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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: residue
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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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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Proving a trigonometric integral by integrating around an ellipse in the complex plain
Prove for \(a,b > 0\) $$\int^{2\pi}_0\frac{dt}{a^2\cos^2 t +b^2\sin^2 t} = \frac{2\pi}{ab}$$ $$\textit{solution}$$ Let us integrate the following function $$f(z) = \frac{1}{z}$$ Around the ellipse $$\oint_{\gamma}f(z)\,dz =2\pi i\,\mathrm{Res}(f,0)$$ The parametrization of the ellipse \(\gamma(t) = a\cos(t)+ib\sin(t)\) $$\oint_{\gamma}f(z)\,dz=\int^{2\pi}_0 \frac{a\sin t+ib\cos t}{a\cos t+ib\sin t} … Continue reading
Posted in Contour Integration
Tagged contour, ellipse, residue, theorem, trigonometric
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Integrating a fraction of exponential and trignometric using rectangular contour
[Ex 41 ] Watson’s complex integration $$\int^\infty_0 \frac{\sin(ax)}{e^{2\pi x}1}\,dx = \frac{1}{4}\coth\left(\frac{a}{2} \right)\frac{1}{2a}$$ $$\textit{solution}$$ By integrating the following function $$f(z) = \frac{e^{iaz}}{e^{2\pi z}1}$$ The function is analytic in and on the contour, indented at the poles of the function Hence by … Continue reading
Posted in Contour Integration
Tagged analysis, complex, contour, exponent, fraction, rectangle, residue, theorem, trignometric
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Integrating around a triangular contour for Fresnel integral
$$\int^\infty_0\frac{\cos(x)}{\sqrt{x}}\,dx =\int^\infty_0\frac{\sin(x)}{\sqrt{x}}\,dx =\sqrt{\frac{\pi}{2}}$$ $$\textit{solution}$$ Consider the following function $$f(z)=z^{1/2}\,e^{iz}$$ Where we choose the principle root for \( z^{1/2}=e^{1/2\log(z)}\). By integrating around the following contour $$\int_{C_r}f(z)\,dz+\int_{r}^R f(x)\,dx+\int_{\gamma}f(z)\,dz+\int^{iR}_{ir}f(x)\,dx = 0$$ Taking the integral around the small quarter circle with $r\to 0$ $$\left … Continue reading
Posted in Contour Integration
Tagged analysis, complex, Fresnel, Integral, residue, theorem, triangle
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Integration related to gamma function using rectangle contour
[Ex 9] Watson’s complex integration $$\int^{\infty}_{\infty}e^{x^2}\,\cos(2ax)dx=e^{a^2}\sqrt{\pi}$$ $$proof$$ Integrate the following function $$f(z) = e^{z^2}$$ Use the following contour Note that the function is entire, hence $$\int^{L}_{L} e^{t^2}\,dt+\int^{L+ai}_{L} e^{t^2}\,dt+\int^{L}_{L+ai} e^{t^2}\,dt+\int^{L+ai}_{L+ai}e^{t^2}\,dt=0$$ For the forth integral use the substitution \(x= tai\) $$\int^{L+ai}_{L+ai}e^{t^2}\,dt=\int^{L}_{L}e^{(x+ai)^2}\,dx=e^{a^2}\int^{L}_{L}e^{x^2}\,e^{2iax}dx$$ Take … Continue reading
Posted in Contour Integration, Gamma function
Tagged analysis, complex, contour, Dawson, Gamma, Integral, rectangle, residue
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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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Contour method for shifted logarithm branch
Prove \( a,b,c,d >0 \) $$\int^\infty_0 \frac{\log(a^2+b^2x^2)}{c^2+d^2x^2}\,dx = \frac{\pi}{cd} \log \frac{ad+bc}{d}$$ Consider the function $$f(z) = \frac{\log(aibz)}{c^2+d^2z^2}$$ We need the logarithm with the branch cut \( y<\frac{a}{b} , x =0 \) . Note that this corresponds to $$\log(a+ibz) = \log\sqrt{(a+y)^2+b^2x^2}+i\theta … Continue reading
Posted in Contour Integration
Tagged analysis, branch, complex, logarithm, rational, residue, shifted, theorem
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Solving Euler sums using Contour integration
Prove that $$\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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