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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Residue at Infinity Question

So, I'm trying to determine a residue of the following at infinity: $$f(z)=\frac{z^6}{(1+z)^3}$$ How would I even begin to approach this? Here are my thoughts; I'm mostly looking for a confirmation ...
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1answer
66 views

Residue integration

Evaluate\begin{align} \int_{-\infty}^{\infty} \frac{\sin(x)}{x^2+4x+5} \, dx \end{align} I set $f(z) = \frac{1}{z^2 + 4z + 5} = \frac{1}{(z-z_0)(z-\bar{z_o})}$, where $z_0 = -2+i$ and $\bar{z_0} ...
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3answers
51 views

How to calculate $I(x)=\int_{-1}^1 \frac{dt}{\sqrt{1-t^2}(t-x)}$ by Residue theorem [$|x|>1$]

How to calculate $\int_{-1}^1 \frac{dt}{\sqrt{1-t^2}(t-x)}$ for $|x|>1$ by Residue theorem? I could do is just as: $$I(x)=\int_{-\pi/2}^{\pi/2}\frac{d\theta}{\sin\theta-x}\\ ...
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1answer
89 views

Using Jordan's Lemma for residue integration

Evaluate \begin{align} \int_{0}^{\infty} \frac{\cos(ax)}{x^2+1} \, dx \end{align} So far, I set $f(z) = \frac{1}{z^2+1}$, and the singularity point are $z = \pm i$. But I am using the top ...
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3answers
544 views

Using residues to evaluate an improper integral

Use residues to evaluate the improper integral \begin{align} \int_{0}^{\infty} \frac{1}{(x^2+1)^2} \, dx \end{align} First, I said $f(z) = \frac{1}{(z^2+1)^2}$. My only question so far is how ...
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1answer
46 views

Finding a residue

I am given $f(z) = \frac{\text{Log} z}{(z^2 + 1)^2}$. The singularity point occurs at $z = i$, with a pole order of $m = 2$. How can I go about finding $\text{Res}_{z=i} f(z)$? The correct answer ...
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42 views

Polynomial Expansion in a Complex Improper Integral

I want to evaluate the integral below by using the Residue theorem. $$\int_{-\infty}^{\infty}\frac{\exp(-iwt)w}{(w-ic)\sqrt{w^2-aw+b}}dw$$ There are branch cut points due to the square rooted term ...
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3answers
330 views

Need help with $\int_0^\infty\frac{\log(1+x)}{\left(1+x^2\right)\,\left(1+x^3\right)}dx$

I need you help with this integral: $$\int_0^\infty\frac{\log(1+x)}{\left(1+x^2\right)\,\left(1+x^3\right)}dx.$$ Mathematica says it does not converge, which is apparently false.
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52 views

Integration by Residue Theorem - Is This Integral equal to Zero?

$$ \int_{-\infty}^{+\infty}\frac{\sin(x)}{2x^{5}-3jx^{3}+2x}dx=0 $$ ...
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2answers
57 views

Calculate the integral $\int_{\varGamma}\frac{3e^{z}}{1-e^{z}}dz$

I am looking to solve $$\int_{\varGamma}\frac{3e^{z}}{1-e^{z}}dz,$$ where $\varGamma$ is the contour $|z|=4\pi/3$. We have been asked first to consider $e^{z}=1$ and $e^{z}=-1$ which I get to be ...
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0answers
33 views

Integration of a complex function having square rooted denominator

How to evaluate this integral ? $$\int_{-\infty}^{\infty}\frac{1}{(w-ia)\sqrt{w^2-ibw+b^2c}}dw$$ where $a$, $b$ and $c$ are real and greater than zero. Does the residue theorem work on it? Please ...
7
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3answers
171 views

How prove this sum$\sum_{k=0}^{\infty}\frac{(-1)^{\frac{k(k+1)}{2}}}{(2k+1)^2}=\frac{\sqrt{2}\pi^2}{16}$

How show that $$\sum_{k=0}^{\infty}\dfrac{(-1)^{\frac{k(k+1)}{2}}}{(2k+1)^2}=\dfrac{\sqrt{2}\pi^2}{16}$$ My idea:I know how to prove the following sum ...
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1answer
46 views

