# Tagged Questions

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### How to calculate complex residues

How would one best calculate the residue of $$f(z)=\frac{z^2}{z^6+1}$$ At its various poles? My method is to use L'hopital to calculate $\lim_{z\to root}(z-a)f(z)$ but this is rather slow and ...
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### Complex analysis integral (maybe using residue theory?)

Trying to compute the following integral: $$\int_{-\pi}^{\pi} \frac{d\theta}{1+\sin^2(\theta)}.$$ This is for a complex analysis course, so I'm trying to find a way to use residue theory of ...
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### Residue Integral: $\int_0^\infty \frac{x^n - 2x + 1}{x^{2n} - 1} \mathrm{d}x$

Inspired by some of the greats on this site, I've been trying to improve my residue theorem skills. I've come across the integral $$\int_0^\infty \frac{x^n - 2x + 1}{x^{2n} - 1} \mathrm{d}x,$$ where ...
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### $\int_\gamma e^{\frac{1}{z^2-1}}\sin{(\pi z)}dz$ on a closed curve of index $N$ with respect to the point $1$.

Let $\gamma$ be a closed curve in the right half plane that has index $N$ with respect to the point $1$. Find $$\int_\gamma e^{\frac{1}{z^2-1}}\sin{(\pi z)}dz$$ This is a problem from an old ...
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### Proper way to set up “Pac-Man” contour integral

I'm trying to evaluate $$\int_0^\infty \frac{x^a}{1+x} \: dx, \: -1<a<0$$ using contour integrals. Actually, I have found the correct answer using a "Pac-Man" contour and residues. My only ...
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### Contour Integral: $\int^{1}_{0}\frac{1}{\sqrt[n]{1-x^n}}dx$

I want to compute: $\int^{1}_{0}\frac{1}{\sqrt[n]{1-x^n}}dx$ for natural $n>1$ using Residue Calculus. I am thinking of using some kind of a keyhole or bone contour that could go around the ...
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### Determine the nature of singularities and calculate the residue of $f(z)=\frac{e^z-\mathrm{sin}z-1}{z^5+z^3}$

$$f(z)=\frac{e^z-\mathrm{sin}z-1}{z^5+z^3},\;\;\;\;\;\;\; \mathrm{Res}[f(z),0]$$ I am having trouble determining the nature of singularities. This is what I managed to do: ...
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### Calculating $\int_0^\pi \sin^2t\;dt$ using the residue theorem

I want to use the residue theorem to calculate $$I:=\int_0^\pi \sin^2t\;dt$$ Since $\sin^2$ is an even function, we've got $$I=\frac{1}{2}\int_0^{2\pi}\sin^2t\;dt$$ The solution of this exercise ...
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### complex variable integral using residue theorem [duplicate]

I am asked to calculate a complex integral. how to compute $\displaystyle \int \limits_{-\infty}^{\infty}\frac{x^4}{1+x^8}dx$ with residue theorem?
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### holomorphic function over the disk that is real on a closed curve must be constant

Let $f$ be holomorphic on $\{z\in \mathbb{C}\mid |z|\leq 3\}$ and real on the boundary of the square $\{z\in\mathbb{C}\mid Re(z)\leq1 \text{ and } Im(z)\leq 1 \}$. Prove $f$ is constant. How to ...
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### Application of Rouché: Equality of a power series and a finite series

Let $f(z) = \sum_0^\infty{a_n z_0^n}$ be a complex power series with radius of convergence $R>0$ and let $z_0 \epsilon \, \mathcal{U}_R(0)$ an arbitrary point. I need to show with $RouchÃ©$ : For ...
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### Complex Integration: theorem of Residues

I am trying to prove a theorem that is doing my head in a bit. I have tried to simplify the problem as much as possible and leave out the details, even though it might look a bit too big. The ...
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### Evaluate by contour integration $\int_0^1\frac{dx}{(x^2-x^3)^{1/3}}$

Evaluate by contour integration [i am learning complex analysis - calculus of residues] $$\int_0^1\frac{dx}{(x^2-x^3)^{1/3}}$$ I tried by taking $x^3$ out from the denominator but that didnt work.
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### Using calculus of residues, prove that $\int z^2log[(z+1)/(z-1)]dz$

