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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14 views

Residue of complex exponential

I need to find the residue of $e^{\frac 1{1-z}}$ using Laurent series. How would I manipulate the function to make it easier? I need to find all singularities and the corresponding residues. I believe ...
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0answers
18 views

Singularities and Residues [Demonstration]

How could I solve the following problem: "Consider $F$, an analytic function, so that $$f(z)=F(\frac{1}{z-1})$$ has a pole. Demonstrate that F(z) is a polynomial." I know that an analytic function ...
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2answers
28 views

what is the residue of $\frac{1}{z(1-cos(z))}$ in $0$?

what is the residue of $\frac{1}{z(1-cos(z))}$ in $0$? I tried using the residual formula $Res_{z=a} f(z) = \lim_{z \rightarrow a}(z-a)f(z)$ But then i get infty? Is there something i'm missing ...
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1answer
24 views

Rational Function Residue Formula Proof [duplicate]

I came across a theorem used to calculate residues of rational functions that states that if f and g are analytic functions at $z_k$ and $g'(z_k)$ isn't 0, then the residue of $f(z)/g(z)$ at $z_k$ is ...
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0answers
16 views

Residue derivation for integral of sin x/x

I am trying to integrate $e^(iz)/z$ over a rectangle contour from $-R$ to $r$ along the real axis and a small semicircle from -r to r and then goes along the real axis to R and then completes the ...
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16 views

Residue Theorem Senior Seminar Theorem [closed]

I am taking a senior seminar class in a week. I really like to learn more about the Residue Theorem. Do you guys have any topic that you think would be fun for me to do a quick research on? I am ...
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1answer
60 views

Calculating Residues and singularities

We haven't done many full examples with residues so just wondered how you would answer the following questions for example: Classify the type of singularity for f and determine the residue of f at ...
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1answer
20 views

How to evaluate $\frac{1}{2\pi i}\int_C \frac{f(z) dz}{(z-z_1)^{m_1}(z-z_2)^{m_2}}$ and the other related contour?

Respected All. I was studying residue theory where I came accross the following problem "If $f$ be analytic in the simply connected domain $D$ and $z_1, z_2$ are two distinct complex point lying in ...
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1answer
40 views

how to calculate $\frac{1}{2\pi i} \int_{\gamma} \frac{2z}{(z-1)^ 4(z-3)}$

How to calculate $\frac{1}{2\pi i} \int_{\gamma} \frac{2z}{(z-1)^ 4(z-3)}dz$ When $\gamma = C_+(0,4)$ and where $\gamma = C_-(0,2)$. I need to use the residuformula which states that is f is ...
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17 views

Find $\text{Res}\left(\dfrac{1}{z(1-e^{-z})},2\pi i\right)$.

How to find Residue of $\dfrac{1}{z(1-e^{-z})}$ at $z = 2\pi i$?
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1answer
45 views

Integral with logarithm - residue

Let $R(x)$ be rational function. It is any general method to calculate $\int_{0}^{\infty}R(x) \log(x)dx$ ? I can do it in special cases, but I am looking for a general method. What should be a minimal ...
3
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2answers
99 views

Integral by residue - “dog bone”

Let $I=\int_{-1}^{1}\frac{x^2 dx}{\sqrt[3]{(1-x)(1+x)^2}}$. I used complex function $f(z)=\frac{z^2}{\sqrt[3]{(z-1)(z+1)^2}}$, which we can define such that it is holomorphic on ...
3
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2answers
36 views

Computing Residue for a General, Multiple-Poled function?

