

legendre_q(3,sqrt(3));

Calculate

legendre_q

legendre_q(8,0.5);

Calculate

legendre_q

legendre_q (5, i);

Calculate

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
load(interpol);
(%o1)         /usr/share/maxima/5.21.1/share/numeric/interpol.mac
(%i2) alpha_1:1;
(%o2)                                  1
(%i3) Fr_1:2^0.5;
(%o3)                          1.414213562373095
(%i4) x0_1:%i*0.577;
(%o4)                              0.577 %i
(%i5) beta_1:(2*(alpha_1+1)/(3+2*alpha_1));
                                       4
(%o5)                                  -
                                       5
(%i6) A_1:(3+2*alpha_1)*(3+2*alpha_1)*(3+2*alpha_1)/((4*alpha_1+8)*(2*alpha_1+5));
                                      125
(%o6)                                 ---
                                      84
(%i7) C2_1:-0.403;
(%o7)                               - 0.403
(%i8) H1p_1(x):=(125.0/1512)/(1-x*x) - 25.0/196;
                                       125.0
                                       -----
                                       1512     25.0
(%o8)                     H1p_1(x) := ------- - ----
                                      1 - x x   196
(%i9) DiffH1p_1(x):= ''(diff(H1p_1(x), x));
                                      0.16534391534392 x
(%o9)                 DiffH1p_1(x) := ------------------
                                                2 2
                                          (1 - x )
(%i10) F1_1(x):=(1-x*x)^(-alpha_1/2)*assoc_legendre_q(8,-1,x);
                               - alpha_1
                               ---------
                                   2
(%o10)     F1_1(x) := (1 - x x)          assoc_legendre_q(8, - 1, x)
(%i11) DiffF1_1(x):= ''(diff(F1_1(x), x));
                                        x + 1   10          9
(%o11)/R/ DiffF1_1(x) := - (45045 log(- -----) x   - 90090 x
                                        x - 1
                x + 1   8           7                x + 1   6           5
 - 135135 log(- -----) x  + 240240 x  + 145530 log(- -----) x  - 218988 x
                x - 1                                x - 1
               x + 1   4          3               x + 1   2
 - 66150 log(- -----) x  + 76464 x  + 11025 log(- -----) x  - 7562 x
               x - 1                              x - 1
             x + 1          4         2
 - 315 log(- -----))/(2304 x  - 4608 x  + 2304)
             x - 1
(%i12) H1_1(x)=C2_1*F1_1(x) +H1p_1(x);

rat: replaced -0.403 by -403/1000 = -0.403

rat: replaced -0.1275510204082 by -25/196 = -0.1275510204082

rat: replaced 0.082671957671958 by 125/1512 = 0.082671957671958
                                      x + 1   9               8
(%o12)/R/ H1_1(x) = (1906079175 log(- -----) x  - 3812158350 x
                                      x - 1
                    x + 1   7               6                    x + 1   5
 - 4574590020 log(- -----) x  + 7878460590 x  + 3694861170 log(- -----) x
                    x - 1                                        x - 1
               4                    x + 1   3              2
 - 5102427330 x  - 1119654900 log(- -----) x  + 845310978 x
                                    x - 1
                  x + 1                             2
 + 93304575 log(- -----) x + 41335552)/(1693440000 x  - 1693440000)
                  x - 1
(%i13) DiffH1_1(x):= DiffF1_1(x)*C2_1+DiffH1p_1(x);
(%o13)          DiffH1_1(x) := DiffF1_1(x) C2_1 + DiffH1p_1(x)
(%i14) H0_1(x):=1.0/(Fr_1*Fr_1) - 1.0/(4*alpha_1+4) + x^2/(4*alpha_1+4);
                                                          2
                           1.0           1.0             x
(%o14)       H0_1(x) := --------- - ------------- + -------------
                        Fr_1 Fr_1   4 alpha_1 + 4   4 alpha_1 + 4
(%i15) DiffH0_1(x):= ''(diff(H0_1(x), x));
                                              x
(%o15)                         DiffH0_1(x) := -
                                              4
(%i16) B_1:(3+2*alpha_1)*(3+2*alpha_1)*(3+2*alpha_1)/((8*alpha_1+20)*(alpha_1+1));
                                      125
(%o16)                                ---
                                      56
(%i17) Fr_2:2^0.5;
(%o17)                         1.414213562373095
(%i18) U1_Al1(zeta):=realpart( (-beta_1/2)*( DiffH1_1(zeta*x0_1)*x0_1 -A_1*DiffH0_1(zeta)/H0_1(zeta)));
                                - beta_1
(%o18) U1_Al1(zeta) := realpart(--------
                                   2
                                                           A_1 DiffH0_1(zeta)
                               (DiffH1_1(zeta x0_1) x0_1 - ------------------))
                                                               H0_1(zeta)
(%i19) Phi1_Al1(y):= B_1/H0_1(y);
                                             B_1
(%o19)                      Phi1_Al1(y) := -------
                                           H0_1(y)
(%i20) Phi2_Al1(y):= -Phi1_Al1(y)/(2.0*H0_1(y))*(125.0/56.0 +  realpart(C2_1*F1_1(y*x0_1)+H1p_1(y*x0_1))) ;
                      - Phi1_Al1(y)  125.0
(%o20) Phi2_Al1(y) := ------------- (-----
                       2.0 H0_1(y)   56.0
                                 + realpart(C2_1 F1_1(y x0_1) + H1p_1(y x0_1)))
(%i21) foo_1(y):= ''(2.0*Phi1_Al1(y)*H0_1(y)*realpart(C2_1*F1_1(y*x0_1)+H1p_1(y*x0_1)));

