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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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11 views

Contour integral involving moduli in the integrand

Long-time lurker, first time questioner. Here is the problem: Let $ a,b \in \mathbb{C} $, with $|b|<1$. Calculate the integral $$ \frac{1}{2\pi i}\int_{|z|=1} \frac{|z-a|^2}{|z-b|^2} \frac{dz}{z}...
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1answer
25 views

contour integration of $I=\int^{\infty}_{-\infty}\frac{\cos x}{x+i}$

Evaluate:$$I=\int^{\infty}_{-\infty}\frac{\cos xdx}{x+i}$$ which means that $$f(z) = \frac{e^{iz}}{z+i}$$ with the simple pole $z=-i$ and since $z=-i$ i'd have to integrate it using the bottom half ...
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1answer
18 views

finding the residue ? Tough question

Find the residue at each of the isolated singularities of the following function on $\mathbb{C}$? $1) \frac{z}{z^2+3z +3}$ $ii) \frac{1}{(z^3 +1)(z+1)^2}$ My attempt : for $1) z^2 +3z +...
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1answer
33 views

Complex integral help: $\oint_C \frac{\sin(z)}{z(z-\pi/4)} dz$

I'm attempting to evaluate the following complex integral: $$\oint_C \frac{\sin(z)}{z(z-\pi/4)} dz, $$ where $C$ is a circle of radius $\pi$ centred on the origin. I have calculated the residues of ...
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0answers
30 views

Eigenvector normalization, poles and residue

I am trying to understand an eigenvector normalization procedure described in an article [1](appendix B). The problem involves a complex valued matrix $\mathbf{Z}$, function of the complex number $s$,...
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1answer
37 views

Compute the integral using complex analysis method

I want to compute this integral $$\int_{C(2i,5)} \frac{z}{e^z-1}dz$$ I was trying to trying to use residue theorem but I could not find residue of this function.
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52 views

Holomorphic coordinates on Riemann surfaces

I have a big problem understanding the meaning of holomorphic coordinates on Riemann surfaces, especially in relation to 1-forms. Holomorphic coordinates on a Riemann surface $X$ is an open set $U \...
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3answers
110 views

evaluate $\int_{|z|=1} \frac{1}{e^z -1-2z}dz$ by using Cauchy's residue theorem

evalulate the integral by using Cauchy's residue theorem $$\int_{|z|=1} \frac{1}{e^z -1-2z}dz$$ MY attempt : $ f(z) =\frac{1}{e^z -1-2z}$, now put $z= 1$ we get $f(z)=-1$ so $$\int_{|z|=1} \frac{...
2
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1answer
28 views

Residue-Calculus: Why does this equation hold?

$\DeclareMathOperator{\Res}{Res}$ $\DeclareMathOperator{\e}{e}$ We did the following equations in our lecture: Calculate $x(t)$ with the help of the residue-calculus: $X(s)=\frac{s^2}{(s+1)^3}$ $...
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0answers
31 views

Residue of a function help

I tried working out the residue of a function $$f(z) = \frac{\sqrt z}{(1+z)^2}$$ at $z = -1$ using the general formula for the residue $$Res(f(z))|_{z=a} = \frac{1}{(m-1)!}\lim_{z \to a} \frac{d^{m-1}}...
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18 views

solving integral with real exponent and real pole with residue theorem

I'm trying to solve this integral: $ \int_{-\infty}^{\infty} \frac{x^3 e^{- \alpha x^2}}{\beta - x} dx$ It looks similar to a complex integral with a pole but notice a few subtleties: The exponent ...
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2answers
33 views

Finding the residue of a function at $z_0$?

I encountered a particular question that led me to question the definition that I was given for a residue, after reviewing the literature I simply want to confirm that my understanding is correct. ...
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2answers
71 views

Integral of real part of $z$ around the unit circle

What is the result of integrating the real part of z (a complex number) anti clockwise around the unit circle? At first glance, I couldn't identify any points within the circle where analyticity ...
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0answers
36 views

How to calculate the arc part when using Residue theorem?