I'm trying to compute the residue of the following function at $a$. I'm having a little trouble seeing which poles are relevant: Compute $\,Res_f(a)$ for the following function: $$f(z) = ...
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1answer
107 views

Partial fraction expansion for non-rational functions

With regard to this answer to an inverse Laplace transform question, I derived the following result: $$\frac1{i 2 \pi} \int_{c-i \infty}^{c+i \infty} ds \, e^{s t} \Gamma(s)^2 = 2 K_0 \left ( 2 ...
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1answer
53 views

Integrate by the method of residue

I want to integrate $$\int_{0}^{\infty}\frac{1}{(1+x)^5}dx$$ by the method of residue, but I have done only problems of simple poles, but this is a problem of fifth order pole. So I am stuck in it. ...
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1answer
53 views

Evaluating $\frac{1}{2\pi j}\int_{c-j\infty}^{c+j\infty}x^{-s}\sigma ^{ms-m} [ \frac{\Gamma ( s )}{\Gamma ( s+2)}]^{m}ds$

I have been trying to solve the problem for $m=3$: $$f(x)=\frac{1}{2\pi j}\int_{c-j\infty}^{c+j\infty}x^{-s}\sigma ^{ms-m}\left [ \frac{\Gamma \left ( s \right )}{\Gamma \left ( s+2 \right )} \right ...
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3answers
99 views

Residue of $\Gamma^{2}$ and $\Gamma^{3}$

Based on wiki, the residues of $\Gamma$ at non positive integers are given by: $$\text{Res}\left ( \Gamma(z),z=-n \right )=\frac{(-1)^{n}}{n!}.$$ I have been trying to find residue for $\Gamma^{2}$ ...
3
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0answers
49 views

Finding residues of rational functions with extremely large powers

$h(z)=\frac{5z^{2015} + 7z^{2010} - 38z^5 + z^4 - 2z^3 + 5}{7z^{2016} + 2z^{2013} - 6z^8 + 3z^7 + z^5 - 4z^2 - z + 111}$ Find the sum of the residues of h at its poles in $C$ How do I find the ...
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1answer
38 views

Prove that for a holomorphic function $f$ on $\mathbb{C}$, if $-f(z) = f(\frac{1}{z})$, then the residue of $f$ at 0 is 0.

Prove that for a holomorphic function $f$ on $\mathbb{C}$, if $-f(z) = f(\frac{1}{z})$, then the residue of $f$ at 0 is 0. Note: $f$ isn't defined on 0. I'm having a bit of trouble getting this ...
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22 views

An application of Rouches Theorem

Let $f$ be an entire function on the complex plane, with Taylor's expansion around zero as $f(z) = \sum_{k=0}^{\infty}c_{k}z^{k}$. Let $N(r)$ be the number of zeroes of $f$ in $D(0, r)$. Show that for ...
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2answers
49 views

Calculate integrals concerning a branch

i am trying to calculate this integral: $$\int_{|z|=5}\frac{dz}{\sqrt{z^2+11}}$$ Using the branch that gives : $\sqrt{36} = -6$ The function has 2 poles at $|z| < 5$, lets call them $\alpha$ and ...
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1answer
44 views

Finding the limit of $a_n$ for $\tan{\frac{\pi z}{2}}$

i need to understand some issues with the prove of the following exercise: Let $f$ be defined as: $$f(z) = \tan{\frac{\pi z}{2}}$$ Its not hard to show that its Taylor series around $z=0$ is from ...
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2answers
261 views

Help with the integral $\int_{0}^{\infty}\log\left(1+\frac{s^{2}}{4\pi^{2}} \log(1+ix)\right ) e^{-2\pi nx}dx$

We have the integral : $$\int_{0}^{\infty}\log\left(1+\frac{s^{2}}{4\pi^{2}} \log(1+ix)\right ) e^{-2\pi nx}dx$$ Where $s$ is a complex parameter, and $n$ is a positive integer. The integral ...
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0answers
17 views

Residue of function (two functions multiplied)

I need to know if I'm doing this exercise correctly: $f(z)$ has only one singularity, which is at $z=0$, and it's a pole of order 7 $f(z)=-f(-z)$ $f(z)$ is analytic in $z=\infty$ $g(z)$ is analytic ...
2
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0answers
57 views

Example of an analytic continuation for a function in integral form

Given $f(z) = \int_{-\infty}^\infty \frac{exp(-t^2)}{z-t}\,dt$, where $Im(z)>0$. Find an analytic continuation to the region $Im(z)<0$. Firstly the solution said that there is a branch cut on ...
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25 views

Residue Theorem: can I say this?

