Multipole Expansion in Electrodynamics: Simplifying with Taylor Series

[h0dgey84bc]

Hi,

I'm just working through some electrodynamics notes, and am a bit stuck following a particular Taylor expansion, the author starts with:

$\frac{1}{R_1}=\frac{1}{r} [1+(\frac{l}{r})^2-2\frac{l}{r}cos(\theta)]^-0.5$

Which he then says by assuming l<<r and expanding we get:

$\frac{1}{R_1}=\frac{1}{r} [1+(\frac{l}{r})cos(\theta)+\frac{1}{2}(\frac{l}{r})^2(3cos(\theta)^2-1)+ \frac{1}{2}(\frac{l}{r})^3(5cos(\theta)^2-3cos(\theta)) ]...$

Just a bit lost at how to do this, what variables am I expanding wrt?

thanks

 

[Meir Achuz]

The variable u=(L/r)

 

[h0dgey84bc]

thanks clem