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 Integrating a cosine log integral around a semicircle contour
 Creating Difficult integrals by the residue theorem
 Proving a trigonometric integral by integrating around an ellipse in the complex plain
 Integrating a fraction of exponential and trignometric using rectangular contour
 Integrating around a triangular contour for Fresnel integral
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Monthly Archives: May 2017
Integrating a cosine log integral around a semicircle contour
Prove that $$\int^1_0 \frac{\cos(\log x)}{x^2+1}\,dx = \frac{\pi}{4}\mathrm{sech}\left( \frac{\pi}{2}\right)$$ First note that $$2 \int^1_0 \frac{\cos(\log x) }{x^2+1}\,dx = \int^\infty_0 \frac{\cos(\log x)}{x^2+1}\,dx$$ Integrate the following function $$f(z) = \frac{e^{i\log(z)}}{z^2+1}$$ Around a semicircle in the upper half place. Where we avoid the branch … Continue reading
Posted in Contour Integration
Tagged circle, contour, cosine, log, rational, residue, semi, theorem
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Creating Difficult integrals by the residue theorem
Theorem Let \( f \) be analytic function in the unit circle \( z\leq 1 \) such that \( f\neq 0\) . Then $$\int^{2\pi}_0f(e^{it})\,dt =2\pi \, f(0) $$ $$\textit{proof}$$ Since the function \(f \) is analytic in and on the … Continue reading
Posted in Contour Integration
Tagged circle, contour, difficult, impossible, Integral, residue, theorem, unit
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