I know that $f(z)$ is analitic everywhere except of $z=0$, where it has a pole of order $k$. Can I say that $f'(z)$ will have a pole of order $k+1$? For example, if $f(z)=\frac{1}{z^k} \rightarrow ...
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2answers
94 views

Residue theorem: When a singularity gives infinite to the residue

What if one of the singularity gives infinity to the residue. Consider this contour; $$X=\int_{\gamma} e^{i(\frac{z^{2}+1}{2z})}\frac{{(z^{2}-1)}^4}{2z^2(z-i)^{3}(z+i)^{3}}dz$$ I have ...
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1answer
12 views

Finding a certain residue

I wish to calculate the value of \begin{equation} \text{Res}\big((z+\pi/4)^2\tan(z);z=\pi/2\big)=\frac{1}{2\pi i}\int_C (z+\pi/4)^2\tan(z)\text{d}z, \end{equation} where $C$ is any counter-clockwise ...
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0answers
82 views

Residue theorem definite trigonometric integration

I am trying to solve this integration $$\int_{0}^{\pi} e^{cos(\theta)} \tan^{3}(\theta)d\theta$$ putting $$z=e^{i\theta}$$ $$\int_{\gamma} ...
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2answers
53 views

Prove that $\oint _{|z|=R} (f-g)' dz = 0$ (Residue Theorem)

I know that $f$ and $g$ have a pole or order $k$ in $z=0$. $f-g$ is holomorph in $\infty$. I need to prove that: $$\oint_{|z|=R} (f-g)' dz = 0$$ Any help? Note: $f$ and $g$ only have a singularity ...
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1answer
94 views

Residue theorem:When a singularity on the circle (not inside the circle)

This is the integration I am trying to solve $$\int_{0}^{\pi} \sin^{2}(\theta)\sec^{3}(\theta)d\theta$$ putting $$z=e^{i\theta}$$ $$\int_{\gamma} \frac{-2{(z^{2}-1)}^2}{i(z-i)^{3}(z+i)^{3}}d\theta$$ ...
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2answers
140 views

Using complex analysis to evaluate $\int_0^\infty\frac{(\ln x)^3}{1+x^2}d x$

Here is my attempt: Let $R>1>r$ and $C$ be the closed curve in $\mathbb{C}$ consists of the following pieces: $$C_1=\{Re^{it}: t\in(0,\pi)\},\quad C_2=[r,R],\quad C_3=\{re^{it}: ...
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0answers
24 views

Contour integral with two branch cuts

I'm trying to solve this integral: \begin{equation} \int_0^\infty d\omega \,\frac{\left(\left(\omega ^2+1\right) \cos (\delta )-2 \omega \right) \log ...
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0answers
29 views

Evaluating this contour integral.

I was reading a paper that had the following integral $\displaystyle\prod_{n=1}^L\oint_{C_n}\frac{dx_n}{2\pi i}\prod_{k<l}^L(x_k-x_l)\prod_{m=1}^L\frac{Q_w(x_m)}{Q^+_\theta(x_m)Q^-_\theta(x_m)}$ ...
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2answers
31 views

What are some books or online resources about the Sommerfeld-Watson method for series summation?

I'm looking for resource recommendations on the Sommerfeld-Watson summation method, i.e. the use of the residue theorem to obtain expressions like $$ \tag 1 \sum_{n \in \mathbb{Z}} g(n) = - \pi ...
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1answer
23 views

Residue theorem and Angle of modular function

Let $f$ be a meromorphic function on the region $Im(z)>0$, $v_p(f)$ be the order of $p$. (The number $n$ such that $\frac{f(z)}{(z-p)^n}$ is holomorphic and non-zero at $p$.) Moreover, assume $f$ ...
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1answer
94 views

Computing Complex Integral to Determine Analytic Continuation of $f(z) = \int_0^\infty {{e^{-zt}} \over {1 + t^2}} dt$

My question is the following: Find the analytic continuation of the function $f(z)$ defined by $$ f(z) = \int_0^\infty {{e^{-zt}} \over {1 + t^2}} dt, \ \ \vert \arg(z) \vert < {1 \over ...
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4answers
91 views

Can I get some guidance on solving $\int_{-\infty}^{\infty} \frac{\sin^2(x)}{x^2} \, dx$?

