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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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Using the residue theorem to compute two integrals

Classify the singular points for the function under the integral and using the residue theorem, compute: (a) $$ \int_{|z-i|=2} \frac{z^2}{z^4 + 8z^2 + 16} \, dz, $$ (b) $$ \int_{|z|=2} \sin\left(\frac{...
GENERAL123's user avatar
1 vote
1 answer
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Compute residue of pole of order $m$ of complicated function

Consider the following function: $$f(z)=\frac{\left(z^{-m}-z^m\right)^2 \left(z^{-n}+z^n\right)}{\omega z-z^2-1}$$ with integers $m,n\geq 0$ and an arbitrary constant $\omega\in \mathbb{C}$ with $\rm{...
papad's user avatar
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Validity of Python-derived solution for contour integral $\oint f(z)f(z-\overline{z})~dz$

$\newcommand{\on}[1]{\operatorname{#1}}$ $$ \mbox{Consider the function:}\quad \on{f}\left(z\right) = \frac{{\rm e}^{tz}}{\left(1 + z^{2}\right)^{3}}\, \left(\sqrt{t} - t\right)\ \ni\ t,z \in \mathbb{...
MASTER DHRUV's user avatar
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Residue of a removable singularity at inifinity

Exercise: Find all the singularities of $$\frac{z^3e^{\frac{1}{z^2}}}{(z^2+4)^2},$$ classify them, and find each residue. I found that $+2i, \ -2i$ are poles of order two. I was able to calculate ...
cor1.1.29's user avatar
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3 votes
1 answer
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Classify the singular points for the function under the integral and using the residue theorem

Classify the singular points for the function under the integral and using the residue theorem, calculate: (a) $\displaystyle\int_{|z|=3} \frac{1 - \cosh z}{z^6 + 2z^5} \, dz $ and (b) $\displaystyle \...
user1718's user avatar
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1 answer
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The function $ f $ is given by the formula $ f(z) = \frac{1 - \cos z}{z^5 + z^7}$

The function $ f $ is given by the formula $$ f(z) = \frac{1 - \cos z}{z^5 + z^7}. $$ (a) Classify the singularity at the point $ z = 0 $ and write down the principal part of the Laurent series ...
user1718's user avatar
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4 votes
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Given the function $f$ with the rule $f(z) = \frac{z \sinh\left(\frac{1}{z^2}\right)}{z^2 + 1}$.

Given the function $f$ with the rule $f(z) = \frac{z \sinh\left(\frac{1}{z^2}\right)}{z^2 + 1}$. (a) Determine and classify the singular points of the function $f$ and calculate the residues at these ...
lolip123's user avatar
2 votes
2 answers
49 views

Computing residue at infinity

Let $g$ be holomorphic function in $G = \{|z| > 100\}$ where $$g (z) = \frac{z^{99}}{\prod_{k=1}^{100} (z-k)}$$ I would like to compute $Res(g,\infty)$. By definition $$Res(g,\infty) = -Res(\frac{1}...
SparklyCape290's user avatar
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3 answers
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What is the residue of $f$ at $\infty$?

How do I compute the residue at $a = \infty$ of the function $\operatorname{f}$ ?: $$ \operatorname{f}: \mathbb{C} \setminus \left\{{\rm i},-{\rm i}\right\} \to \mathbb{C},\quad z \mapsto \frac{1}{z^{...
2GR8's user avatar
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2 answers
96 views

Obtaining the residues of $\frac {(z-1)^2} {(e^z - 1)^3}$

I want to calculate the residues of $$f(z) = \frac {(z-1)^2} {(e^z - 1)^3}.$$ I already know that the isolated singularities are of the form $2\pi i \cdot \mathbb Z$, and that they are poles of order $...
Minerva's user avatar
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1 answer
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Use residues to verify the integral $\int_0^{\infty} \frac{x^{1/2}}{(x+1)^2} dx$ [duplicate]

I found this question on page $285$ of the book COMPLEX ANALYSIS FOR MATHEMATICS AND ENGINEERING by J, Mathews. I tried to draw the contour and calculate the contour integral and each path integral,...
Tommaso Zou TommasoZou's user avatar
6 votes
0 answers
174 views

