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Homework Help: Series expansion of function of two variables

  1. Mar 18, 2012 #1
    This was stated in a lecture:
    For r < 1 we can make a series expansion of [itex]f(r,u)[/itex] in terms of powers of r where:

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

    Where [itex]P_n(u)[/itex] 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 [itex]f(r,u)[/itex]. I'm probably just missing some prerequisite maths knowledge so if anyone could point me in the right direction, I'd appreciate it.
  2. jcsd
  3. Mar 19, 2012 #2
    You know why there is a general series expansion of the form [itex] \sum a_n t^n [/itex], 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).
  4. Mar 19, 2012 #3
    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 learnt 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.
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