University of Florida/Egm6321/F10.TEAM1.WILKS/Mtg37

Mtg 37: Thurs, 17Nov09

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[[media: Egm6321.f09.mtg36.djvu | P.36-4]] continued $$ P_n(x) \ $$ $$ P_1(x)=x \ $$ $$ Q_n(x) \ $$ From [[media: Egm6321.f09.mtg18.djvu  |  P.18-1 ]]: HW show $$ Q_1(x)=\frac{1}{2}xlog \left ( \frac{1+x}{1-x} \right ) -1 = x \tanh ^{-1}x-1 \ $$ ref [[media: http://books.google.com/books?id=9Cg3HWCnCjAC&printsec=frontcover&dq=differential+equations+billingham&ei=pGR4SpPVLojSMpb07Qw#v=onepage&q=&f=false | K p33 ]] for $$ Q_2, Q_3...\ $$ $$ Q_0(x)= \tanh ^{-1}(x) \ \ $$ is odd HW: Use Eq(2) to show when $$ Q_n \ $$ is even or odd, depending on "n" HW: Plot $$ \left \{ P_0, P_1,...,P_4 \right \} \ $$ and $$ \left \{ Q_0, Q_1,...,Q_4 \right \} \ $$ Legendre function $$ L_n(x)=P_n(x) \ $$ or $$ Q_n(x) \ $$ solution of Legendre solution

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for $$ n \ne m \ $$ for $$ L_n =P_n, \left \langle L_n,L_n \right \rangle = \left \langle P_n,P_n \right \rangle = \frac{2}{2n+1}  \ $$ for $$ \left \langle L_n,L_n \right \rangle = \left \langle P_n,Q_n \right \rangle \ =  $$ HW $$ \left \langle L_n,L_m \right \rangle =  \left \langle P_n,P_m \right \rangle \ =  $$ [[media: Egm6321.f09.mtg33.djvu  | Eq.(3) P.33-1]] $$ \left \langle L_n,L_m \right \rangle = \left \langle P_n,Q_m \right \rangle \ = 0 $$ for $$ n \ne m \ $$ Proof: Legendre equation, [[media: Egm6321.f09.mtg14.djvu | Eq.(1) P.14-2 ]]: 2) $$ \left [ (1-x^2)y' \right ]'+n(n+1)y=0 \ $$   Where $$ \Rightarrow \ \left [ (1-x^2)L_n' \right ]'+n(n+1)L_n=0 \ $$  Multiply by $$ L_m \ $$ and integrate from  -1 to +1: $$ \int_{-1}^{1} L_m \left [ (1-x^2)L_n' \right ]' \, dx + n(n+1) \int_{-1}^{1} L_mL_n \, dx=0 \ $$ Where  $$ L_m \left [ (1-x^2)L_n' \right ]' = \alpha\ \ $$

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Integrate $$ \alpha\ \ $$ by parts: Interchange n and m: Eq(1)-Eq(2): $$ \left [ n(n+1)-m(m+1) \right ] \left \langle L_n,L_m \right \rangle =0 $$ Where $$ \left [ n(n+1)-m(m+1) \right ] \ne 0 $$ since $$ n \ne m \ $$ $$ \Rightarrow \ \left \langle L_n,L_m \right \rangle =0 \ $$ when $$ n \ne m \ $$ [[media: http://books.google.com/books?id=9Cg3HWCnCjAC&printsec=frontcover&dq=differential+equations+billingham&ei=pGR4SpPVLojSMpb07Qw#v=onepage&q=&f=false | cf.K.p41]]