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.