rat: replaced 0.577 by 577/1000 = 0.577

rat: replaced -0.403 by -403/1000 = -0.403

rat: replaced -0.1275510204082 by -25/196 = -0.1275510204082

rat: replaced 0.082671957671958 by 125/1512 = 0.082671957671958

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786
                                                                             11
(%o21) foo_1(y) := (4.464285714285714 (741724899620869378121796486996078681 y
              2                                         577 y
 sqrt(332929 y  + 1000000) (atan2(577 y, - 1000) - atan(-----) - %pi)
                                                        1000
                                          9              2
 + 7574782194501581116776447858489798438 y  sqrt(332929 y  + 1000000)
                              577 y
 (atan2(577 y, - 1000) - atan(-----) - %pi)
                              1000
                                           7              2
 + 29031893253004367050799515694452800000 y  sqrt(332929 y  + 1000000)
                              577 y
 (atan2(577 y, - 1000) - atan(-----) - %pi)
                              1000
                                           5              2
 + 50769125747525708318584807200000000000 y  sqrt(332929 y  + 1000000)
                              577 y
 (atan2(577 y, - 1000) - atan(-----) - %pi)
                              1000
                                           3              2
 + 38418615447231297096000000000000000000 y  sqrt(332929 y  + 1000000)
                              577 y
 (atan2(577 y, - 1000) - atan(-----) - %pi)
                              1000
                                                        2
 + 8876601654102000000000000000000000000 y sqrt(332929 y  + 1000000)
                              577 y
 (atan2(577 y, - 1000) - atan(-----) - %pi)
                              1000
                                          10              2
 - 2570970189327103563680403767750706000 y   sqrt(332929 y  + 1000000)
                                           8              2
 - 23681651611442049468605076676349388000 y  sqrt(332929 y  + 1000000)
                                           6              2
 - 78981841855515216067805804248800000000 y  sqrt(332929 y  + 1000000)
                                                      2
 - 5715474186240000000000000000000000000 sqrt(332929 y  + 1000000)
    2                                                           2
 + y  (16028853347520000000000000000000000000000 sqrt(0.332929 y  + 1)
                                                       2
 - 60161744582671382560000000000000000000 sqrt(332929 y  + 1000000))
    4                                                          2
 + y  (3947525835840000000000000000000000000000 sqrt(0.332929 y  + 1)
                                                        2
 - 112645908725371144582586400000000000000 sqrt(332929 y  + 1000000))
                                                            2
 + 12530880000000000000000000000000000000000 sqrt(0.332929 y  + 1)))
                                               4                2
/(- 30948602552985600000000000000000000000000 y  sqrt(0.332929 y  + 1)
                                               2                2
 - 185917133116108800000000000000000000000000 y  sqrt(0.332929 y  + 1)
                                                             2
 - 279214387200000000000000000000000000000000 sqrt(0.332929 y  + 1))
(%i22)  S_FoP_Al1(y):= ''(integrate( foo_1(y),y,0,1))  ;

