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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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Tag Archives: rational
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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Contour integration for a rational function of cos and cosh
$$ \int^{\infty}_{\infty} \frac{\cos(ax)}{\cosh(x)} \,dx = \pi \, \mathrm{sech} \left( \frac{\pi a}{2}\right)$$ $$\textit{proof}$$ Consider $$f(z) = \frac{e^{iaz}}{\sinh(z)}$$ If we integrate around a contour of height \( \pi \) and stretch it to infinity we get By taking \( T \to \infty … Continue reading
Posted in Contour Integration
Tagged contour, cosine, function, hyperbolic, Integral, rational, rectangle
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Contour method for shifted logarithm branch
Prove \( a,b,c,d >0 \) $$\int^\infty_0 \frac{\log(a^2+b^2x^2)}{c^2+d^2x^2}\,dx = \frac{\pi}{cd} \log \frac{ad+bc}{d}$$ Consider the function $$f(z) = \frac{\log(aibz)}{c^2+d^2z^2}$$ We need the logarithm with the branch cut \( y<\frac{a}{b} , x =0 \) . Note that this corresponds to $$\log(a+ibz) = \log\sqrt{(a+y)^2+b^2x^2}+i\theta … Continue reading
Posted in Contour Integration
Tagged analysis, branch, complex, logarithm, rational, residue, shifted, theorem
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