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.
2,672
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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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2
votes
1
answer
65
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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.
...
- 1,280
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0
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32
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Help with contour integral for a reproducing kernel [closed]
If we assume $f$ is analytic and this integral makes sense on the unit disc $U$, then I'm trying to show this is a weighted Bergman kernel, but I'm stuck here:
$$
\frac{4}{\pi}\int_{U}\frac{f\left(...
2
votes
2
answers
96
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Residue Theorem Integral of $\cos(e^z)/\sin^2(z)$
I need help with the following integral:
$$
\frac{1}{2\pi i }\int_{|z| = 4} \frac{\cos(e^z)}{\sin^2(z)} dz.
$$
I have tried finding the residues, which gave me 0 through my (possibly erroneous) ...
- 31
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0
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21
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Confusion in computing the residue of Veneziano Amplitude function: a general statement for computing a composite function residue.
Premise: I'm an undergraduate student, so I'm only considering the math perspective of the following problem and not its physical relevance nor significance.
I'm currently studying Euler's Gamma ...
2
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0
answers
53
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Evaluate $\int_0^\infty\frac{\sin x^p}{x^p}dx$ with residue theorem
Evaluate $I=\int_0^\infty\frac{\sin x^p}{x^p}dx\,\,(p>\frac12)$.
I was able to solve it by converting the integral to a gamma function integral. We have $$I=\Im\left(\int_0^\infty\frac{e^{i x^p}}{x^...
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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....
- 501
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2
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57
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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 ...
- 11
3
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2
answers
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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 ...
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2
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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 ...
- 370
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votes
1
answer
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How to integrate $\int_0^1 x^{a-1} (1-x)^{-a} dx$
I am stuck on how to integrate the following: $$\int_0^1 x^{a-1} (1-x)^{-a} dx$$ where $a \in (0,1)$. I am aware that this is a variant of the Euler gamma/beta functions and will be equal to $\pi/\sin(...
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Calculating Fourier transform with residue, last step
I got a rather specific question for when calculating Fourier transform with residue, hoping someone understand what I'm looking for and can help me!
My solution is rather long, but it's only the last ...
- 551
4
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2
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169
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Solving $\int_{0}^{\infty}{\frac{\cos(tx^n)}{x^n+a}\, dx}$ via residues
I was trying to evaluate $\int_{0}^{\infty}{\frac{\cos(tx^n)}{x^n+a}\, dx}$, with a semicircle in the upper half plane we have :
$$\oint{\frac{e^{itz^n}}{z^n+a}\, dz}=\int_{-\infty}^{\infty}{\frac{e^{...
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A doubt on solving a problem involving residues [duplicate]
Let $D=\{ z \in \mathbb{C}: |z|<2\}$ and $f:D\rightarrow\mathbb{C}$ be a complex-valued function such that $f$ is analytic at all points of $D$ except a simple pole at $z=1.$ Given that $f$ has ...
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How do I find the residues of log(zeta(s))/s?
In On the Number of Primes Less Than a Given Magnitude
Riemann gives the following relationship between the zeta function and his prime-power counting function:
$$\Pi(x)=\frac{1}{2\pi i} \int_{a-\...
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2
votes
2
answers
221
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Partial fraction decomposition of $\frac{1}{(x(x+1)(x+2)...(x+n))^2}$
In view of this question, I am trying to find the partial fraction decomposition of $$\frac{1}{(x(x+1)(x+2)...(x+n))^2}$$ where $n\in\mathbb{N}$
Since every $k$, $k=-n,...,-2,-1,0$ is a pole of order ...
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Complex integration - showing that the arc integral vanishes using the estimation lemma
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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Evaluating $\oint\frac{dw}{(w-w_+)(w-w_-)} $ over the unit circle, where $w_\pm=z\pm\sqrt{z^2-1}$ and $z\in\Bbb{C}$ with $|w_+|\neq|w_-|$
The following problem arises in E. N. Economou's ''Green's Functions in Quantum Physics'', 3e.
Evaluate the following integral over the unit circle
$$
\oint\frac{dw}{(w-w_+)(w-w_-)}
$$
where
$w_\pm = ...
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2
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1
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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 ...
- 21
2
votes
2
answers
94
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Contour Integral around the unit circle $C$: $\oint_C \frac{e^z-1}{\sin^3(z)}dz$
Studying once again for my last attempt at the complex analysis qualifying exam. I'm a bit confused as to what to do with this contour integral, where $C$ is the unit circle.
$$\oint_C \frac{e^z-1}{\...
- 470
2
votes
4
answers
65
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Residue when computing $\int_{\gamma} \frac{\cos(z)}{z-1} dz$
I'm stumbled upon a question when computing Residue.
