# Tagged Questions

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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### Argument principle for meromorphic forms on Riemann surfaces

Let $X$ be a compact Riemann surface and $D \subset X$ a compact domain with boundary $\partial D$. Let $\omega$ be a meromorphic $1$-form in a neighborhood of $D$ which does not have neither zeros, ...
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### Laurent expansion - Faster technique

I'm currently preparing for an exam in complex analysis. There is a type of exercise, where I need to compute Laurent expansions about different places. However, my ...
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### Integral Calculus with non integer power using Cauchy integral's theorem or Residue theorem

I want to calculate the following integral: $$I(\beta)=\int_0^\infty \frac{\sin(x)}{ x^{\beta+1}}dx$$ for $0 \leqslant\beta<1$ I used comparison test for improper integrals to be sure that these ...
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### Proving that an infinite series equal a finite series

Suppose we have a function $f(z)$, which has $m$ isolated singularities, which are non-integers (say, $z_1$, $z_2$,...,$z_m$). Define $H(z):=\frac{\pi f(z)}{\sin(\pi z)}$. Assume that there exists a ...
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### Residue at all integers of complex function involving sines

Given $H(z)=\frac{\pi f(z)}{\sin(\pi z)}$, where $f(z)$ is some function which has isolated singularities only at non-integers, is it correct to calculate its residue at $z=k$ (for $k\in\mathbb{Z}$) ...
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### calculating curvilinear integral by residue theorem

Calculate the following integral by transposing to a curve integral and then using the residue theorem: $\displaystyle \int_{0}^{2\pi}{\frac{e^{int}}{C-e^{it}}dt}, \qquad |C|\ne1, n\in \mathbb N$. ...
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### residue theorem, integral of inverse quadratic not working

$$\int_{-\infty}^\infty\frac{dx}{3x^2+0.4x+10}$$ roots of $3x^2+0.4x+10$ $= z_1,z_2=-\frac{1}{15}\pm 1.825i$ Using $\mathrm{res} = \frac{1}{z_1-z_2}=-0.274i$ ans: $2\pi i\cdot -0.274i=1.72$ the ...
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### Evaluate $\int _{ 0 }^{ \infty }{ \frac { { x }^{ n }-1 }{ \ln { x }}} dx$ using residue theorem.

$$\int _{ 0 }^{ \infty }{ \frac { { x }^{ n }-1 }{ \ln { x }}} dx$$ I couldn't solve this problem using the residue theorem. Can anyone help me get the answer? I know the steps like taking the ...
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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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### How do i prove that $Res(f(z)e^\frac{1}{z};0)=\sum_{n=0}^\infty \frac{a_n}{(n+1)!}$ with $f(z)=\sum_{n=0}^\infty a_nz^n$

$f(z)=\sum_{n=0}^\infty a_nz^n$ (around $0$) I need to prove that $Res(f(z)e^\frac{1}{z};0)=\sum_{n=0}^\infty \frac{a_n}{(n+1)!}$ I know that $Res(z^ne^\frac{1}{z},0)=\frac{1}{(n+1)!}$ but I don'...
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### Closed form for $\int_0^1 d u \, \frac{1}{u + \lambda} \ln \left(\frac{1 + u}{1 - u} \right)$

The parameter $\lambda$ is complex and it's not on the real axis. There are some similar cases: Help me evaluate $\int_0^1 \frac{\log(x+1)}{1+x^2} dx$ Evaluate $\int_0^1 \frac{\ln(1+bx)}{1+x} dx$ ...
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### How do i prove that $Res(F_n,0)=\frac{1}{(n+1)!}$ with $F_n(z)= z^ne^\frac1z$

I have $F_n(z)= z^ne^\frac1z$ and i've to prove that $Res(F_n,0)=\frac{1}{(n+1)!}$ And i know that $\sum_{n=0}^\infty \frac{1}{n!} \frac{1}{z^n} = e^\frac{1}{z}$ but i don't know how to procede ...
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### Trying to solve improper integral

I've been trying to solve this $$\int_{-\infty}^\infty {\sin(x)\over x+1-i }dx$$ using residue theorem. I've tried using a square contour pi, pi+pii, -pi+pii, pi and half a circle but with the ...
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### Improper integral complex analysis $\int_{-\infty}^\infty \frac{e^{ax} \, dx}{\cosh(x)}$

I tried the following problem but I don't think I got the right answer. I checked it by substituting $a=\frac{1}{2}$ into the integral and putting that through Wolfram Alpha but it didn't match the ...
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### How to find $L = \int_0^1 \frac{dx}{1+{x^8}}$

Let $L = \displaystyle \int_0^1 \frac{dx}{1+{x^8}}$ . Then $L < 1$ $L > 1$ $L < \frac{\pi}{4}$ $L > \frac{\pi}{4}$ I got some idea from this video link. But got stuck while evaluating ...
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### Solution Ahlfors 4.5.3 3(f) [closed]

Using the Redidue Theorem, find $$\int_0^\infty \frac{x \text{sin}(x)}{x^2+a^2} dx$$ I don't know how to proceed.
### Evaluation of the principal value of $\int\limits_{-\infty}^\infty \frac{\sin 2x}{x^3} \, dx$
I'm trying to evaluate an integral $\int\limits_{-\infty}^\infty \frac{\sin 2x}{x^3}\,dx$ using Cauchy's theorem. Considering an integral from $-R$ to $-\epsilon$, then a semicircular indentation ...