Finding mistake in contour integral. $f(z)=\frac{\exp{(-1+i)z}}{z \cdot z^{1/2}}$

I'm trying to calculate the following real integral: $$I=2\int_{0}^{\infty}\frac{e^{-x^2}\sin(x^2)}{x^2}\mathop{\mathrm{d}x} = \int_{0}^{\infty}\frac{e^{-t}\sin{t}}{t^{3/2}}\mathop{\mathrm{d}t}.$$ I ...
Josemi's user avatar
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1 answer
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Integration over Hankel contour

In a write-up by Paul Garrett, he claims that he can apply the Residue Theorem to the equality $$ \zeta\left(s\right) = \frac{1}{\Gamma\left(s\right)\left(1-{\rm e}^{2\pi{\rm i}s}\right)} \lim_{\...
cxrlo's user avatar
  • 50
0 votes
1 answer
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Residue theorem: Choice of contour

after first learning about the Residue-theorem we now apply it in Quantum Field Theory but now I have realised that I have forgotten a lot of important details. I don't understand when the closed ...
Gogoman96 X's user avatar
4 votes
2 answers
287 views

Is there a way of finding a closed-form expression for $\int_0^\infty\frac{k^2J_0(k)^2\,\mathrm{d}k}{k^4+\left( k^2+x^2 \right)^2}$, $x\in\mathbb{R}$?

I am trying to find a closed-form expression for the following improper integral $$ \int_0^\infty \frac{k^2 J_0(k)^2 \, \mathrm{d}k}{k^4 + \left( k^2 + x^2 \right)^2} \, , $$ where $x \in \mathbb{R}$....
Siegfriedenberghofen's user avatar
1 vote
0 answers
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Theorem requirements not satisfied but answer is correct?

In Ron Gordon's answer here, he uses the residue theorem to compute the sum $$\sum_{k=-\infty}^{\infty} \frac{(-1)^{k} (2 k-1)}{k^2-k+1}$$ The theorem for the case of infinite sums states that Let $f ...
Max0815's user avatar
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5 votes
0 answers
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Zeta Lerch function. Proof of functional equation.

so I'm trying to prove the functional equation of Lerch Zeta, through the Hankel contour and Residue theorem, did the following. In the article "Note sur la function" by Mr. Mathias Lerch, a ...
Nightmare Integral's user avatar
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24 views

The real part of the Stieltjes transform.

Suppose that $\mu$ is a probability measure with pdf f. Consider the Stieltjes transform of $\mu$ being $$G(z) = \int\frac{\mu(d\tau)}{\tau -z}.$$ We know that the imaginary part of $G(z)$ has an ...
H L's user avatar
  • 1
1 vote
1 answer
46 views

Single formula for infinitely many residues of a complex function

I have the following complex function $\frac{(z-1)^2}{(\exp(z) - 1)^3}$. Is it possible to obtain a single formula to express the residues of the function at the third order poles $z=2k\pi i$, where $...
CharlesG's user avatar
1 vote
0 answers
67 views

Using residue theorem to calculate the integral, am I wrong?

$$ \mbox{I encountered an integral}\quad \int_{0}^{2\pi}\frac{cos(x-ai)}{sin(x-ai)}dx $$ It is not a difficult problem, but my result is different from the answer. ...
Andrew_Ren's user avatar
1 vote
3 answers
170 views

Computing integral using complex analysis $\int_{-\infty}^{\infty}\frac{x}{(\sinh(x)-i)}dx$

I couldn't find related problem, so here I am. I have this integral: $$ \int_{-\infty}^{\infty}\frac{x}{\sinh(x)-i}dx=I $$ that i need to compute using complex analysis. I am given the answer is $\pi$...
Fabio I's user avatar
  • 11
2 votes
1 answer
96 views

Convert the Laplace transform of the Bessel function to a Fourier transform

I want to calculate the Fourier transform of the function $f(t)$, defined as $f(t)=0$ if $t<0$ and $f(t)=J_{n}(t)$ if $t\ge0$, in which $J_{n}(t)$ is the Bessel function of the first kind. That is, ...
Lucas Bitencourt's user avatar
1 vote
2 answers
77 views