Using calculus of residues, how can it be proven that $$\int z^2\log\left[\frac{z+1}{z-1}\right]\;dz$$ taken round the circle $\left\vert z\right\vert=2$ has the value $\frac{4\pi i}{3}$?
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### Conformal mapping, Residue

I have solved the second question till $z=0$ is a simple pole... But it's residue is mentioned as $2$. How come the laurent series expansion helps in finding $a_{-1}$, is there any other way to ...
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### Understanding Poles and Zeros of a complex function

I am learning complex analysis and need help understanding and relating the concept of poles and zeros of a complex function. I understand that if a complex function $f(z)$ has an isolated ...
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### How to calculate the Residue?

I need help how this $I_r$ is calculated. It says residue but can any one help me to reach the final results.
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### A difficult integral evaluation problem

How do I compute the integration for $a>0$, $$\int_0^\pi \frac{x\sin x}{1-2a\cos x+a^2}dx?$$ I want to find a complex function and integrate by the residue theorem.
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### Quick question on infinite complex integral

I'm supposed to find: $$\int_0^{\infty} \frac{(\ln x)^2}{1+x^2} dx$$ I start of by finding: $$I_1 = \oint \frac{(\ln z)^2}{1+z^2} dz$$ $$= \oint \frac{(\ln z)^2}{(z+i)(z-i)} dz$$ Now I ...
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### Singularities of a function

Given $f(z):=\operatorname{Log}(\frac{z-2}{z-3})$, $\operatorname{Log}:\mathbb{C}\setminus\mathbb{R_{\le 0}}\to \mathbb{C}$. Is in $z_0=3$ a essential singularity of f? I'm not sure what is correct... ...
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### help with integral using residue theorem

I'm solving the following integral: $$I=\int_{-\infty}^{+\infty}\frac{x\cdot\cos x}{x^2-2x+10}\mathrm{d}x$$ I used the residue theorem as follows: ...
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### Question about Residues at Poles

I've been asked to find the residues of the following function; $$f(z) = \frac{1}{(z^2+1)(z-1)^2}$$ So, I've got that the singularities are $\pm i$ and $1$, with order 1 and 2 respectively. I was ...
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### Classification of Singularities and Residues

I'm kind of stumped on a question here. I've been asked to determine and classify the singularities of; $$f(z) = \frac{z^3}{(1+z)^3}$$ To me, it's pretty obvious that a singularity will occur when ...
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### Laurent-Series Expansion and Residues

I was asked to find a Laurent-series expansion for $f(z) = z^{-1} \sinh(z^{-1})$ about the point 0, and to classify the singularity at 0. Now, that wasn't too bad - with some manipulation of the ...
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### Computing the contour integral of $\frac{\log(z)}{z^2 +a^2}$.

I'm still a bit insecure when it comes to complex analysis and I wondered if you guys could take a look at my solution to this problem. Let $a > 0$ and define $$f(z) = \frac{\log(z)}{z^2 +a^2}$$ ...
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### How to compute residue of $f(z)=z^3e^{\frac{1}{z}}$?

How to compute residue of $$f(z)=z^3e^{\frac{1}{z}}$$ I find there is an essential singularity in $z=0$, and I have $$f(z) = \sum_0^\infty \frac{z^{3-k}}{k!}$$ but how to compute residue in 0 ?
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### what order is the pole at 0?

What order is the pole at $z=0$? $$\int\frac{\sin(3z)-3\sin(z)}{\sin(z)(\sin(z)-3)} dz$$ And do I calculate this this way: $$\frac{1}{(n-1)!} \lim_{z\to 0} \frac{d^{n-1}}{dz^{n-1}} f(z)$$ P.S. I ...
I want to prove that $$\sum_{j=1}^n \frac{1}{\left[\text{cos} \left( \frac{j \pi}{2n+1} \right)\right]^4}=\frac{8n(n+1)(n^2+n+1)}{3}$$ for $n \in \mathbb{N}$ by using the residue theorem. Which ...