User:Egm6321.f10.team4.Yoon/Mtg40

=EGM6321 - Principles of Engineering Analysis 1, Fall 2010=

[[media: 2010_11_23_15_05_23.djvu | Mtg 40:]] Tue, 23 Nov 10

= Historical development of Legendre functions =

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Historical Development of $$\displaystyle \{ P_n \} $$ Plan: - Atraction of spheres - Newtonian potential in spherical coordinate - Power series => $$\displaystyle \{ P_n \} $$ - binomial theorem - 2 recurrence relations - Legendre differential equation

Legendre: Attraction of spheres



Newtonian potential: $$\displaystyle V=\frac{1}{r_{PQ}}$$ Recall: Force = grad $$\displaystyle V (\Rightarrow \frac{1}{r^2})$$ Express $$\displaystyle r_{PQ} $$ in terms of spherical coodinate of P and Q

= Generating function for Legendre polynomials =

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Recall


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$$  \displaystyle (r_{PQ})^2 = (r_P)^2 + (r_Q)^2 - 2r_Pr_Q \cos \gamma $$     (3)
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Now power series of potential energy.

= Power series of potential energy =

Goal: Power series of $$\displaystyle V$$ in terms of $$\displaystyle \rho$$.

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King p.35:

General binomial theorem
n integers(negative, positive)

Generalize (2) to case n=r=real numbers

Note: