Page 538 - Electromagnetics

1 2 1 + x 3
Q 2 (x) = (3x − 1) ln − x (E.132)
4 1 − x 2

1 3 1 + x 5 2 2
Q 3 (x) = (5x − 3x) ln − x + (E.133)
4 1 − x 2 3
1 4 2 1 + x 35 3 55
Q 4 (x) = (35x − 30x + 3) ln − x + x (E.134)
16 1 − x 8 24
2 1/2
1
P (x) =−(1 − x ) =− sin θ (E.135)
1
2 1/2
1
P (x) =−3x(1 − x ) =−3 cos θ sin θ (E.136)
2
2
2
2
P (x) = 3(1 − x ) = 3 sin θ (E.137)
2
3 3
1 2 2 1/2 2
P (x) =− (5x − 1)(1 − x ) =− (5 cos θ − 1) sin θ (E.138)
3
2 2
2
2
2
P (x) = 15x(1 − x ) = 15 cos θ sin θ (E.139)
3
3
3
2 3/2
P (x) =−15(1 − x ) =−15 sin θ (E.140)
3
5 3 2 1/2 5 3
1
P (x) =− (7x − 3x)(1 − x ) =− (7 cos θ − 3 cos θ) sin θ (E.141)
4
2 2
15 15
2 2 2 2 2
P (x) = (7x − 1)(1 − x ) = (7 cos θ − 1) sin θ (E.142)
4
2 2
3
2 3/2
3
P (x) =−105x(1 − x ) =−105 cos θ sin θ (E.143)
4
4
2 2
4
P (x) = 105(1 − x ) = 105 sin θ (E.144)
4
Functional relationships

0, m > n,
m
P (x) = (E.145)
2
n m (1−x ) d n+m (x −1) n
2 m/2
(−1) , m ≤ n.
n
2 n! dx n+m
n
2
1 d (x − 1) n
P n (x) = (E.146)
n
2 n! dx n
m
d R n (x)
m
m
2 m/2
R (x) = (−1) (1 − x ) (E.147)
n
dx m
(n − m)! m
m
−m
P (x) = (−1) P (x) (E.148)
n n
(n + m)!
n
P n (−x) = (−1) P n (x) (E.149)
Q n (−x) = (−1) n+1 Q n (x) (E.150)
m
m
P (−x) = (−1) n+m P (x) (E.151)
n
n
m
m
Q (−x) = (−1) n+m+1 Q (x) (E.152)
n
n

1, m = 0,
m
P (1) = (E.153)
n
0, m > 0.
|P n (x)|≤ P n (1) = 1 (E.154)
© 2001 by CRC Press LLC