I want to compute the integral $\int_{\gamma} \frac{\cos(z)}{z-1} dz$ with help of residue.
My solution is rather short since I can see directly ...
- 551
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1
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61
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How to find the residue at $z_0=\exp(i\pi/3)$ of $f(z)=\frac{z^{2}}{z^{4}+z^{2}+1}$ using the limit definition?
I know $z_0$ is a simple pole so I just want to evaluate the limit $\lim_{z\to z_0} (z-z_0)f(z)$.
(The answer is $\frac{1}{12}(3 - i \sqrt{3})$ and I'm not getting it using the limit way. I got the ...
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let f be analytic in $Ball_2\left(0\right)$ and f is odd. let $U=\left\{z\in \mathbb{C}|1<\left|z\right|<2\right\}$, prove equality
Sorry for title being not full, could not write it all:
let f be analytic in $Ball_2\left(0\right)$ and f is odd.
let $U=\left\{z\in \mathbb{C}|1<\left|z\right|<2\right\}$
Prove:
$\exists f\in ...
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2
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Find residue with limit $\lim _{z\to 0}\frac{d^2}{dz^2}\left(\frac{e^zz}{\sin\left(z\right)}\right)$ without Taylor series
$$\lim _{z\to 0}\frac{d^2}{dz^2}\left(\frac{e^zz}{\sin z}\right)=\lim _{z\to 0}\left(\frac{d}{dz}\left(\frac{e^z\:z+e^z}{\sin z}-\frac{ze^z\cos z}{\sin^2 z}\right)\right)$$
Hi, can someone help me ...
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Compute double complex integral: $\int_{|\zeta|=2} \int_0^{2\pi} \frac{\zeta}{\zeta+\sin\theta} d\theta d\zeta$.
I am trying to compute the following integral: $$\int_{|\zeta|=2} \int_0^{2\pi} \frac{\zeta}{\zeta+\sin\theta} d\theta d\zeta.$$
I know that if $\zeta \in \mathbb R$ with $|\zeta|>1$, then the ...
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Where and what kinds of singularities do these functions have?
I am supposed to compute the singularities and their kind of $f$ in $\mathbb{C}$ for the following functions, furthermore I shall compute $\int_{|z|=4}f(z)dz$:
a) $\displaystyle f(z)=\frac{\sin(z)}{e^...
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Showing that, under additional assumptions, an entire function is not a polynomial
I'm not believing the claim (below) that is to be proven true.
Show that if $f$ is entire, not zero on circles of natural radius centred at the origin and $\oint_{|z| = n}\frac{1}{f(z)}\mathrm{d}z \...
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On a Nearly Separable PDE in Toroidal Coordinates and a Possible Fourier Transform Approach
Background
I have recently come very close to an analytical solution to a particular PDE related to my research. In particular, solving a very specific case of the Navier-Stokes equations in a torus ...
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votes
2
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Computing an Integral Involving Rational and Bessel Functions
I tried to compute the following integral by using Contour integration method.
$$
\int_0^{\infty}\frac{x^2}{x^4+1}J_0(ax) dx
$$
where $J_0$ is Bessel function of the first kind and $a$ is a ...
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0
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Use Laurent series to compute the residue
I know you have to use the Laurent series to solve this problem, since the function has an essential singularity.
$$ \oint\limits_{\gamma} e^{\frac{1}{z}}\,\mathrm{d}z = \lim\limits_{n \rightarrow \...
- 11
2
votes
1
answer
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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....
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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 ...
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Residue calculus: Integrals vanish
Let $a>0, \omega > 0$. We want to prove that $$\frac 1 {2 \pi i} \int_{\omega - i \infty}^{\omega + i \infty} \frac{a^z}{z(z+1)} dz = (0 \text{ if } a \in [0,1) \text{ and } 1-\frac 1 a \text{ ...
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0
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Looking for a specific zeta function.
I am looking for a zeta function
$$ f(s) = \sum \frac{1}{a_n^s}$$
Where $a_n$ is a sequence of distinct positive integers,
such that
$f(s)$ is analytic for all $Re(s) > 1$
$f(s)$ has a simple ...
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1
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1
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Find the analiticity domain of a complex function
I need to study the domain of analyticity of this function:
$$ f(z) = \frac{\sqrt{(z-3)(z^2-4)}}{2z^2}\sin z$$
and compute the integral over the unitary circle $\gamma: \theta \to e^{i\theta}, \theta \...
4
votes
4
answers
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Evaluating $\int_{0}^{\infty}\frac{\sin^2(x)}{x^2(1+x^2)}\,dx$
As the title says I’m wondering what is wrong with my solution process in evaluating
$$\int_{0}^{\infty}\frac{\sin^2(x)}{x^2(1+x^2)}\,dx$$
Here is what I do:
$$I=\int_{0}^{\infty}\frac{\sin^2(x)}{x^2(...