Computing $\int_0^{2\pi}(1+2\cos t)^n\cos nt\ \mathrm{d}t$

I'd like to calculate the following integral on the interval $[0,2\pi]$: $$ I=\int_0^{2\pi}(1+2\cos t)^n\cos nt\ \mathrm{d}t = 2\pi. $$
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2answers
174 views

Integral $\int_{0}^{2\pi}\log|e^{i \theta}-1|d \theta$

Consider $$\int_{0}^{2\pi}\log|e^{i \theta}-1|d \theta$$ Is it equal to $0$ ? Why ? Any hint ?
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1answer
122 views

Using Cauchy integral formula to calculate $\int_\gamma \frac{\cos{z}}{z^n}$

Let $\gamma(\vartheta)=\mathrm{e}^{i\vartheta},\,\vartheta\in[0,2\pi]$, and consider the integral $$I(n)=\int_\gamma \frac{\cos{z}}{z^n},$$ where $n\in \{0,2,4,6,...\}$. Is there any way to prove ...
4
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0answers
62 views

Can we use a sum of residues to develop an asymptotic expansion for this unknown function?

In the course of solving a particular physical problem, I have derived a relationship between two unknown functions: $$ f(s) = \frac{s \sinh{\frac{\pi s}{2}}}{2 \pi i \beta} \int_{-c- i ...
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0answers
16 views

A question on Residues and a family of functions

I have the following: $$\alpha(t)=\int_{\left|z\right|=1}z^{k}\frac{f^{\prime}_{t}(z)}{f_{t}(z)}\,dz$$ where $f_{t}(z)$ is a family of entire functions depending on $t \in \Delta$ and $k \geq 0$. By ...
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3answers
77 views

Integral $\int_{0}^{\infty}e^{-ax}\cos (bx)\operatorname d\!x$

I want to evaluate the following integral via complex analysis $$\int\limits_{x=0}^{x=\infty}e^{-ax}\cos (bx)\operatorname d\!x \ \ ,\ \ a >0$$ Which function/ contour should I consider ?
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1answer
62 views

Evaluating a few complex integrals on the unit circle

I'm stuck on a few of these, but I have most of the details worked out: (i) $\int_{|r|=1}(z^2-4)^{-1}\,dt=\int_{0}^{2\pi}ie^{i\theta}(e^{2i\theta}-2)^{-1}\,d\theta$ (ii) ...
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0answers
42 views

Integrating $\oint_\Gamma \cos(\log|z|)\cosh(\text{Arg}(z))\text{Arg}(z)e^{is(z-1)}dz$ using residue calculus.

I'm trying to use the residue calculus to evaluate $$\oint_\Gamma \cos(\log|z|)\cosh(\text{Arg}(z))\text{Arg}(z)e^{is(z-1)}dz,$$ where $s>0$, and where $\text{Arg}$ is the principal argument, ...
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3answers
123 views

Confusion about a way to compute a residue at a pole

Suppose I have a function of the form $$f(x)=\frac{1}{(x-a)(x-b)^2(x-c)^3}$$ Clearly, I have a simple pole at $a$, and poles of order 2,3 at $b,c$, respectively. By definition, the residue at ...
2
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1answer
52 views

$\sum_{i=1}^{n}\operatorname{Res}(f,z_{i}) + \operatorname{Res}(f,\infty) = 0$

Let $f \in H(\mathbb{C}- \{ z_{1}, \dots, z_{n} \})$. I need a proof of the fact that $$\sum_{i=1}^{n}\operatorname{Res}(f,z_{i}) + \operatorname{Res}(f,\infty) = 0.$$ Where can I find it ?
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1answer
261 views

What is the residue of $f(z)=\tan{z}$ at any of its pole ? Is the solution correct?

The residue of $$f(z)=\tan{z}$$ at any of its pole is, $$f(z)=\tan{z}=\frac{(z-\frac{\pi}{2})(\tan{z})}{(z-\frac{\pi}{2})}$$ $$\begin{align} \left({\operatorname{Res} {f(z)=\tan{z}; ...
0
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2answers
172 views

Prove a function has a removable singularity at $z=0$.