Suppose the function $f(z)$ has singularities and I want to calculate the integral $$\int_{-\infty}^{\infty}f(z)dz=?$$ I use Residue theorem $$\oint_cf(z)dz=2\pi i\sum\text{Res}(f,z_0)$$ And $$\...
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2answers
97 views

Calculate the value of $\int_0^\infty \frac{\sqrt{x}\cos(\ln(x))}{x^2+1}\,dx$

I'm asked to evaluate the integral $\displaystyle\int_0^\infty \frac{\sqrt{x}\cos(\ln(x))}{x^2+1}\,dx$. I tried defining a funcion $f(z)=\frac{e^{(1/2+i)\operatorname{Log}(z)}}{z^2+1}$, taking $\...
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1answer
74 views

Estimating complex integral

I am reading chapter 12 of "Lectures on the Riemann Zeta function" by H. Iwaniec. I am stuck at understanding the computation done at the beginning of page 45. We want to estimate the following ...
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1answer
26 views

Laurent series/isolated singularity

I want to classify the singularities of $$ f(z)=\frac{\sin(2z)}{(z-1)^3}$$ The Taylor series is: $\sin(2z)=\sum_{k=0}^{\infty}\frac{(-1)^k}{(2k+1)!} 2^{2k+1} z^{2k+1}$ So: $ \frac{\sum_{k=0}^{\...
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0answers
661 views

Proof without words of $\oint zdz = 0$ and $\oint dz/z = 2\pi i$

I found this visual "proof" of $\oint zdz = 0$ and $\oint dz/z = 2\pi i$ quite compelling and first want to share it with you. But I have a real question, too, which I will ask at the end of this post,...
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1answer
44 views

Connection between sums and integrals over closed paths and their areas

Is there a deeper connection between the following identies involving sums or integrals over a closed path (resp. circle) resp. the area enclosed – e.g. a general principle that is underlying ...
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38 views

Residues of $\frac{\pi cot(\pi z)}{(z+1)(z+2)}$

I need to find the residue at $z= -1$ and $z=-2$ of the following function: $$f(z) = \frac{\pi cot(\pi z)}{(z+1)(z+2)}$$ I now that $f(z)$ has a double pole at $z=-1$ because $sin(-\pi) = 0 $. Now ...
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1answer
29 views

Residue and Laurent Series, is this valid?

something with the Laurent series is confusing me, first I'll give a background of what I think I know. If $z_0$ is an isolated singularity of a function $f$ we can find the $Res(f, z_0)$ by finding ...
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2answers
64 views

How to use complex analysis to evaluate the trigonometric integral $\int_{0}^{2 \pi}{\frac {\cos \left( x \right) }{2+\cos \left( x \right) }}$ [duplicate]

So I have the integral below which I am meant to evaluate using complex analysis. I was thinking to transform this integral and evaluate using the Residue theorem, but the process is incredibly messy ...
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0answers
42 views

Cauchy Principal Value and Residue Theorem: Apparent Contradiction

The following formula is well known and I already understood one of its proofs: $$ \frac{1}{x\pm i\epsilon} = \mathrm{CH} \frac{1}{x} \mp i\pi\delta(x), $$ where the limit $\epsilon \to 0$ is implied ...
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2answers
42 views

Solving integral using residues

I'm trying to find the value of the integral $\int_0^{2\pi}\frac{\cos^2u}{2-\sin u}du$ using the substitution: $\cos u =\frac{1}{2}(z+\frac{1}{z})$ and $\sin u = \frac{1}{2i}(z-\frac{1}{z})$. Making ...
4
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1answer
86 views

Residue of $f(z)=\frac{z}{\sin{\left(\frac{\pi}{z+1}\right)}}$ in all isolated singularities

I have this complex function: $$f(z)=\frac{z}{\sin\left(\frac{\pi}{z+1}\right)}$$ I'd like to compute residues in all isolate singularities. If I'm not mistaken $f$ has poles in $z=\frac{1}{k}-1$ and ...
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0answers
26 views

How do I prove the following statement to do with formal power series?

