Q2(x) (3x2-1)h/+x)3 Q3(x)=(5x3-3x)ln (E133) Q4(x)=1(35x4-30-22、,(1+ Pl(x)=-(1-x2)2=-sin6 (E.135) P(x)=-3x(1-x2)2=-3cos6sin (E.136) P2(x)=3(1-x2)=3sin2 (E137) P(x)=-2(5x2-1)-x35o6-1)in E.138) P2(x)=15x(1-x2)=15cos6sin26 (E.139) P3(x)=-15(1-x2)312=-15sin36 (E140) P(x)=-7(7x3-3x)(1-x2)2=-5(7cos6-3cos)sinb(E141) P2(x)=(7x2-1)(1-x2)==(7 P2(x)=-105x(1-x2)312=-105cos6sin36 (E143) P4(x)=105(1-x2)2=105sin16 (E144) Functional relationships m >n. -1a, (E145) Pn(x)=2"n! dxn Rm(x)=(-1)=(1-22d8n x)= (n +mi pn(r) (E148) Pn(-x)=(-1)Pn(x) Qn(-x)=(-1)y+gn(x) (E.150) Pn(-x)=(-1)”+mPm(x) (E151) Qm(-x)=(-1)+m+gm(x) (E152) (1)= 0.m>0 Pn(x)≤Pn(1) @2001 by CRC Press LLCQ2(x) = 1 4 (3x 2 − 1)ln1 + x 1 − x − 3 2 x (E.132) Q3(x) = 1 4 (5x 3 − 3x)ln1 + x 1 − x − 5 2 x 2 + 2 3 (E.133) Q4(x) = 1 16(35x 4 − 30x 2 + 3)ln1 + x 1 − x − 35 8 x 3 + 55 24 x (E.134) P1 1 (x) = −(1 − x 2 ) 1/2 = − sin θ (E.135) P1 2 (x) = −3x(1 − x 2 ) 1/2 = −3 cos θ sin θ (E.136) P2 2 (x) = 3(1 − x 2 ) = 3 sin2 θ (E.137) P1 3 (x) = −3 2 (5x 2 − 1)(1 − x 2 ) 1/2 = −3 2 (5 cos2 θ − 1)sin θ (E.138) P2 3 (x) = 15x(1 − x 2 ) = 15 cos θ sin2 θ (E.139) P3 3 (x) = −15(1 − x 2 ) 3/2 = −15 sin3 θ (E.140) P1 4 (x) = −5 2 (7x 3 − 3x)(1 − x 2 ) 1/2 = −5 2 (7 cos3 θ − 3 cos θ)sin θ (E.141) P2 4 (x) = 15 2 (7x 2 − 1)(1 − x 2 ) = 15 2 (7 cos2 θ − 1)sin2 θ (E.142) P3 4 (x) = −105x(1 − x 2 ) 3/2 = −105 cos θ sin3 θ (E.143) P4 4 (x) = 105(1 − x 2 ) 2 = 105 sin4 θ (E.144) Functional relationships Pm n (x) =  0, m > n, (−1)m (1−x2)m/2 2n n! dn+m (x2−1)n dxn+m , m ≤ n. (E.145) Pn(x) = 1 2nn! dn(x 2 − 1)n dx n (E.146) Rm n (x) = (−1) m(1 − x 2 ) m/2 dm Rn(x) dxm (E.147) P−m n (x) = (−1) m (n − m)! (n + m)! Pm n (x) (E.148) Pn(−x) = (−1) n Pn(x) (E.149) Qn(−x) = (−1) n+1Qn(x) (E.150) Pm n (−x) = (−1) n+m Pm n (x) (E.151) Qm n (−x) = (−1) n+m+1Qm n (x) (E.152) Pm n (1) =  1, m = 0, 0, m > 0. (E.153) |Pn(x)| ≤ Pn(1) = 1 (E.154)