University of Florida/Egm6321/F10.TEAM1.WILKS/Mtg41

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Review for exam 2

- Historical development - Legendre functions
Question: How to obtain ${\displaystyle P_{n}\ }$ based on known ${\displaystyle P_{n-1},P_{n-2},...\ }$ ? - 2 recurring relationships. Same technique in power series.
Solution: Frobenius method
Question: Find a differential equation governing all ${\displaystyle \left\{P_{n}\right\}\ }$ ? - Legendre differential equations
2 families of homogeneous solutions:
- Legendre functions= ${\displaystyle \left\{P_{n}\right\}\ }$ + ${\displaystyle \left\{Q_{n}\right\}\ }$

${\displaystyle L_{n}=P_{n}\ }$ or ${\displaystyle L_{n}=Q_{n}\ }$

Newtonian potential is solution of Laplace equation

i.e., ${\displaystyle \Delta \ \left({\frac {1}{r}}\right)=0\ }$

${\displaystyle {\frac {1}{r}}={\frac {1}{r_{PQ}}}={\frac {1}{r_{Q}}}\left(1-2\mu \ \rho \ +\rho \ ^{2}\right)^{-{\frac {1}{2}}}\ }$

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${\displaystyle ={\frac {1}{r_{Q}}}\sum _{n}P_{n}(\mu \ )\rho \ ^{n}\ }$ , where ${\displaystyle =\rho \$ :={\frac {r_{P}}{r_{Q}}}\ }

${\displaystyle \Rightarrow \ {\frac {1}{r}}=\sum _{n}P_{n}(\mu \ ){\frac {r_{P}^{n}}{r_{Q}^{n+1}}}=\sum _{n}H_{n}\left(\mu \ ,r_{P},r_{Q}\right)\ }$ , where ${\displaystyle H_{n}\left(\mu \ ,r_{P},r_{Q}\right)=H_{n}\left(x,y,z\right)\ }$

${\displaystyle \Delta \ {\frac {1}{r}}=0=\sum _{n}\Delta \ H_{n}(x,y,z)\ \Rightarrow \ \Delta \ H_{n}=0}$
Where this argument is based on the power series
Laplace equations in a sphere
axisymmetrical case P.29-1
separation of variables P.30-1
General solution of axisymmetrical Laplace equations in a sphere

${\displaystyle \psi \ (r,\theta \ )=\sum _{n}(A_{n}r^{n}+B_{n}r^{n+1})(C_{n}P_{n}+D_{n}Q_{n})\ }$

Where ${\displaystyle A_{n}r^{n}+B_{n}r^{n+1}\ }$ can be found on P.31-2

and ${\displaystyle C_{n}P_{n}+D_{n}Q_{n}\ }$ can be found on P.32-1

and ${\displaystyle \mu \ =\sin \theta \ \ }$

References