Let f(t) be a formal power series with ord(f) = 1 and let g(t) be the inverse of f. Then b0 = 0 and bn = (Res(1/(f(t))^n))/n where n>0 I am completely lost in understanding how I should show that ...
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0answers
18 views

How do I solve this question to do with residue of a formal power series?

If f(t) is a formal power series with ord(f) = 1, then it follows that Res(f'(t)/f^n(t)) equals 0 when n is not equal to 1, but equals 1 when n is equal to 1. I know that residue is defined for ...
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0answers
28 views

contour integral logarithm

When calculating integrals like $\int_{0}^{\infty} R(x)log(x) dx$ with R(x)=P(x)/Q(x) a rational function i use the keyhole contour as in the example 4 of this link https://en.wikipedia.org/wiki/...
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1answer
110 views

Calculate $\int_0^1 \frac{1}{\sqrt{x(1-x)}} dx$ using residue calculus

I'd like to calculate $$\int_0^1 \frac{1}{\sqrt{x(1-x)}} dx,$$ using residue calculus. I was given a hint to consider the function $$f(z) = \frac{1}{z\sqrt{1-\frac{1}{z}}}. $$ I thought I was ...
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0answers
26 views

Shifting roots in infinite sums of polynomials

Define $$P(n)=(n-r_1)(n-r_2)...(n-r_k)$$ where $r_i\in\mathbb{Q}$ and $r_i\neq r_j$ for $i\neq j$. Now, define $$Q(n)=(n-(r_1+m))(n-r_2)...(n-r_k)$$ where $m\in\mathbb{Z}$ and $r_1+m\neq r_j$ for $...
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2answers
48 views

On the residues of $f(z)=(z-z_0)^{-n}$

Define $f_{n}(z)=(z-z_0)^{-n}$ where $n$ is a positive integer. Notice that $f_n$ has a pole of order $n$ at $z=z_0$, for every positive integer $n$. My question: Does $f$ has residue $0$ at $z=z_0$ ...
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1answer
39 views

Laurent serie of $ \frac { \cos z}{ \sin z + \sinh z - 2z}$

I'm working on an example given in my book of complex analysis: $$ \frac { \cos z}{ \sin z + \sinh z - 2z}$$ but I can't figure out how he finded the residue in 0. The few steps he is showing make ...
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1answer
38 views

Question about integrating a Laurent series

I want to expand the function $ f(z) = \frac {z-1}{z^2 -2z -3} $ in $ 0 < |z+1| < 4$ Then I want to use the result to evaluate this integral $ \int_C \frac {z-1}{(z+1)(z-3)} dz $ in $ C : |z+...
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1answer
48 views

A question about the Residue of $h=fg$

Let $f$ and $g$ be two functions (not necessarily analytic) of the complex variable $z$ such that for some $\varepsilon >0$ : 1) $f$ is continuous on $0<\left\vert z\right\vert <\varepsilon $...
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3answers
111 views

Evaluating $\int_{-\infty}^{\infty}\frac{e^{ax}}{\cosh{x}}dx $ using contour integration

Let $a \in \mathbb{C}$ with $-1 <$ Re $a < 1$. By considering a rectangular contour with corners at $R, R + i\pi, -R+ i\pi, -R,$ show that $$\int_{-\infty}^{\infty}\frac{e^{ax}}{\cosh{x}}...
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1answer
43 views

Integration using residues $\int z^2 \log [(z+1)/(z-1)] dz$

$\int z^2 \log [(z+1)/(z-1)] dz$ taken around circle $|z|=2$ I am taking residues at $\pm 1$. This gives me 0 as the value of integral. Is this correct. How do I modify the integral to get value ...
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1answer
54 views

Integration of a multivalued function

The integral is: $$I=\int_1^2\frac{\sqrt{(x-1)(2-x)}}{x^2}dx$$ To solve this problem I integrate over a path $C$ that surrounds clockwise the branch cut, so the integral becomes: $$I=\frac{1}{2}\oint_{...
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1answer
66 views

$\int_{0}^{2\pi}\frac{d\theta}{a+b\sin\theta}$ where $a,b>0$ [duplicate]

$\int_{0}^{2\pi}\dfrac{d\theta}{a+b\sin\theta}$ where $a,b>0$ Supposedly a very simple residue theorem problem but I'm stuck with the pole(s). Let $z=e^{i\theta}$ and $d\theta=\frac{dz}{iz}$. ...
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1answer
51 views

When would I use Cauchy's Integral Formula over Residue

Just a quick question I've been wondering about. When would I use Cauchy's Integral Formula over the Residue Theorem to solve complex integration problems with poles? To me it seems that Residue ...
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1answer
37 views

When to resolve into partial fractions for applying Cauchy's integral formula?

