Tag Archives: analysis

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

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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

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Integrating a function around three branches using a semi-circle contour

[Ex] Watson’s complex integration $$\int^{\pi/2}_{0}\cos(nt)\cos^m(t)\,dt=\frac{\pi \Gamma(m+1)}{2^{m+1}\Gamma\left(\frac{n+m+2}{2}\right)\Gamma\left(\frac{2-n+m}{2}\right)}$$ $$\textit {solution}$$ Let us integrate the following function $$f(z) = z^{n-m-1}\left(1+z^2\right)^m$$ We choose the principle logarithm where $$\log(z) = \log|z|+\mathrm{Arg}(z)$$ Note that the function \(z^{n-m-1} = e^{(n-m-1)\log(z)}\) will have a branch cut on the … Continue reading

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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 \(x= t-ai\) $$\int^{-L+ai}_{L+ai}e^{-t^2}\,dt=\int^{-L}_{L}e^{-(x+ai)^2}\,dx=-e^{a^2}\int^{L}_{-L}e^{-x^2}\,e^{-2iax}dx$$ Take … Continue reading

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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 $$f(z)=e^{z^{-1}}z^{n-1}$$ Now we integrate the function along a circle of radius 1 The contour encloses a pole at \(z = 0\) $$\oint_{|z|=1}e^{z^{-1}}z^{n-1} dz=2\pi i\mathrm{Res}(f(z),0) $$ Now we … Continue reading

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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 $$f(z) = \frac{\log(a-ibz)}{c^2+d^2z^2}$$ 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

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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 non-negative integers By integration around a large circle \( |z| = \rho \) Note that $$\oint f(z)\,dz = 2\pi … Continue reading

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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

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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 multi-valued defined as $$\log(z) = \ln|z|+i\theta +2k\pi i$$ This maps the complex plain more than once … Continue reading

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