Series expansion of function of two variables

[JesseC]

This was stated in a lecture:
"
For r < 1 we can make a series expansion of $f(r,u)$ in terms of powers of r where:

$$f(r,u) = \frac{1}{\sqrt{1+r^2-2ru}} = \sum^{\infty}_{n=0}r^nP_n(u)$$
"

Where $P_n(u)$ is a function of u (and is actually the Legendre polynomials). This was stated without real explanation. I don't understand how you can just 'see' this series expansion from the form of $f(r,u)$. I'm probably just missing some prerequisite maths knowledge so if anyone could point me in the right direction, I'd appreciate it.

 

[clamtrox]

You know why there is a general series expansion of the form $\sum a_n t^n$, right?

The relation you were given is just one possible definition of the Legendre polynomials. There are other definitions, and then the problem becomes to show that these definitions are equivalent. I think you can find most of the proofs for example in Arfken&Weber (although I don't have it near me right now, so I can't check).

 

[JesseC]

You know why there is a general series expansion of the form $\sum a_n t^n$, right?.

Not really. I have an idea of when you can and can't use a series solution for a 2nd order DE (essential singular points). I don't know the mathematical reason 'why' there should be a series expansion of a particular function or solution. I've just learned how to apply it!

I found this website very helpful: http://www.serc.iisc.ernet.in/~amohanty/SE288/l.pdf and I think I'm a but clearer on this particular problem.