phsopher
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I was reading a paper where the following integral appears:
I = \int_0 ^{\pi}dt\sqrt{k^2 + \sin^2t}
In the limit k^2 \ll 1 the authors present the following approximation
I \approx 2 + \frac{k^2}{2}\left(\ln{\frac{1}{k^2}} + 4\ln 2 + 1\right).
I'm trying to reproduce this result but with no luck. Any idea how it should go? I've plotted both expressions as a function of k and they indeed agree for small k.
I = \int_0 ^{\pi}dt\sqrt{k^2 + \sin^2t}
In the limit k^2 \ll 1 the authors present the following approximation
I \approx 2 + \frac{k^2}{2}\left(\ln{\frac{1}{k^2}} + 4\ln 2 + 1\right).
I'm trying to reproduce this result but with no luck. Any idea how it should go? I've plotted both expressions as a function of k and they indeed agree for small k.
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