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2
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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 ...
0
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0
answers
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Finding Cauchy principal value of contour integral
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 ...
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2
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51
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Calculate improper integral using contour
I'm trying to calculate integral with contour method.
\begin{equation}
\int_{-\infty}^{\infty}\frac{\cos x}{4x^2-\pi^2}\mathrm{d}x
\end{equation}
I checked the singularity points of function $f(z)=\...
- 149
1
vote
1
answer
48
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An easy way to calculate the residue of $\tan^2z$
I need to calculate the value of residue of $f(z) = \tan^2z$ at $z_k=\frac{\pi}{2} + \pi k$. I know that these points are poles of the second order, so the common way is to calculate residue by the ...
2
votes
1
answer
50
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Complex Contour Integration along a Circular Arc and Residues
Recently I stumbled upon this answer in which the author derives a very nice identity for when one does not integrate along a whole circle but rather only a circular arc. Their proof is concise but ...
- 65
1
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1
answer
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Calculate $\int_0^\infty \frac{\sqrt{t}}{1+t^2} dt$ [duplicate]
I'm thinking of using the residue theorem. The residues are at $\pm i$ and I know how to calculate them but I'm stuck on finding a contour that fits? How do you find the correct contour in general?
- 143
2
votes
1
answer
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Formula for residues of a simple pole
If we have The rational function:
$$f(z)=\frac{\phi(z)}{\psi(z)},~~~~ \phi(z_0)\neq 0 , ~\psi(z_0)=0 , ~\psi'(z_0)\neq 0$$ and the point at $z_0$ is a simple pole, then the residue can be calculated ...
- 163
2
votes
3
answers
92
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Where is the mistake $\oint_C \frac{e^\frac{1}{z}}{z^2-1}dz$
Evaluate $\oint_C \frac{e^\frac{1}{z}}{z^2-1} dz$ where $C$ is the locus of $z$ satisfying $|z-1|<3/2$ (answer: $iπ/e$).
My attempt at an answer comes nowhere close to this given result:
The two ...
- 608
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votes
6
answers
190
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Calculate $\int_0^\infty \frac{\ln(t)}{1+t^3}dt$
How can I evaluate
$$\int_0^\infty \frac{\ln(t)}{1+t^3}dt$$
I've seen similar posts with $1+t^2$ instead of $1+t^3$. But I'm not sure whether they could help me in this case. I'm thinking of using ...
- 143
1
vote
2
answers
85
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Solving the integral $\int_{0}^{2\pi}\frac{dt}{(a+b\cos(t))^2}$
For solving the iuntegral: $\int_{0}^{2\pi}\frac{dt}{(a+b\cos(t))^2}$ I will make it a complex one using the substitusion: $e^{it}=z$ So the integral becomes:
$$\frac{4}{i}\oint_V \frac{zdz}{(bz^2+2az+...
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votes
2
answers
102
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How to calculate $\int_{-\infty}^{\infty}\frac{\cos^{3}x}{x^2+1}\mathrm{d}x$ with the help of complex function?
I have met this improper integral:
$$\int_{-\infty}^{\infty}\frac{\cos^{3}x}{x^2+1}\mathrm{d}x$$
I tried to use residues but it doesn't work. The singularity point of function $f(z)=\frac{\cos^{3}z}{1+...
- 149
1
vote
1
answer
98
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How to calculate the residue of $f(s)=\frac{\zeta(2s)}{\zeta(s)}\frac{x^{s-1/2}}{s-1/2}$
Let $x\in\mathbb{R},\,x>1$, and
$$f(s)= \frac{\zeta(2s)}{\zeta(s)}\frac{x^{s-1/2}}{s-\frac 12}$$
where $s=\sigma+it\in\mathbb{C}$. How do you calculate the residues of $f(s)$ in the critical strip (...
- 373
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votes
2
answers
58
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Solving $\int_0^\infty \frac{x\sin (3x)}{x^2+1} dx$ using the residue theorem
Before I proceed here's a quick disclaimer: English is not my first language, some of the terminology I used might be incorrect.
I tried solving it and my final solution was not a real number (which ...
- 3
0
votes
0
answers
46
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Integrate $\csc(1/z)$ on a contour through $0$
$\gamma$ is the (triangle) contour $i\longrightarrow-i\longrightarrow1\longrightarrow i$.
$\def\rmd{\mathop{}\!\mathrm{d}}$
Using Mathematica to evaluate the directional limit at $0$ on $\gamma$ ...
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