Let $f$ be a holomorphic function on $\mathbb{C}\smallsetminus \{0\}$. Suppose $\int_{|z|=1}z^nf(z)\,dz=0$ for any $n=0,1,2,\ldots$. Prove that $f$ has a removable singularity at $z=0$. How to prove? ...
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1answer
52 views

Find the residues of $f(z) = \left( \frac{z-1}{z+1}\right)^{\frac{1}{2}}\frac{1+z} {1+z^{2}}$

Consider the function $$f(z) = \left( \frac{z-1}{z+1}\right)^{\!\frac{1}{2}}\frac{1+z} {1+z^{2}}$$ I want to calculate the residues of $f$ in $\{+i,-i\}$. Using the usual techniques, we have that ...
3
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2answers
62 views

Using residue to find a complex integral

Given the following: $$\int_{\varGamma_R} {z\,dz\over e^{2\pi iz^2}-1}, \ \ \ \varGamma_R=\{z\in \Bbb C:|z|=R\},\quad n<R^2<n+1,\,n\in\Bbb N.$$ I want to use the residue for this, but I can't ...
4
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1answer
159 views

Integral Using Harmonic Functions

Evaluate the integral: $$\int^{2 \pi}_0 \dfrac{\cos^2 \theta}{|2e^{i\theta}-z|^2} \, d \theta \qquad \mbox {when} \, |z| \neq 2.$$ Now, I thought about trying to change this to look like a Poisson ...
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2answers
94 views

About the integral $\int_{0}^{+\infty}\frac{\sin(ax)\,dx}{x(x^2+1)}$

I need to prove the following identity: $$\forall a>0,\qquad\int \limits_{-\infty}^{+\infty}\frac{\sin(ax)\,dx}{x(x^2+1)}=\pi(1-e^{-a}).$$ I think it can be proven using Laurent series. I tried ...
4
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1answer
56 views

Quarternionic Analysis

What is/are the current understanding/opinions about Quarternionic Analysis as a generalization of Complex Analysis with respect to a "Quarternionic Residue Calculus" (if such a thing exists)? i.e. ...
14
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0answers
370 views

Integrating $\int_0^\infty\frac{\log (1+z^2)}{e^z-1}dz$ using residue calculus.

I've been looking at how to integrate the following definite integral using the residue calculus, but can't seem to get my thoughts together. I know the $\log$ term is a multivalued function and the ...
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2answers
70 views

$\int_{0}^{\infty}\frac{\cos2\pi x}{x^4+x^2+1}dx=-\frac{\pi}{2\sqrt{3}}\mathrm{e}^{-\pi\sqrt{3}}$

Can somebody help me out with the following integral? Prove that: $\int_{0}^{\infty}\frac{cos2\pi x}{x^4+x^2+1}dx=\frac{-\pi}{2\sqrt{3}}e^{-\pi\sqrt{3}}$ I have already determined the ...
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4answers
58 views

Integrating $t^{2r-1} / t^{2k} (1+t^2)^{r+1}$

Let $k$ and $r$ be natural numbers such that $1 \leq k \leq r$. I want to calculate $$ \int_0^\infty \frac{t^{2r-1}}{t^{2k}(1+t^2)^{r+1}} dt. $$ Since the integrand is an odd function the standard ...
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0answers
88 views

Pole on path of integration.

Upon evaluating $$\int_0^\infty \frac{1}{1+z^5}dz$$ using the Residue Theorem, why isn't the pole at $z=e^{\pi i}$ taken under consideration in the summation of the residues? We were taught that ...
4
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4answers
296 views

Calculating $\int_0^\infty \frac {\sin^2x}{x^2}dx$ using the Residue Theorem.

I am trying to compute the following integral using the Residue Theorem but am quite stuck: $$\int_0^\infty \frac{\sin^2x}{x^2}dx$$ I have tried applying Jordan's lemma, having written $\sin(x)$ as ...
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1answer
51 views

Integral $\int_{0}^{+\infty}\frac{t \sin(t)}{t^{2}+b^{2}}dt$

I want to solve the integral $$\int_{0}^{+\infty}\frac{t \sin(t)}{t^{2}+b^{2}}dt$$ Which function and contour should I consider ?
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2answers
148 views

$\int_{-\infty}^\infty \frac{e^{ax}}{1+e^x}dx$ with residue calculus

I'm trying to compute $\displaystyle \int_{-\infty}^\infty \frac{e^{ax}}{1+e^x}dx$, $(0<a<1)$ Let $f$ denote the integrand. I'm using the rectangular contour given by the following curves: ...
16
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2answers
427 views

Show $\int_0^{\pi/3} \big((\sqrt{3}\cos x-\sin x)\sin x\big)^{1/2}\cos x \,dx =\frac{\pi\sqrt{3}}{8\sqrt{2}}. $

I have run a FORTRAN code and I have obtained strong evidence that $$\int_0^{\pi/3} \!\! \big((\sqrt{3}\cos\vartheta-\sin\vartheta)\sin\vartheta\big)^{\!1/2}\!\cos\vartheta \,d\vartheta ...
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1answer
102 views

Residue of a simple pole. Why are they different?

