Tag Archives: ellipse

Proving a trigonometric integral by integrating around an ellipse in the complex plain

Prove for \(a,b > 0\) $$\int^{2\pi}_0\frac{dt}{a^2\cos^2 t +b^2\sin^2 t} = \frac{2\pi}{ab}$$ $$\textit{solution}$$ Let us integrate the following function $$f(z) = \frac{1}{z}$$ Around the ellipse $$\oint_{\gamma}f(z)\,dz =2\pi i\,\mathrm{Res}(f,0)$$ The parametrization of the ellipse \(\gamma(t) = a\cos(t)+ib\sin(t)\) $$\oint_{\gamma}f(z)\,dz=\int^{2\pi}_0 \frac{-a\sin t+ib\cos t}{a\cos t+ib\sin t} … Continue reading

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