Tagged Questions

Questions on the evaluation of integrals along a locus in the complex plane.

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Obatin $\int_{\gamma_1}F\cdot dl =\int_{\gamma_2}F\cdot dl$

Let $F = (F_1,F_2)$ be a $C^1$ vector field such that all its components are continuously differentiable in $\Omega$. Assume that $\frac{\partial F_1}{\partial y}=\frac{\partial F_2}{\partial x}$ Let ...
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Integration of 1/(square root of a third order polynomial) on the complex plane

I have to compute the integral $$\oint_{a,b} \frac{d \lambda}{\sqrt{(\lambda-\lambda_1)(\lambda-\lambda_2)(\lambda-\lambda_3)}},$$ Here is the picture of the integration contours and cuts: The "...
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Divergence Theorem: Conditions for the boundary integration to vanish?

Consider the Divergence Theorem for example in two dimensions, in the upper right quadrant of Euclidean space: $$\int_0^\infty dx \int_0^\infty dy ~\vec\nabla\cdot\vec F=\oint_C ds~\vec n\cdot\vec F$$...
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How to linearlize level curves at a saddle point

Let $f(x,y)$ be a real-valued function on a domain $D$ in $\mathbb{R}^2$, and let $(x_s, y_s)$ be a saddle point of $f(x,y)$ in $D$. That is to say, \begin{align} \frac{\partial f}{\partial x}(x_s, ...
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Contour integral $\int_{-\infty}^{\infty}e^{-iax}/(-b+\cos(x))\mathrm dx$ with $a>0$ and $0<b<1$

The integral is $$\text{PV}\int_{-\infty}^{\infty}\frac{e^{-iax}}{(-b+\cos(x))}\, dx$$ with $a>0$ and $0<b<1$. This integral stems from the Fourier transform of a Green's function in ...
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Evaluate $\int_{-\infty}^{\infty} \frac{(1-\cos { y } )}{\mid{y}\mid^{1+\alpha}}dy$ [closed]

How do I evaluate the following integral? $$\int_{-\infty}^{\infty} \frac{(1-\cos { y } )}{\mid{y}\mid^{1+\alpha}}dy=\frac{\pi}{\Gamma(1+\alpha)\sin(\frac{\pi\alpha}{2})}$$ Thank you in advance. ...