勒让德多项式的正交性 正交性 ∫(x)2(x)=02(9O)(s9im=0.(k≠1) 模 P(x)P(x)dx= P(cos O)P(cos 0)sin 0d0=N 2l+1 正交性应用例题 P(x)dx Po(P(xdx=OONo =25lo xP(x)dx= R(x)P(x)dx=S, N2=35, 厂x)k-」(+)=号2N2+号50N
勒让德多项式的正交性 ( ) ( ) 0, (cos ) (cos )sin 0, ( ) 0 1 1 P x P x d x P P d k l k l = k l = + − 模 正交性 2 1 2 ( ) ( ) (cos ) (cos )sin 2 0 1 1 + = = = + − l Pl x Pl x dx Pl Pl d Nl 正交性应用例题 ,0 2 0 ,0 0 1 1 1 1 Pl (x)dx = P (x)Pl (x)dx = l N = 2 l − + − 3 ,1 2 2 1 ,1 1 1 1 1 1 ( ) ( ) ( ) xPl x dx = P x Pl x dx = l N = l − + − 2 3 ,0 0 2 1 3 ,2 2 2 3 0 1 3 2 2 1 1 2 1 1 x Pl (x)dx = ( P + P )Pl dx = l N + l N − + −
勒让德多项式模的计算 u(x,)=∑nP(x)y 2rx+r ∑。(x)∑。1(x)k=∫ dx 11-2rx+r ∑∑“∫(/(k n(1-2rx+r2)1 2r ∑∑1N=-m(1-n)-m1+ N22=∑ +(1 n+1 2k+1 2k+1
勒让德多项式模的计算 0 2 1 2 1 ( , ) ( ) rx r u x r P x r l l − + = = 2 1 0 0 1 1 1 1 2 ( ) ( ) rx r d x P x r P x r d x k k l l − + = − − 1 1 2 1 0 0 1 ln(1 2 ) | 2 1 ( ) ( ) + − − + = − − + rx r r r P x P x dx l k l k [ln(1 ) ln(1 )] 2 1 , 0 0 r r r r l k Nk l k = − − − + + n k n k n k r k r n r n N r 2 2 2 0 2 1 2 1 ( 1) 1 1 + = + − + + = 2 1 2 2 + = k Nk