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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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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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Integral of rational function with cosine hyperbolic function using rectangle contour
$$ \int^{\infty}_{\infty} \frac{1}{(5 \pi^2 + 8 \pi y + 16y^2) }\frac{\cosh\left(y+\frac{\pi}{4} \right)}{\cosh^3(y)}dy=\frac{2}{\pi^3}\left(\pi \cosh\left(\frac{\pi}{4} \right)4\sinh\left( \frac{\pi}{4}\right) \right)$$ $$\textit{proof}$$ Consider $$f(z) = \frac{\sinh(z)}{z \sinh^3(z\pi/4)}$$ If we integrate around a contour of height \( \pi \) and stretch it to infinity we … Continue reading
Posted in Contour Integration
Tagged analysis, complex, cosine, hyperboic, rectangle
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Contour integraion of a rational function of logarithm and exponential
$$\int_{0}^\infty \frac{\log(x)\cos(x)}{(x^2+1)^2}\,dx = – \frac{\pi \mathrm{Ei}(1)}{4e}\frac{\pi}{4e}$$ $$\textit{proof}$$ Consider the following function $$f(z) = \frac{\log(z) }{(z^2+1)^2}e^{iz}$$ Now consider the principle logarithm where $$\log(z) = \logr+i \theta \,\,\, , \theta \,\in (\pi , \pi]$$ Consider the following contour Then by … Continue reading
Posted in Contour Integration
Tagged contour, cosine, exponential, fraction, Integral, proof, residue, theorem
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