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- 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 multi-valued defined as $$\log(z) = \ln|z|+i\theta +2k\pi i$$ 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 $$\int^{1}_{0} \sqrt{x}\sqrt{1-x}\,dx = \frac{\pi}{8}$$ $$\textit{proof}$$ Consider the function $$f(z) = \sqrt{z-z^2} = e^{\frac{1}{2}\log(z-z^2)}$$ Consider the branch cut on the x-axis $$x(1-x)\geq 0\,\, \implies \, 0\leq x \leq 1 $$ Consider \( w= z-z^2 \) then $$\log(w) = \log|w|+i\theta,\,\, … Continue reading
Contour integraion of a rational function of logarithm and exponential
$$\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) = \log|r|+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}_{c-i\infty}\Gamma(a+t)\Gamma(b-t) 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(b-z) 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 non-positive integer where … Continue reading