# Questions tagged [residue-calculus]

Questions on the evaluation of residues, on the evaluation of integrals using the method of residues or in the method's theory.

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### Integral Residue Calculation

Integral: $I =\int^{\infty}_{-\infty}\frac{\cos x-1}{x^2(x^2+a^2)}\mathrm dx$ where $a \in \mathbb{R}$ and $a > 0$. Method 1: We observe that there is a removable singularity at $x=0$. Thus a ...
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### (Fake proof) Counterclockwise contour integral of identity function around unit circle is $-2\pi i$

First, the result is obviously false by Cauchy's integral formula, given that the identity function is one of the simplest analytic functions and has no singularities. So the contour integral is zero. ...
1 vote
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### Partial fractions trick, repeated roots [closed]

Do you know how can one extend this trick to find partial fractions coefficients when the roots of the denominator are repeated? From now, I'm just interested in the cases when the roots are algebraic....
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### Poles of $f(z)$ when $f(z)$ has a zero and a singularity at the same point

I want to determine the poles, and their orders, of $f(z)$.1) $$f(z) = \frac{1+e^{i\pi z}}{(z-1)^2(z+1)^2}$$ The solution says that $f(z)$ has two simple poles at $z = +1$ and $z= -1$, but to me ...
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### Closed form of $\int_{-\infty}^\infty \frac{\sin (\pi x)}{(x^2 - 7x + 10)(x^2 + 1)} dx$

Find the closed form for the integral $$\mathcal I= \int_{-\infty}^\infty \frac{\sin (\pi x)}{(x^2 - 7x + 10)(x^2 + 1)} dx$$ My attempt In order to solve this integral, what I first consider is using ...
1 vote
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### Closed form for $\int_{-\infty}^\infty \frac{(x+1)\sin (3x)}{(x^2 + 6x + 10)^2} dx$

Find the closed form for $$\mathcal{I} = \int_{-\infty}^\infty \frac{(x+1)\sin (3x)}{(x^2 + 6x + 10)^2} dx$$ My attempt In order to find the closed form for this integral, what I first thought was ...
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### Complex integration - showing that the arc integral vanishes

I came across the following integral: $$\int_{- \infty}^{\infty} \frac{x^2}{((x-t)^2 + \delta^2 )^2((x + t)^2 + \delta^2)^2} \textrm{d}x.$$ I understand that if we turn this into a complex integral ...
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### Computing the residue of given function and appliance of residue theorem

Given the function: \begin{align*} f:\mathbb{C}\setminus\left\{i,i+\frac{1}{\pi} \right\} \to \mathbb{C}, \quad z\mapsto \cos\left(\frac{1}{z-i}\right) \cdot \frac{1}{z-i-\frac{1}{\pi}} \end{align*} 1....
1 vote
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### How do I change the Integral?

I found this other thread: Calculation of Complex Integral using residue theorem Now I wonder, how I can get from the Integral over $[0,2 \pi]$ to the circle Integral over Gamma. I understood, that ...
1 vote
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### Determining the residue of a function by computing its laurent expansion

I need to find the residue of $$f(z) = \frac{e^z\sin(z)}{z(1-\cos(z))}$$ at $z = 0$ via its Laurent series expansion. First of all, I tried to expand all the functions via Taylor series at $z = 0$. I ...
Consider a triangle on the complex plane, these three points are $z=1,\omega,\bar{\omega}$, here $\omega=e^{\frac{2}{3}\pi i}$. The boundary of triangle is $\partial T$, whose direction is ...