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assoc_legendre_p-assoc_legendre_q-diff-legendre_p-legendre_q

f(x):= assoc_legendre...

f1(x):=''(diff(f(x),x));

f1(0);

Calculate

assoc_legendre_p-diff-legendre_p

z([x]):= assoc_legend...

z(0);

k([x]):=diff(assoc_le...

Calculate

assoc_legendre_p-legendre_p

assoc_legendre_p(5,3,...

Calculate

assoc_legendre_p-diff-legendre_p

z([x]):= assoc_legend...

z(0);

k([x]):=diff(assoc_le...

Calculate

assoc_legendre_p-diff-legendre_p

z:assoc_legendre_p(5,...

diff(assoc_legendre_p...

Calculate

assoc_legendre_p-assoc_legendre_q-diff-legendre_p-legendre_q

f(x):= assoc_legendre...

f1(x):=''(diff(f(x),x));

f1(0);

Calculate

assoc_legendre_p-assoc_legendre_q-diff-legendre_p-legendre_q

f(x):= ((x*x-1)^(-0/2...

f(-%i);

f1(x):=''(diff(f(x),x));

Calculate

assoc_legendre_p-assoc_legendre_q-diff-legendre_p-legendre_q

f(x):= ((x*x-1)^(-0/2...

f(-%i);

f1(x):=''(diff(f(x),x));

Calculate

assoc_legendre_p-assoc_legendre_q-diff-legendre_p-legendre_q

f(x):= ((x*x-1)^(-0/2...

f(-%i);

f1(x):=''(diff(f(x),x));

Calculate

assoc_legendre_p

Run Example
(%i1)z:assoc_legendre_p(5,0,x);
                               5              4                            2
                     63 (1 - x)    315 (1 - x)              3   105 (1 - x)
(%o1) - 15 (1 - x) - ----------- + ------------ - 70 (1 - x)  + ------------
                          8             8                            2
                                                                            + 1
(%i2) k:diff(assoc_legendre_p(5,0,x),x);
                                   4              3
                        315 (1 - x)    315 (1 - x)               2
(%o2)   - 105 (1 - x) + ------------ - ------------ + 210 (1 - x)  + 15
                             8              2
(%i3) 
Run Example
? assoc_legendre_p;

 -- Function: assoc_legendre_p (<n>, <m>, <x>)
     The associated Legendre function of the first kind of degree <n>
     and order <m>.

     Reference: Abramowitz and Stegun, equations 22.5.37, page 779,
     8.6.6 (second equation), page 334, and 8.2.5, page 333.


(%o1)                                true
(%i2) 
Run Example
f(x):= ((x*x-1)^(-0/2))*assoc_legendre_p(5,0,x);
                                 - 0
                                 ---
                                  2
(%o1)           f(x) := (x x - 1)    assoc_legendre_p(5, 0, x)
(%i2) f(-%i);
                   5               4                              2
        63 (%i + 1)    315 (%i + 1)               3   105 (%i + 1)
(%o2) - ------------ + ------------- - 70 (%i + 1)  + -------------
             8               8                              2
                                                              - 15 (%i + 1) + 1
(%i3) f1(x):=''(diff(f(x),x));
                                          4              3
                               315 (1 - x)    315 (1 - x)               2
(%o3) f1(x) := - 105 (1 - x) + ------------ - ------------ + 210 (1 - x)  + 15
                                    8              2
(%i4) f1(-%i);
                   4               3
       315 (%i + 1)    315 (%i + 1)                2
(%o4)  ------------- - ------------- + 210 (%i + 1)  - 105 (%i + 1) + 15
             8               2
(%i5) g(x):= ((x*x-1)^(-0/2))*assoc_legendre_q(5,0,x);
                                          - 0
                                          ---
                                           2     0
(%o5)                    g(x) := (x x - 1)    "Q" (x)
                                                 5
(%i6) g(-%i);
                      %i - 1    5          4              %i - 1    3
(%o6)/R/ - (945 log(- ------) %i  + 1890 %i  - 1050 log(- ------) %i
                      %i + 1                              %i + 1
                                            2             %i - 1
                                   - 1470 %i  + 225 log(- ------) %i + 128)/240
                                                          %i + 1
(%i7) g1(x):=''(diff(g(x),x));
                             x + 1   6        5             x + 1   4        3
(%o7)/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
(%i8) simplify(g1(-%i));
                                        1 - %i
                             135 log(- --------) - 212 %i
                                       - %i - 1
(%o8)               simplify(----------------------------)
                                          4
(%i9) 

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