I am trying to evaluate: $$I = \int_{-\infty}^{\infty} \frac{\sin^2(x)}{x^2} \, dx.$$ Using a contour semi-circle (upper plane), I can get: $$ \oint_{C} f(z) \,dz = \oint_{C} \frac{1 - e^{2iz}}{z^2} ...
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1answer
19 views

Can inverse fourier transform be formulated in terms of residue?

Today I ran into a peculiar problem when trying to perform the inverse fourier transform of $\frac{1}{a+jw}$ where a is some number $$ \mathcal{F^{-1}}(\frac{1}{a+jw}) = ...
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1answer
66 views

Residue Theorem for Laplace Transform

I need to know what's the Residue Theorem for a Laplace Transform. Does anyone know the name or something, so I can search it? I couldn't find anything. For example, if I have this two equations: ...
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1answer
19 views

integration of an open curve about isolated singularities

I know if I integrate a circular arc of an angle $\theta < 2\pi$ about an isolated singularity of the complex funciton I would get a fraction $\frac{\theta}{2\pi}$ of the residue of that ...
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1answer
41 views

contour integral in a region where the function doesn't have any poles

What is the value of the following contour integral? The contour is a circle with radius $0.5$ around $z=i$ point: $|z-i|<\frac{1}{2}$ $$\oint_C\frac{dz}{2-\sin z}$$ I think it is $0$ because ...
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1answer
54 views

How can I calculate the integral $ \int_{\left| z \right| = r} \frac{dz}{(z-a)^n(z-b)^n} $ [closed]

How can I calculate the integral? $$ \int_{\left| z \right| = r} \frac{dz}{(z-a)^n(z-b)^n} $$ for $ \left| a \right| < r < \left| b \right|$ and $ m, n > 1$ I tried to use the cauchy ...
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2answers
147 views

Integrating around simple pole and semicircle

Let $f$ be a holomorphic function on $\mathbb{C}$ with simple pole at $z_0$. Then if $\Gamma$ is a circle around $z_0$ oriented counter-clockwise with radius $r$ and $r\rightarrow 0$, then ...
4
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1answer
64 views

Check my answer - complex analysis, using residue and rouche's theorem

I was asked the following questions and I am unsure of my solutions, any advice would be appreciated, maybe there is a better way of doing this. Question: We are given $f(z)=2z-\sinh (z)$ defined on ...
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1answer
20 views

Bounding quantities that appear after using the residue theorem

for an exercise using the residue theorem I need to prove that this term $$\left|\dfrac{e^{R+it}-e^{-R-it}}{\left(e^{R+it}+e^{-R-it}\right)^2}\right|$$ tends to zero as $R\to\infty$. It's clear that ...
0
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3answers
66 views

How can I compute the residue of $\frac{\sinh(z)\sin(\omega z)}{\cosh^2(z)}$ at $z=\frac{i\pi}{2}$?

I need to compute the residue of $$f(z)=\frac{\sinh(z)\sin(\omega z)}{\cosh^2(z)},$$ where $\omega\in\mathbb{R}$ is a parameter, at $z=\frac{i\pi}{2}$, but can I do it without computing the integral ...
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2answers
81 views

Pole on a contour. Problem with integration

I have a problem with calculation of the complex integral $$\int_{|z|=1}\frac{z^2+3z+2i}{(z+4)(z-1)}dz$$ Apparently integrand has a pole in $1$ lying on our circle. What can I do? I cant use Cauchy ...
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1answer
28 views

Finding the limit of a function with sines and cosines by using the taylor expansion

I need to find the residue of a second order pole $z=0$, the residue works out to the following: $$\lim_{z\to 0}\frac{2z\sin{z^2}-2z^3\cos{z^2}}{\text{sin}^2{z^2}}$$ My professor said it's ...
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0answers
29 views

Higher order residues using series

For example if we have any $f(z)$ with a singularity at $z=0$ but it is of order 12 Instead of using the limit definition, suppose $f(z)$ is in the form of: $$f(z) = a_0z + a_1z^2 + a_2z^3 + ... + ...