Questions tagged [cauchy-integral-formula]

In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary of the disk, and it provides integral formulas for all derivatives of a holomorphic function. Cauchy's formula shows that, in complex analysis, "differentiation is equivalent to integration"

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Evaluate $\int_{\mathcal{C}}\frac{e^z}{z^2}\,\mathrm{d}z$

How can I solve this problem using Cauchys integral formula $$\int_{\mathcal{C}}\frac{e^z}{z^2}\,\mathrm{d}z$$ When $$C:|z| = 1$$ I believe I should get something like this$$\int_{|z|=a} \frac{f(z)}{z-...
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Evaluate $\int_{\mathcal{C}}\frac{\sin z}{z-\pi/2}\,\mathrm{d}z$

Problem.1) Evaluate $\displaystyle \int_{\mathcal{C}}\frac{\sin z}{z-\pi/2}\,\mathrm{d}z$, given that $\mathcal{C} : |z| = \pi/2$
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Why this integral vanishes for $t\leq 0$?

I am studying semigroup theory and in the proof of Hille-Yosida theorem, we have the integral $$\int_{\omega'-i\infty}^{\omega'+i\infty} \frac{e^{\lambda t}}{\lambda^3}R(\lambda;A)A^3u \, d\lambda$$ ...
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Is the Cauchy's Theorem valid for a path with multiple points?

Let $f$ be analytic in a finite simply-connected region $\mathcal{R}$ described by the inside of a closed curve $C$ (that can contain multiple points) and on its boundary $C$. Then $$ \oint_C f \...
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Basic Complex Integration Question on a circle

I'm trying to find $$\int \dfrac{dz}{(z-1)^3}$$ on the circle $|z|=2$. I'm pretty new to this, so my instincts were to use Cauchy's theorem for this, namely that the integral of an analytic function ...
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$\int _C \frac{\exp(z^2)}{z^2\left(z-1-i\right)}dz$, $C$ consists of $|z|=2$ counterclockwise and $|z|=1$ clockwise.

$\int _C \frac{\exp(z^2)}{z^2\left(z-1-i\right)}dz$, $C$ consists of $|z|=2$ counterclockwise and $|z|=1$ clockwise. Not sure where to begin and was wondering if anyone could give me a hint. I ...
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Let $γ$ be a counter-clockwise path around the unit circle centred at zero. Compute the path integral$\int \frac{(z+1)^{2020}}{z}dz$

$\int \frac{(z+1)^{2020}}{z}dz$ My solution : The path is a closed circle of centre $a = 0$ , radius, $ r=1$ so $ C(a,r) = C(0,1)$ so using Cauchy's integral formula, let $f(z) = (z+1)^{2020}$ and $\...
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Integral for a piecewise smooth contour in complex plane

Let $C$ be the simple closed contour in a complex plane. If $P(z)$ is a polynomial with no root on the curve $C$, show that if $f(z)=\frac{P'(z)}{P(z)^2}$: $$ \int_C \frac{P'(z)}{P(z)^2} dz = 0$$ $\...
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Characterizing functions which satisfy Cauchy integral formula

Charecterize all complex valued functions $f$ satisfying below statement: Suppose $C$ is a simple closed curve inside a simply connected domain $D$ and for any point $a$ inside $C$, $$2i\pi f(a)=\...
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How can I compute the following complex integral?

Let $f(z)=Re(z)$ and $\gamma=\{|z|=1\}$ be the unit circle oriented counterclockwise. Compute $\int_\gamma \frac{f(z)}{z-1/2}~dz$ From the lecture we know that $f(z)=\frac{z+z^{-1}}{2}$ on $\gamma$. ...
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How do I compute the following integral using Cauchy's integral formula?

Let $f$ be an analytic function on a neighborhood $\{|z|\leq 1\}$ and $\gamma$ is a unit circle oriented counterclockwise. Show that for $0<|z|<1$ $$2\pi i f(z)=\int_\gamma \frac{f(w)}{w-z}~dw-\...
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Is it possible to calculate $\int_0^\pi\frac{dx}{2+\sin^2x}$ without residue theorem

I came across the following integral: Calculate $\int_0^\pi\frac{dx}{2+\sin^2x}$. I know that you probably can solve it using residue theorem, but since we haven't proved it yet, I tried another ...
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Using Cauchy Formula to find $\int_{|z|=2}\frac{e^z}{z(z-3)}dz$

I am a bit confused on how I would use Cauchy's to find the integral: $$\int_{|z|=2}\frac{e^z}{z(z-3)}dz$$ Any help would be appreciated! Edit: $z = 0$ inside $z = 2$, so it is a singularity. $$\int_{|...
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Show that $f$ given by a power series has no roots in $B_r^{\mathbb{C}}(0)$ [closed]

One has a convergent power series $f(z)= 1+ \sum_{n=1}^\infty a_n z^n$ on $B_R^{\mathbb{C}}(0)$ (open ball) for some $R>0$, $~0<\rho<R$, $~M_\rho(f):=\max\{|f(z)|:|z|=\rho\}$ and $r:=\frac{\...
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How to evaluate the following contour integral?