We'll show you two way of calculation of the Residue in consideration. $$f(z) = \frac{z\sin(z)}{1-\cos(z)}$$ I'm interested to calculate the residues in $2\pi$ and $-2\pi$. I choose one of ...
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1answer
103 views

Find the order of the poles of $\dfrac{z}{\cos z}$

I know that they are simple poles, but how can you find this? The usual equation that I have for finding the order of poles which is, $$ \displaystyle\lim_{z\to z_0} (z-z_0)^{n}f(z), $$ and seeing ...
3
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1answer
143 views

Would like help with a contour integral.

Disclaimer: the knowledge I have about contour integration is solely from the book "Mathematical Methods in the Physical Sciences" by Mary L. Boas. I am trying to understand how the following ...
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3answers
116 views

Question Residues -integral at Complex Analysis

How can i find the integral below , which transformation should i do ? İ think i need to get $sin$ and $cos $ but i can't see $$\int\limits^{+\infty}_{-\infty} \frac{ \exp\left({ax}\right)} ...
3
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1answer
159 views

Choice of branches for contour integration.

Suppose I have the following function of a complex variable $$f(z)=\log(z)(z^2+1)^{1/2}.$$ Wolfram Alpha tells me the branch cuts of $f(z)$ are $z\leq 0$ (presumably for the logarithmic term), and ...
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2answers
49 views

Residue of a a complex quotient

I have the following Laurent expansion corresponding to the function: $$f(z)=\frac{z+2}{z^2-4iz-3}$$ $$f(z)=\left(-1+\frac{1}{2}i\right)\sum_{n=1}^\infty ...
8
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3answers
284 views

How to show $\int^{\infty}_{-\infty}\frac{\sin(ax)}{x(x^2+1)}dx=\pi(1-e^{-a})$? ($a\ge0$)

$$\int^{\infty}_{-\infty}\frac{\sin(ax)}{x(x^2+1)}dx=\pi(1-e^{-a}), \ a\ge0$$ I tried to solve but came up with $\pi(2-e^{-a}) $. Could you tell me where did I do the mistake? if $x=z$ then ...
1
vote
1answer
35 views

Carry out integral by using Cauchy's theorem

I have kind of a silly question, which probably has an easy answer which I should know myself, but here goes. Say we want to integrate $$ \int_{-\infty}^\infty dx \frac{1}{(x^2 + 1)(x - 1 - i)}. $$ If ...
3
votes
2answers
160 views

How to compute the integral $\int_0^\infty\frac{x}{e^x+1}dx$ using the Residue theorem.

How to compute the integral $\int_0^\infty\frac{x}{e^x+1}dx$ using the Residue theorem, just as the title says. I have used rectangles, circles to do, but without any progress. By changing variable ...
3
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0answers
108 views

How to calculate this residue

how to calculate this residue $$Res(\frac{Ln z}{z(z+1)},0).$$ Is it $\infty$? And if this could not be calculated, then how to calculate $$\int_0^\infty \frac{x}{e^x+1}dx$$ by changing variables ...
4
votes
2answers
105 views

Intuition behind the residue at infinity [duplicate]

The residue at infinity is given by: $$\underset{z_0=\infty}{\operatorname{Res}}f(z)=\frac{1}{2\pi i}\int_{C_0} f(z)dz$$ Where $f$ is an analytic function except at finite number of singular points ...
2
votes
1answer
61 views

Integral using residue theorem

We have the following problem given: $$ \int_{-\infty}^\infty \frac{\cos(t)^2}{t^4 + 5 t^2 + 4} \, \mathrm dt. $$ I thought that I could solve it using the residue theorem and by arguing that the ...
1
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1answer
65 views

Limit of characteristic function

I have a characteristic function defined as the following: $$\phi(\frac{t}{N})^N = (\alpha E_{\alpha+1}(-\frac{iLt}{N}))^N$$ where $E_n(z)$ is the $E_n$ function having the form $E_n(z) = ...