Suppose I have to calculate $\int \frac{f(z)}{(z-a)(z-b)}dz$ around a curve in which both $a$ and $b$ lie inside. Should I apply Residue theorem like this: Residue at $a$= $\frac{f(a)}{a-b}$ ...
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2answers
47 views

Residue at infinity of $\frac{e^{z^2}}{z^{2n+1}}$

$$\frac{e^{z^2}}{z^{2n+1}}$$ Am I right that limit as z approaches infinity does not exist? So its residue at infinity is equal to $c_{-1}$ of Laurent series. How am I supposed to get Laurent series ...
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1answer
47 views

Find complex residue and Laurent series expansion

$$ w = \sin(z) * \sin(\frac{1}{\:z}) $$ special point is $$ z_0 = 0 $$ $$ \lim _{z\to 0}\left(\sin\left(z\right)\cdot \:\sin\:\left(\frac{1}{z}\right)\right) $$ isn't exist, next I have to decompose ...
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0answers
25 views

Limit of a complex residue

I was solving the residue for $\frac{e^{1/x}}{x^2}$ at x = 0 found it to be 0 using Laurent series expansion. This doubt stuck in my mind Does this limit $\lim_{p \to 0}Res \frac{e^{1/x}}{x^2(x-p)}...
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0answers
30 views

How to get correct residue of complicated function with exponentials and associated contour integral?

I am trying to calculate the improper integral: $$ I=\int_{-\infty}^{+\infty} f(x) dx $$ with $$ f(x)=\frac1{8\pi^3}\frac{x^2 \sqrt{1+x^2}}{1+e^\sqrt{1+x^2}}.$$ The function $f(x)$ has poles at $x_0=\...
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2answers
37 views

Improper Integral Residue Theorem

I'm stuck on a question involving evaluating improper integrals using the residue theorem. Here's what I'm trying to evaluate: $\int_{-\infty}^{\infty} \frac{1}{(1+x^2)^3} dx$ To begin with, we can ...
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1answer
39 views

Is the residue theorem the correct approach for this integral?

I want to calculate $$\int_{\gamma} \frac{\sin z}{z^2 + 1}dz$$ where $\gamma$ is the upper-half circle of radius 2 centered at the origin starting at 2. I know that since $\gamma$ is not a closed ...
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2answers
51 views

Missing $i$ while evaluating $\int_{-\infty}^{\infty}\frac{e^{iz}}{(z^2+2z+2)^2}$ using residue theorem

Okay, first I'm a bit ashamed to ask because I already asked a question yesterday about a similar question (it's from far not the same integral though), but I'm missing an $i$ somewhere in the process,...
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2answers
28 views

Residue of a product of series

I need to find the residue of $f(z)=\frac{e^{\frac{1}{z}}}{z-1}$ in $z=0$. To do this, I proceeded to find the Laurent series of $f$ which is: $\sum_{n=0}^{\infty} \frac{z^{-n}}{n!}\sum_{k=0}^{\...
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0answers
19 views

How to apply the method of steepest descent to solve this integral?

I want to reproduce Eq. (11) of this paper. It is the result of solving the integral $$ \int_{0}^{\Lambda} \text{d}q \frac{q (e^{i q r} - e^{-iqr})}{q^2 - x} $$ (where $\Lambda = \pi$ and $x = \omega/\...
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0answers
20 views

Classify the singularities of $f'(z)$ and its residue

I want to know which singularity does $f'(z_o)$ have if $f(z)$ has a singularity in zo. I know that if $f(z)$ has an essential singularity in $z_o$, then its Laurent series has infinite negative ...