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From a fraction with infinite sum in denominator to partial fractions?

  1. Oct 19, 2012 #1
    From a fraction with infinite sum in denominator to partial fractions??

    I am currently studying a course on Perturbation Methods and in particular an example considering the following integral [tex] \int_{0}^{\frac{\pi}{4}} \frac{d\theta}{\epsilon^2 + \sin^2 \theta}. [/tex]

    There's a section of the working where, having used the Taylor expansion of sin near 0 and using sin θ ≈ θ together with substitution θ=εu, we get the following fraction for the integrand
    [tex] \frac{1}{\epsilon^2 + \epsilon^2 u^2 - \frac13 \epsilon^4 u^4 + \cdots}.[/tex]
    This then in both my lecture notes and a book I'm following becomes
    [tex] \frac{1}{\epsilon^2}\left( \frac{1}{1+u^2} + \frac{\epsilon u^4}{3(1+u^2)^2} + \cdots \right). [/tex]

    Can anyone see how these are equal?
     
  2. jcsd
  3. Oct 19, 2012 #2

    mathman

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    Re: From a fraction with infinite sum in denominator to partial fractions??

    It looks like using a variation on 1/(1-x) = 1 + x + x2 + ... after factoring out 1/{ε2(1+u2)}.

    Also second term numerator should be (εu)4
     
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