User:EGM6321.F10.TEAM1.WILKS/Mtg41

=EGM6321 - Principles of Engineering Analysis 1, Fall 2009= Mtg 41: Tues, 1Dec09

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Review for exam 2 - Historical development - Legendre functions Question: How to obtain $$ P_n \ $$ based on known $$ P_{n-1}, P_{n-2},... \ $$ ? - 2 recurring relationships. Same technique in power series. Solution: Frobenius method Question: Find a differential equation governing all $$ \left \{ P_n \right \} \ $$ ? - Legendre differential equations 2 families of homogeneous solutions: - Legendre functions= $$ \left \{ P_n \right \} \ $$ +  $$ \left \{ Q_n \right \} \ $$ $$ L_n = P_n \ $$ or $$ L_n = Q_n \ $$ Newtonian potential is solution of Laplace equation i.e., $$ \Delta\ \left ( \frac{1}{r} \right ) = 0 \ $$ $$ \frac{1}{r} = \frac{1}{r_{PQ}} = \frac{1}{r_Q} \left ( 1-2 \mu\ \rho\ + \rho\ ^2 \right )^{- \frac{1}{2}} \ $$

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$$ = \frac{1}{r_Q} \sum_{n} P_n ( \mu\ ) \rho\ ^n \ $$, where $$ = \rho\ := \frac{r_P}{r_Q} \ $$ $$ \Rightarrow \ \frac{1}{r} = \sum_{n} P_n ( \mu\ ) \frac{r_P^n}{r_Q^{n+1}} = \sum_{n} H_n \left ( \mu\, r_P, r_Q \right ) \ $$ , where $$ H_n \left ( \mu\ , r_P, r_Q \right ) = H_n \left ( x,y,z \right ) \ $$ $$ \Delta\ \frac{1}{r} = 0 = \sum_{n} \Delta\ H_n(x,y,z) \ \Rightarrow \ \Delta\ H_n = 0$$ Where this argument is based on the power series Laplace equations in a sphere axisymmetrical case [[media: Egm6321.f09.mtg29.djvu |  P.29-1 ]] separation of variables [[media: Egm6321.f09.mtg30.djvu | P.30-1]] General solution of axisymmetrical Laplace equations in a sphere $$ \psi\ (r, \theta\ ) = \sum_{n} (A_nr^n+B_nr^{n+1})(C_nP_n+D_nQ_n) \ $$ Where $$ A_n r^n + B_n r^{n+1} \ $$ can be found on [[media: Egm6321.f09.mtg31.djvu | P.31-2]] and $$ C_n P_n + D_n Q_n \ $$ can be found on [[media: Egm6321.f09.mtg32.djvu | P.32-1]] and $$ \mu\ = \sin \theta\ \ $$