### Related

##### assoc_legendre_p-assoc_legendre_q-diff-legendre_p-legendre_q

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

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

f1(0);

Calculate

##### assoc_legendre_p-legendre_p

z:assoc_legendre_p(5,...

Calculate

##### assoc_legendre_p-assoc_legendre_q-diff-legendre_p-legendre_q

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

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-assoc_legendre_q-diff-expand-legendre_p-legendre_q

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

expand(f(-%i));

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

Calculate

##### assoc_legendre_p-diff-legendre_p

z:assoc_legendre_p(5,...

k:diff(assoc_legendre...

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-diff-legendre_p

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

z(0);

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

Calculate

##### assoc_legendre_p-assoc_legendre_q-diff-expand-legendre_p-legendre_q

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

expand(f(-%i));

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

Calculate

##### assoc_legendre_p-diff-legendre_p

z:assoc_legendre_p(5,...

k:diff(assoc_legendre...

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) ```

### Related Help

Help for Assoc_legendre_p