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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: contour
Euler reflection formula proof using contour integration
$$\int_{0}^{\infty}\frac{x^{\alpha}}{x+1}\,dx= \pi \csc(\pi \alpha)$$ $$\textit{proof}$$ Consider the following function $$f(z) = \frac{z^{\alpha}}{1+z} = \frac{e^{\alpha \log(z)}}{1+z}$$ As we know the function \( \log(z) \) is multivalued defined as $$\log(z) = \lnz+i\theta +2k\pi i$$ This maps the complex plain more than once … Continue reading
Posted in Beta function, Contour Integration, Gamma function
Tagged analysis, complex, contour, Euler, formula, proof, reflection
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Solving an integral using Dogbone contour
Prove that $$\int^{1}_{0} \sqrt{x}\sqrt{1x}\,dx = \frac{\pi}{8}$$ $$\textit{proof}$$ Consider the function $$f(z) = \sqrt{zz^2} = e^{\frac{1}{2}\log(zz^2)}$$ Consider the branch cut on the xaxis $$x(1x)\geq 0\,\, \implies \, 0\leq x \leq 1 $$ Consider \( w= zz^2 \) then $$\log(w) = \logw+i\theta,\,\, … Continue reading
Contour integraion of a rational function of logarithm and exponential
$$\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]$$ Consider the following contour Then by … Continue reading
Posted in Contour Integration
Tagged contour, cosine, exponential, fraction, Integral, proof, residue, theorem
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Bromwich contour integration of the gamma function
$$\frac{1}{2\pi i}\int^{c+i\infty}_{ci\infty}\Gamma(a+t)\Gamma(bt) s^{t}\,dt= \frac{\Gamma(a+b)}{(1+s)^{a+b}}s^a$$ $$\textit{proof}$$ Consider the following function $$f(z) = \Gamma(z+a)\Gamma(bz) s^{z}$$ Suppose that \(a,b \in \mathbb{R} \) and \(a < b \). Note that the Gamma function has a pole of order 1 at each nonpositive integer where … Continue reading