I'm reading section 17 of Mathematical Physics by Donald H. Menzel on Harmonic functions.(adsbygoogle = window.adsbygoogle || []).push({});

They start with newtons law of gravitation (although the following method can be aplied to any potential field with a 1/r dependence.)

see: http://en.wikipedia.org/wiki/Legendre_polynomials#Applications_of_Legendre_polynomials_in_physics

[tex]dV=-\frac{GdM}{r_{12}}=\frac{GdM}{(r_2^2-2r_1r_2 cos \gamma + r_1^2)^{1/2}}[/tex]

Where:

[tex]R_{12}[/tex]=The distance from P1 to the unit volume

[tex]R_{1}[/tex] is the distrance from the orgin to the unit volume

[tex]R{2}[/tex] is the distance from P1 to the orgin.

note that the law of cosines was used to express [tex]R_{12}[/tex] in terms of [tex]R_1[/tex] and [tex]R_2[/tex]

The substitutions:

[tex]cos \gamma = \mu [/tex]

[tex]\frac{r_1}{r_2}=\Beta_1[/tex]

are made giving:

[tex]dV=-\frac{GdM}{r_2} \left[ 1 - \beta (2 \mu - \beta) \right]^{-1/2} [/tex]

This is expanded using the more general form of the binomial theorem:

[tex]dV=-\frac{GdM}{r_2} \left[ 1 + \frac{1}{2}\beta (2\mu - \beta ) + \frac{1*2}{2*4}\beta^2(2\mu-\beta)^2+... \right] [/tex]

Now here is I where I get lost.If you expand and collect the terms (powers of [tex] \beta [/tex]) then supposedly the coefficients are:

[tex]P_l(\mu )=\frac{(2l)!}{2^l(l!)^2} \left[ u^l - \frac{l(l-1)}{2(2(l-1)}\mu^{l-2}+\frac{l(l-1)(l-2)(l-3)}{2 * 4(2l-1)(2l-3)}\mu^{l-4}... \right][/tex]

and apparently the numerical coefficients can be represented as follows:

[tex]\frac{(2l)!}{2^l(l!)^2}=\frac{(2l-1)(2l-3)...1}{l!}[/tex]

This then gives:

[tex]dV=-\frac{GdM}{r_2} \sum_{l=0}^{\inf}P_l(\mu ) \left( \frac{r_1}{r_2} \right)^l[/tex]

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# Newtons Law of Gravity Legendre Polynomial & Harmonic functions

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