- #1
megzaz
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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 \int_{0}^{\frac{\pi}{4}} \frac{d\theta}{\epsilon^2 + \sin^2 \theta}.
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
\frac{1}{\epsilon^2 + \epsilon^2 u^2 - \frac13 \epsilon^4 u^4 + \cdots}.
This then in both my lecture notes and a book I'm following becomes
\frac{1}{\epsilon^2}\left( \frac{1}{1+u^2} + \frac{\epsilon u^4}{3(1+u^2)^2} + \cdots \right).
Can anyone see how these are equal?
I am currently studying a course on Perturbation Methods and in particular an example considering the following integral \int_{0}^{\frac{\pi}{4}} \frac{d\theta}{\epsilon^2 + \sin^2 \theta}.
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
\frac{1}{\epsilon^2 + \epsilon^2 u^2 - \frac13 \epsilon^4 u^4 + \cdots}.
This then in both my lecture notes and a book I'm following becomes
\frac{1}{\epsilon^2}\left( \frac{1}{1+u^2} + \frac{\epsilon u^4}{3(1+u^2)^2} + \cdots \right).
Can anyone see how these are equal?