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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 function around three branches using a semicircle contour
[Ex] Watson’s complex integration $$\int^{\pi/2}_{0}\cos(nt)\cos^m(t)\,dt=\frac{\pi \Gamma(m+1)}{2^{m+1}\Gamma\left(\frac{n+m+2}{2}\right)\Gamma\left(\frac{2n+m}{2}\right)}$$ $$\textit {solution}$$ Let us integrate the following function $$f(z) = z^{nm1}\left(1+z^2\right)^m$$ We choose the principle logarithm where $$\log(z) = \logz+\mathrm{Arg}(z)$$ Note that the function \(z^{nm1} = e^{(nm1)\log(z)}\) will have a branch cut on the … Continue reading
Posted in Beta function, Contour Integration
Tagged analysis, circle, complex, contour, halfcircle, integration, semi
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Integral of arctan and log using contour integration
$$\int^\infty_0\frac{\log\left(x^2+1 \right)\arctan^2\left(x\right)}{x^2}\,dx = \frac{\pi^3}{12}+\pi \log^2(2)$$ Lemma $$\int^\infty_0 \frac{\log^3(1 + x^2)}{x^2}\,dx = \pi^3+ 3 \pi \log^2(4)$$ Start by the following $$\int^{\infty}_0 x^{p}(1+x)^{s1} dx= \frac{\Gamma(1p)\Gamma(ps)}{\Gamma(1s)}$$ Let \( x\to x^2 \) $$\int^{\infty}_0 x^{2p+1}(1+x^2)^{s1} dx= \frac{\Gamma(1p)\Gamma(ps)}{2\Gamma(1s)}$$ Let \( p = 3/2 \) $$\int^{\infty}_0 \frac{1}{x^2(1+x^2)^{1s}} … Continue reading
Posted in Beta function, Contour Integration
Tagged arctan, branch, contour, cut, halfcircle, integration, logarithm, residue, theorem
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Integrating along the unit circle
Prove that $$\int^{2\pi}_0e^{\cos \theta}\cos(n\theta \sin \theta)\,d \theta=\frac{2\pi}{n!}$$ $$\textit{proof}$$ Consider the following function $$f(z)=e^{z^{1}}z^{n1}$$ Now we integrate the function along a circle of radius 1 The contour encloses a pole at \(z = 0\) $$\oint_{z=1}e^{z^{1}}z^{n1} dz=2\pi i\mathrm{Res}(f(z),0) $$ Now we … Continue reading
Posted in Contour Integration
Tagged analysis, circle, complex, contour, integration, residue, theorem, unit
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Solving Euler sums using Contour integration
Prove that $$\sum_{n=1}^\infty \frac{H_n}{n^2} = 2\zeta(3)$$ $$\textit{proof}$$ Consider the function $$f(z) = \frac{(\psi(z)+\gamma)^2}{z^2}$$ Note that \( f \) has poles at nonnegative integers By integration around a large circle \( z = \rho \) Note that $$\oint f(z)\,dz = 2\pi … Continue reading
Posted in Contour Integration, Euler sum
Tagged analysis, complex, contour, Euler, integration, residue, sums
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Bromwich contour integration of the gamma function
$$\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