# Tagged Questions

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

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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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### Certain type of integrals $\int_0^{\pi}d\theta\sin\theta \frac{1}{x+i\epsilon - \sqrt{y+z\cos\theta}},$

I would like to do the following integral $$\int_0^{\pi}d\theta\sin\theta \frac{1}{x+i\epsilon - \sqrt{y+z\cos\theta}},$$ where the $i\epsilon$ has been added to avoid some possible divergencies. ...
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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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### What kind of contour can be used for evaluating this integral?

I have the following integral: $\displaystyle\int\limits_{0}^\infty\frac{\arctan(\eta_2+x)}{(x+\eta_1)^2+\eta_3^2}\ d x$ where $\eta_1$, $\eta_2$ and $\eta_3$ are real numbers. If the integrand were ...
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### Evaluating $\int_0^{2\pi} e^{2i\theta} f(e^{i\theta})\,d\theta$ using Cauchy's Theorem

If $f(z)$ is analytic on the disk $|z| \leq 2$, evaluate $$\int_0^{2\pi} e^{2i\theta} f(e^{i\theta})\,d\theta.$$ This problem comes from Mathematical Methods in the Physical Sciences 2nd edition by ...
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### Approaching a contour integral with singularities on each axis

How do I solve an integral like this using complex methods? $$\int_{0}^{\infty} \frac{\ln(x)}{\left(x^2 + 2\right)\left(x^2 + 1\right)}dx.$$ I tried using two semi circles in the upper half plane ...
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### Calculate $\int_{\gamma}zdz$ by using Cauchy's integral formula

How can I calculate $$\int_{\gamma}zdz~\text{with }\gamma:[0,1]\rightarrow\mathbb{C},t\mapsto te^{2\pi i t}$$ by using Cauchy's integral formula? The line $\gamma$ isn't even closed. Has anyone a hint?...
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### Extend $g(z)$ holomorphically from $\mathrm{Re}(z) > 0$ to $\mathrm{Re}(z) \geq 0$.

If I have holomorphic function $g(z)$ defined on $\mathrm{Re}(z) > 0$ and I extend in a holomorphic way to $\mathrm{Re}(z) \geq 0$, how do I get $g(0)$ purely in terms of stuff in the right half ...
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### Solving an improper integral contour integral, calculated via Wolfram but in need of analytic derivation possibly

In my studies of dynamical systems I have just encountered this supposedly tough looking improper integral, which is (not really relevant for my predicament) the Melnikov function, with the integral ...
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### A Contour Integral

I'm interested in computing the integral: $$- \frac{1}{2 \pi} \int_{- \infty}^{\infty} dE \; \frac{e^{-iEt}}{E^2 - \omega^2 + i\epsilon}.$$ I have two small queries: How does one choose the ...
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### Dumbbell Contour? $\int_0^1 \log(x)\log(1-x)dx$ via complex methods.

Having evaluated this integral via the power series and various approaches via special functions, I'm now curious if there is a direct way to compute this integral by taking a slit along $[0,1]$ and ...
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### Very tricky complex integral, with poles on both sides of the real line,

I am trying to evaluate$$\int_{-\infty}^{\infty} \frac {x^2 -x^4}{1-x^6}\,dx,$$ which is an old exam problem. There is a special note on this problem that reads: Note: Your answer need not be a ...
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### Let C be the circle $|z|=1$. Evaluate $\int_{c}\frac{e^{2\pi z}}{(2z+1)^3}dz.$

Let C be the circle $|z|=1$. Evaluate $$\int_{c}\frac{e^{2\pi z}}{(2z+1)^3}dz.$$ Any idea, suggestion, advice or solution.
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### Help, why these are two different results of integral of $\sqrt{z}$ on unit circle depending the choice of Branch cut

everyone, I want test the effect of different choice of branch cut for contour, So I find a simple function, i.e. $\sqrt{z}$ with $z=re^{i\theta}$ on 1st Branch as $$I=\oint_{UnitCircle}{\sqrt{z}dz}$$ ...
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### contour integral z/conjugate(z)

I am trying to calculate: $$\int_C \frac{z}{\bar{z}}dz$$ where C is the upper semicircles of the circles centred in (0,0) of radii 1,2 joined at their intersections with the real axis by the real ...
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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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### Evaluating $\int\limits_0^\infty \! \frac{x^{1/n}}{1+x^2} \ \mathrm{d}x$

I've been trying to evaluate the following integral from the 2011 Harvard PhD Qualifying Exam. For all $n\in\mathbb{N}^+$ in general: $$\int\limits_0^\infty \! \frac{x^{1/n}}{1+x^2} \ \mathrm{d}x$$ ...
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### Use the Residue Theorem to evaluate the integral:

$$\int_{0}^{∞} \frac{\sqrt{x}}{x^2+2x+5} dx$$ I'm thinking of using the "keyhole" contour, but I'm not sure how to proceed from there. Please help! Thanks!
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### A variation of Ahmed's integral

Given that the closed form exist, evaluate the following Integral: $$\displaystyle \int\limits_{0}^{1} \frac{(x^2+4)\sin^{-1}x}{x^4-12x^2+16} \, dx$$
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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. ...
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### Real integral using a contour integral

I am going to calculate $\int_{-\infty}^{\infty}\dfrac{x \sin \pi x}{x^2+2x+5}dx$ So I have to compute the following limit $\lim_{R \to \infty}\int_{C_1}\dfrac{z \sin \pi z}{z^2+2z+5}dz$ where $C_1$ ...
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Question.Let $Ω$ be a bounded Connected on $R^3$ with smooth boundary $\partial{Ω}$.Let $u$ be a harmonic function on $Ω$ with continuous derivatives on $Ω\cup \partial{Ω}$ prove that. \iint_V \ {\...