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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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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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The most ugly looking integral
Prove the following $$I= \log \left\{\frac{\Gamma(b+c+1) \Gamma(c+a+1)\Gamma(a+b+1)}{\Gamma(a+1) \Gamma(b+1) \Gamma(c+1) \Gamma(a+b+c+1)} \right\}$$ where $$I = \int_0^1 \frac{(1x^a)(1x^b)(1x^c)}{(1x)(\log x)}dx$$ $$\textit{proof}$$ First note that since there is a log in the denominator that gives as an idea to use differentiation under … Continue reading
Posted in Digamma function, Gamma function
Tagged difficult, Digamma, Gamma, Hard, Integral, proof
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