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786
(%o22) S_FoP_Al1(y) := 4.464285714285714
  1
 /
 [                                         11              2
 I  (741724899620869378121796486996078681 y   sqrt(332929 y  + 1000000)
 ]
 /
  0
                              577 y
 (atan2(577 y, - 1000) - atan(-----) - %pi)
                              1000
                                          9              2
 + 7574782194501581116776447858489798438 y  sqrt(332929 y  + 1000000)
                              577 y
 (atan2(577 y, - 1000) - atan(-----) - %pi)
                              1000
                                           7              2
 + 29031893253004367050799515694452800000 y  sqrt(332929 y  + 1000000)
                              577 y
 (atan2(577 y, - 1000) - atan(-----) - %pi)
                              1000
                                           5              2
 + 50769125747525708318584807200000000000 y  sqrt(332929 y  + 1000000)
                              577 y
 (atan2(577 y, - 1000) - atan(-----) - %pi)
                              1000
                                           3              2
 + 38418615447231297096000000000000000000 y  sqrt(332929 y  + 1000000)
                              577 y
 (atan2(577 y, - 1000) - atan(-----) - %pi)
                              1000
                                                        2
 + 8876601654102000000000000000000000000 y sqrt(332929 y  + 1000000)
                              577 y
 (atan2(577 y, - 1000) - atan(-----) - %pi)
                              1000
                                          10              2
 - 2570970189327103563680403767750706000 y   sqrt(332929 y  + 1000000)
                                           8              2
 - 23681651611442049468605076676349388000 y  sqrt(332929 y  + 1000000)
                                           6              2
 - 78981841855515216067805804248800000000 y  sqrt(332929 y  + 1000000)
                                                      2
 - 5715474186240000000000000000000000000 sqrt(332929 y  + 1000000)
    2                                                           2
 + y  (16028853347520000000000000000000000000000 sqrt(0.332929 y  + 1)
                                                       2
 - 60161744582671382560000000000000000000 sqrt(332929 y  + 1000000))
    4                                                          2
 + y  (3947525835840000000000000000000000000000 sqrt(0.332929 y  + 1)
                                                        2
 - 112645908725371144582586400000000000000 sqrt(332929 y  + 1000000))
                                                            2
 + 12530880000000000000000000000000000000000 sqrt(0.332929 y  + 1))
                                               4                2
/(- 30948602552985600000000000000000000000000 y  sqrt(0.332929 y  + 1)
                                               2                2
 - 185917133116108800000000000000000000000000 y  sqrt(0.332929 y  + 1)
                                                             2
 - 279214387200000000000000000000000000000000 sqrt(0.332929 y  + 1)) dy
(%i23) 
Run Example
legendre_q (5, i);
                    i + 1   5         4              i + 1   3         2
(%o1)/R/ (945 log(- -----) i  - 1890 i  - 1050 log(- -----) i  + 1470 i
                    i - 1                            i - 1
                                                            i + 1
                                                + 225 log(- -----) i - 128)/240
                                                            i - 1
(%i2) x0:-0.74*%i;
(%o2)                              - 0.74 %i
(%i3) F1(x):=legendre_q (5, x);
(%o3)                          F1(x) := "Q" (x)
                                           5
(%i4) DiffF1(x):= ''(diff(F1(x), x));
                                 x + 1   6        5             x + 1   4
(%o4)/R/ DiffF1(x) := (315 log(- -----) x  - 630 x  - 525 log(- -----) x
                                 x - 1                          x - 1
                3             x + 1   2                    x + 1        2
         + 840 x  + 225 log(- -----) x  - 226 x - 15 log(- -----))/(16 x  - 16)
                              x - 1                        x - 1
(%i5) a:DiffF1(x0);

rat: replaced -0.04038511243215 by -625/15476 = -0.04038511243215

rat: replaced 647.425577312 by 278393/430 = 647.4255813953488

rat: replaced -347.36456840544 by -237250/683 = -347.364568081991
                                  1 - 0.74 %i
               12752187500 log(- -------------) - 23767802375 %i
                                 - 0.74 %i - 1
(%o5)/R/       -------------------------------------------------
                                   909029288
(%i6) c:realpart(1.678/(F1(x0) - 0.592*x0*a)),numer;

rat: replaced -0.74 by -37/50 = -0.74

rat: replaced 0.5 by 1/2 = 0.5

rat: replaced -0.5 by -1/2 = -0.5

rat: replaced 0.5 by 1/2 = 0.5

rat: replaced 0.33333333333333 by 1/3 = 0.33333333333333

rat: replaced 0.25 by 1/4 = 0.25

rat: replaced 0.2 by 1/5 = 0.2

rat: replaced 1.1000745665743609E-9 by 1/909029288 = 1.1000745665743609E-9

rat: replaced -0.43808 by -1369/3125 = -0.43808

rat: replaced 1.678 by 839/500 = 1.678
(%o6) - (2860033397370000000 (9292198558968750000 (%pi + 1.867451995038426)
 + 4510925222621773053 (- %pi - 1.867451995038426) - 8.8721865732720937E+18))
/(expt(9292198558968750000 (%pi + 1.867451995038426)
 + 4510925222621773053 (- %pi - 1.867451995038426) - 8872186573272093100, 2)
 + 399539.5785361568)
(%i7) C2:float(c);
(%o7)                         - 0.18968977717912
(%i8) C2:-0.05623;
(%o8)                              - 0.05623
(%i9) plot2d(realpart(C2*F1(y*x0)-0.045),[y,0,1.0]);
plotplot2d(realpart(C2*F1(y*x0)-0.045),[y,0,1.0]);

Related Help

Help for Legendre_q