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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: complex
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 $r\to 0$ $$\left … Continue reading
Posted in Contour Integration
Tagged analysis, complex, Fresnel, Integral, residue, theorem, triangle
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Integrating a function around three branches using a semicircle contour
[Ex] Watson’s complex integration $$\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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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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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 $$\oint f(z)\,dz = 2\pi … Continue reading
Posted in Contour Integration, Euler sum
Tagged analysis, complex, contour, Euler, integration, residue, sums
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Integral of rational function with cosine hyperbolic function using rectangle contour
$$ \int^{\infty}_{\infty} \frac{1}{(5 \pi^2 + 8 \pi y + 16y^2) }\frac{\cosh\left(y+\frac{\pi}{4} \right)}{\cosh^3(y)}dy=\frac{2}{\pi^3}\left(\pi \cosh\left(\frac{\pi}{4} \right)4\sinh\left( \frac{\pi}{4}\right) \right)$$ $$\textit{proof}$$ Consider $$f(z) = \frac{\sinh(z)}{z \sinh^3(z\pi/4)}$$ If we integrate around a contour of height \( \pi \) and stretch it to infinity we … Continue reading
Posted in Contour Integration
Tagged analysis, complex, cosine, hyperboic, rectangle
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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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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