Show that $\int_{(0, 2\pi)} \frac{\cos(2\theta) d\theta}{5+3 \cos(\theta)} = \frac{\pi}{18}$ using residue theory

Let $\theta=e^{iz}$ with $C : |z| <= 1$ it implies $d \theta = \frac{dz}{iz}$ and then we can get $\cos \theta = \frac{1}{2} (z+\frac{1}{z})$ and $\cos(2\theta) = \frac{1}{2}(z^2+\frac{1}{z^2})$ so ...
Ocean's user avatar
  • 105
0 votes
1 answer
64 views

Procedure to solve $\mathcal{P}\int_{-\infty}^{+\infty} \frac{\cos (\alpha x)}{x^2-1} dx$

I recently solved a complex analysis exercise that required to solve this integral using residue theory, but I don't know where to check for the correctness of the result I got, so I thought of ...
deomanu01's user avatar
  • 113
1 vote
2 answers
55 views

Singularities of complex function $\frac{e^{\frac{1}{z}}}{z^2}$

I know that the function $ e^{\frac{1}{z}}$ has an essential singularity in the origin, but I want to study the function $$ f(z) = \frac{e^{\frac{1}{z}}}{z^2} $$ Especially using the corollary that ...
Hamza Amine's user avatar
2 votes
0 answers
28 views

Calculating $\text{ord}(0,\frac{1}{\sin^2(z)}-\frac{1}{z^2})$

Is the following procedure valid? $$\begin{align*} \operatorname{ord}\left(0,\frac{1}{\sin^2(z)}-\frac{1}{z^2}\right) &=\operatorname{ord}\left(0,\frac{z^2-\sin^2(z)}{(z\sin(z))^2}\right)\\[9 pt] &...
J P's user avatar
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1 vote
0 answers
29 views

$\int_{0}^{\infty} \frac{x^{m-1}}{x^n+1} dx$ [duplicate]

Show that $\large\int_{0}^{\infty} \frac{x^{m-1}}{x^n+1} dx= \frac{\frac{\pi}{n}}{\sin(\frac{m}{n}\pi)}$; $n>m$ Given $z \in \mathbb{C}; z^n\neq-1$, I can define $f(z)= \frac{z^{m-1}}{z^n+1}$ and ...
J P's user avatar
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1 vote
1 answer
46 views

Taylor-Laurent series expansions

I'm having some issues finding how to series expand some complex functions that my professor gave past years in exams. For example, in this exercise, it is asked to find the first two terms of the ...
deomanu01's user avatar
  • 113
0 votes
0 answers
49 views

How to expand a complex function around the point at infinity?

I came across a problem that asked to expand the function $$ f(z) = \frac{1-e^{2iz}}{z^2} $$ both around the point $z=0$ and $z=\infty$. The correct expansion around $z=0$ should be $$ f(z) = -\sum_{k=...
deomanu01's user avatar
  • 113
0 votes
0 answers
63 views

Evaluate $\int_0^\infty \frac{\cos x}{(x^2+1)^2} \, dx$ [duplicate]

Evaluate $$\int_0^\infty \frac{\cos x}{(x^2+1)^2} \, dx \tag{1}$$ It's just setting this one up that's the hardest part. Finding the right contour is what is most tricky to me. My first move is to ...
Grigor Hakobyan's user avatar
3 votes
2 answers
113 views

Residue of $f(z)=\frac{e^{\frac{1}{z}}}{z-z^2}$ at $z=0$

I am trying to calculate the residue of $ f(z)=\frac{e^{\frac{1}{z}}}{z-z^2}$ at $z=0$ by noticing that $$\frac{e^{\frac{1}{z}}}{z}\frac{1}{1-z}=\left(\frac{1+\frac{1}{z}+\frac{(\frac{1}{z})^2}{2!}+ \...
J P's user avatar
  • 893
2 votes
0 answers
101 views

Contour integral $\oint_{c}\frac{1}{z \left(1-(z+1) \sqrt{\frac{r}{z}}\right)}dz$

