# Tagged Questions

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### How to compute $\int_C {e^{3z}-z\over (z+1)^2z^2}$?

I am asked to compute the integral $$\int_C {e^{3z}-z\over (z+1)^2z^2}$$ where $C$ is a circle with the center at the origin and radius ${1 \over 2}$. My approach was to separate the integral as a ...
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### Work done by gravitational force

In my calculus class we learned about line integrals, and for homework we have exercise to find work done by gravitational force on material dot with mass $m$ which follows path of the elipse ...
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### Evaluate $\int\limits_0^\pi \frac{\sin^2x}{2-\cos x}\ \mathrm dx$ by complex methods

find integral $$\int\limits_0^\pi \frac{\sin^2x}{2-\cos x}dx$$ what I had in mind is to use Euler formula, to turn it into a complex integral and change the limits of integration from $-\pi$ to ...
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### Complex integral square

Let $\alpha$ be the closed curve along the square with vertices at $1, i, -1, -i$. Give an explicit parametrization for $\alpha$ and calculate $$\frac{1}{2\pi i}\int_\alpha\frac{dz}{z}$$ I ...
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### Residues of Complex Functions

I need to find the residues of $f$ at the isolated singular points, namely $z=1,z=0$. Where $f(z)=\dfrac{2z+1}{z(z+1)}$. I already have that the residue at $z=0$ is $1$, and I know I need to do ...
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### Calculating ${\int_{-\infty}^{\infty} \frac{\cos(\omega x)}{x^{2} + 25}\,{\rm d}x}$ using contour integration

I want to calculate $\displaystyle{% \int_{-\infty}^{\infty}\frac{\cos\left(\omega x\right)}{x^{2} + 25}\,{\rm d}x\,, \quad}$ for $\omega \in \mathbb{R}$ I thought of integrating along the line ...
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### Inverse FT of $Z(\omega) = a [- \frac{1}{i\omega}+\pi \delta(\omega)]$ (Contour integration)

Given is the Fourier transform of some function $z(t)$: $Z(\omega) = a [- \frac{1}{i\omega}+\pi \delta(\omega)]$ I now want to invert the tranform using contour integrals. How can I do that? I ...
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### Show that $\int_0^\infty e^{-x^2} \sin{2xb}\, dx =e^{-b^2}\int_0^b e^{x^2} \, dx, \, b>0,$?

Show that $$\int_0^\infty e^{-x^2} \sin{2xb}\, dx =e^{-b^2}\int_0^b e^{x^2} \, dx, \, b>0,$$ I need help. I did the following steps: Apply Cauchy's Theorem, being $\varphi (x) = e^{-z^2}$ analytic ...
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### The solution of the contour integral for $\epsilon =+1$

I understand the solution for $\epsilon =-1$. And I am trying the solve this question for $\epsilon =+1$. This is important for me. I want really to learn perfectly because I am continuously seein' ...
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### Integral of $z^{n} \log z$ on the unit circle under two assumptions

I'm asked to calculate $\int_{|z| = 1} z^{n} \log z dz$ in two ways: (1) if $\log 1 = 0$; (2) if $\log (-1) = i \pi$. I understand it means that in case (1) I have to work with the principal ...
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### Contour integration using Cauchy's integral formula

I need to show that $$\int_{-\infty}^\infty\frac{\sin^2(x)}{x^2+1}dx=\frac{\pi}{2}\left(1-\frac{1}{e^2}\right)$$ but I don't really know why I'm not getting the result using contour integration ...
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### Complex contour integrations

Consider the appropriate contour integral (circle $\oint=e^z$, show that $$\int^{2\pi}_{0}e^{cos\theta}cos(sin\theta +\theta)d\theta = 0$$ A more thorough explanation would be for the better.
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