I am new to complex analysis and am trying to figure out the following contour integral: $$ I = \oint_{\scr C}\frac{1}{z\left[1 - \left(z + 1\right) \sqrt{r/z}\,\right]}\,{\rm d}z $$ where $0<r<...
plywood98's user avatar
0 votes
1 answer
50 views

Using analytic continuation for calculating $\int_{-\infty}^{\infty} \frac{x^2e^{ix}}{(1+x^2)^2}dx$

I have the following integral $\int_{-\infty}^{\infty} \frac{x^2e^{ix}}{(1+x^2)^2}dx$; the complex function $f(z)=\frac{z^2e^{iz}}{(1+z^2)^2}$ has only one pole at $\mathbb{H}=\{z \in \mathbb{C} \mid ...
J P's user avatar
  • 893
0 votes
0 answers
30 views

Residue's theorem- analytic continuation

Let $U$ be an open set and $A \subset U$ a finite set ; let $\gamma$ be a simple loop and null homotopic in $U$ such that $tr(\gamma) \cap A= \varnothing$. Let $f: U -A \to \mathbb{C}$ be an ...
J P's user avatar
  • 893
0 votes
0 answers
65 views

Calculus of residues; $\frac{f'}{f}$

Let $U \subseteq \mathbb{C}$ be an open set, $z_0 \in U$ and $f: U-\{z_0\} \to \mathbb{C}^*$ an analytic function such that $\DeclareMathOperator{\ord}{ord} \ord(z_0,f) \in \mathbb{Z}^*$. Then $z_0$ ...
J P's user avatar
  • 893
0 votes
1 answer
32 views

Inverse Laplace Transform of $e^{-as}/s^2$ for $a>0$

I am trying to compute the inverse Laplace transform of $$F(s) = \frac{e^{-as}}{s^2}$$ for $a > 0$. I computed it as follows: $$\mathcal{L}^{-1}_{s\to t} \left\{\frac{e^{-as}}{s^2}\right\} = \text{...
idk31909310's user avatar
0 votes
1 answer
44 views

Contour integration for option pricing formula of Lewis (2002)

I am looking at the paper by Alan Lewis on option formula for exponential Levy processes and he derives the call option formula to be $$C(S,K,T) = -Ke^{-rT}/2\pi\int_{iv_1-\infty}^{iv_1+\infty} e^{-...
ihammer's user avatar
6 votes
2 answers
190 views

How to calculate $\int_{0}^{2\pi}\frac{\sin^{2}k}{(1+a\cos k)\sqrt{1+b\cos k}}dk$ (elliptic integral)?

My goal is to calculate $$\int_{0}^{2\pi}\frac{\sin^{2}k}{(1+a^2+2a\cos k)\sqrt{1+b^2+2b\cos k}}dk,$$ where $a\neq b$ and $b\neq 1$. But we may simplify it into $\int_{0}^{2\pi}\frac{\sin^{2}k}{(1+a\...
ZJX's user avatar
  • 337
5 votes
2 answers
603 views

Taylor expansion gives a wrong estimate of a residue?

Consider the function $F(z) = e^{iz} / (z^2+1)^2$. To compute the residue of $F$ at $z_0=I$, one can compute the limit of the function $\frac{d}{dz} \left\{ (z-z_0)^2 \, F(z) \right\}$ as $z \to z_0$. ...
poukrakk's user avatar
4 votes
2 answers
112 views

Calculating $\int_{0}^{2\pi} \frac{\sin(\theta) + \cos(2\theta)}{2 + \sin(2\theta)}d\theta$ using Cauchy's Residue Theorem

Calculate the following integral using the residue theorem. $$\int_{0}^{2\pi} \frac{\sin(\theta) + \cos(2\theta)}{2 + \sin(2\theta)}d\theta$$ This was my attempted method: Let $z=e^{i\theta}$, $dz=...
Kaf's user avatar
  • 155
3 votes
0 answers
73 views

Residue of $ze^{\frac{1}{z}}$

I am trying to calculate the residue of $ze^{\frac{1}{z}}$, here's what I got: We have a singularity at $z=0$. We know that $e^w=\sum_{n=0}^\infty \frac{w^n}{n!}$ so $e^{\frac{1}{z}}=\sum_{n=0}^\infty ...
Luke's user avatar
  • 99
0 votes
0 answers
40 views

Trying to calculate $\int_{\vert z \vert = 2} \frac z{\cos z}dz$ [duplicate]

Trying to calculate $$\int\limits_{\vert z \vert = 2} \frac{z}{\cos z}dz$$ but running into a lot of issues. I decided to try and use $$\int\limits_{\vert z \vert = 2} \frac{z}{\cos{z}}dz = 2 \pi i \...
robert lewison's user avatar
3 votes
2 answers
56 views

Issue with evaluating the residue at a double pole

I need to find the residue of the function $f(z)=\frac{A(z)}{B{{(z)}^{2}}C(z)}$ at a zero ${{\bar{z}}_{B}}$ of $B(z)$. The functions $A(z)$, $B(z)$, and $C(z)$ are not elementary polynomials, and have ...
Sharat V Chandrasekhar's user avatar
2 votes
1 answer
79 views

Residue of inverse function using Lagrange inversion

Let $v(z)$ be a infinite power series of terms $z^k$ where $k>0$ with coefficients $v_k$ and a analytic function except at z=0 and $V(z)$ be the inverse of $v(z)$,then show that $Res_0(V(z)^{-k})=k\...
Will Silva's user avatar
4 votes
1 answer
128 views

How to calculate this definite-Integral? Is it possible to use residue theorem?

I'm seeking assistance with the following integral: $$ \int_{0}^{\infty}\frac{1}{\left(1 + cx\right)^{a}\, \left(1 + dx\right)^{b}\,}\,{\rm d}x\quad\mbox{where}\quad \left\{\begin{array}{rcl} {\...
Henry's user avatar
  • 85
0 votes
3 answers
74 views

Q. If $C$ is the circle $|z|=1$ taken with positive orientation, evaluate $\displaystyle\oint_C\dfrac{e^{\sin z}}{z^4}dz$ [closed]

$\displaystyle\oint_C\dfrac{e^{\sin z}}{z^4}dz$ my answer is coming out to be zero I did it using the residue theorem, the residue is coming out to be zero at the point $z=0$ is this the correct way? ...
Manasvi's user avatar
2 votes
2 answers
102 views

Find $\int_0^\pi \frac{8 \, d \theta}{5+2 \cos \theta}$

Find $\int_0^\pi \frac{8 \, d \theta}{5+2 \cos \theta}$ Let $z = e^{i \theta}$. Then $dz = \frac{d \theta}{iz}$ \begin{align} \begin{split} 8 \int_0^\pi \frac{\, d \theta}{5+2 \cos \theta}...
Grigor Hakobyan's user avatar
0 votes
1 answer
51 views

evaluate $\int_{|z-e|=2} \frac{1}{(z-1)\log z}dz$

evaluate $\int_{|z-e|=2} \frac{1}{(z-1)\log z}dz$ we know that $\log(z) = \sum_{n=1}^{\infty} \frac{(-1)^{n-1}(z-1)^n}{n}$ so $\int_{|z-e|=2} \frac{1}{(z-1)\log z}dz = \int_{|z-e|=2} \frac{1}{(z-1)(\...
Confused's user avatar
2 votes
1 answer
42 views

Computing Cauchy-Principal value with residue theorem on the positive line?

I'm trying to compute the following integral \begin{equation} \mathcal{PV}\int_0^{+\infty}dy\frac{y}{\sqrt{1+y^2a^2}{(y-b)}} \end{equation} with $a$ and $b$ positive. I have tried to compute an ...
MohamedSU's user avatar
0 votes
1 answer
59 views

Complex Integral and Residue involving multiple Branch Points inside the contour

I encountered the integral: $$\oint_{|z|=1} \frac{f(z) \ dz}{\sqrt{(z-a)(z-b)}} \ \ \ \ \text{with} \ \ \ |a|,|b| < 1$$ So that the branch points are inside the contour. I am not adding the ...
prikarsartam's user avatar

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