I have recently studied Cauchy's integral formula which states that if $f:\Omega\to \mathbb C$ be a holomorphic function and $C$ be a positively oriented simply closed curve whose interior is also ...
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A question about the condition of Cauchy's theorem

Theorem 10.12 from Rudin's Real and Complex Analysis says: Suppose $F \in H(\Omega),F'$ is continuous in $\Omega$ .Then for every closed path $\gamma$ in $\Omega$ ,$\int_{\gamma}F'(z)dz=0$. And a path ...
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Analatic functions with two variables applied to square matrices

I understand if square matrix $A=P_{1}^{-1}D_{1}P_{1}$, where $D_{1}$ is a diagonal matrix, we can write $f(A)$ as $P_{1}^{-1}f(D_{1})P_{1}$. My question is how to define $f(A,B)$ where $A=P_{1}^{-1}...
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Compute $\int_{\Gamma} \frac{\cos(z)} {z^2(z-3)}\, dz$ along the contour indicated in Fig

Here is the problem. How do I choose the correct denominator? Compute $$ \int_{\Gamma} \frac{\cos(z)} {z^2(z-3)}\, dz $$ along the contour indicated in Fig below. Here is my answer. $$ \begin{align} \...
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Justification of interchanging limit and integral in the proof of the derivative of Cauchy integral formula

On Stein and Shakarchi's Complex Analysis book, in the course of proving that for $f$ a holomorphic (i.e. differentiable) function on an open set, and $C$ a circle whose interior is also contained in ...
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Calculating the integral $I(w)=\int_\mathbb{R}\exp(-\pi(x+w)^2)dx$ for $w\in\mathbb{C}$

Suppose $I(w)=\int_\mathbb{R}\exp(-\pi(x+w))^2dx$ with $w\in\mathbb{C}$. There are two parts to this problem : i) I have to show that $I(w)$ converges uniformly on compact sets $K\subset\mathbb{C}$ ii)...
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How to calculate $\int_{\partial D(0;a)} \frac{dz}{(z-a)(z-b)}$

How to calculate $\displaystyle\int_{\partial D(0;r)} \frac{dz}{(z-a)(z-b)}$ when $|a|<r<|b|$?. My first idea: I tried to separate by partial fractions, so for some $A,B$ \begin{equation} \...
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Complex integral of $\int_{-\pi}^{\pi}\frac{\cos\theta\,d\theta}{\csc \alpha+\cos\theta}$ [closed]

A current $I_1$ flows in a circular circuit of radius a and a current $I_2$ flows through a very long straight conductor in the same plane of the circular circuit (see the figure). From the laws of ...
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3 votes
1 answer
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Closed rectifiable curve with arbitary winding number.

This is an exercise from Conway's Functions of One Complex Variable(page 83, exercise 2). Give an example of a closed rectifiable curve $\gamma$ in $\mathbb C$ s.t. for any integer $k$ there is a ...
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Cauchy's theorem with a non closed curve

I'm reading Wavelets - tools for science and technology by Jaffard, Meyer and Ryan and I have a problem with a statement of section 10.4.1. They say that there is a form of Cauchy's theorem that says :...
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Integral of a complex function (contour integral)

This is the integral: $$\int_{\gamma}z^2\frac{f'(z)}{f(z)}dz$$ we know that: $\gamma$ is a simple closed curve; $f$ is analytical in $R$ (the region bounded by the curve) and in all the points of the ...
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Why do holomorphic functions on a disk admit so many different representation formulae?

Consider a holomorphic function $f$ on the unit disk that extends continuously to the boundary circle, i.e. $f \in H(\mathbb{D}) \cap C(\overline{\mathbb{D}})$ with $f|_{\partial \mathbb{D}} = f_1$. ...
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2 answers
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Using the Cauchy integral formula show that $\oint_{ |z|=2} \dfrac { e^zdz}{(z-1)^2(z-3)}= -\dfrac 3 2je\pi$

Using the Cauchy integral formula show that $$\oint_{ |z|=2} \dfrac { e^zdz}{(z-1)^2(z-3)}= -\frac 3 2je\pi.$$ My work is in this picture. The answer is $-\frac 3 2je\pi$. I used the singularities ...
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Holomorphic function being identically zero

I've got this question that I've been trying to prove using no Taylor/Laurent series, no use of Residue theorem, only using the Cauchy's integral formula but have not made any progress on. ...
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Cauchy integral problem

Cauchy's integral formula states that if $f$ is holomorphic inside and on a positively contour $\gamma$, then if $a$ is inside $\gamma$ we have: $$ \tag{1} f(a)=\frac{1}{2 \pi i} \int_{\gamma} \frac{f(...
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Using Residues theorem to do inverse Z transform "nearly works".

I have tried to build the habit of using the Residue theorem to do inverse Z transform where ever the table entries are not clearly associated with the function. I have a long example here which "...
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1 answer
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Is this solution using Cauchy's integral formula correct?

Cauchy's integral formula is: $$f(z_0)n(\gamma,z_0)=\frac{1}{2\pi i}\oint_\gamma\frac{f(z)}{z-z_0}dz$$ The problem I had to solve was: $$\int_\gamma \frac{dz}{(z^2+9)(z+9)};\gamma:|z|=4$$ My thoughts ...
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Second order of boundary value problem system using repeated Cauchy integral?

Let $(S)$ be the folowing boundary problem such that : $$(S)\begin{cases} y''=f(x,y(x)), \quad a<x<b \\ y(a)=\alpha,\quad y(b)=\beta \end{cases}$$ We call $ y\in C^2([a,b],\mathbb{R})$ every ...
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Cauchy's integral theorem using differentiation under the integral sign

I came across the following extremely simple proof of Cauchy's integral theorem for analytic functions: https://www.jstor.org/stable/27642307?seq=1#metadata_info_tab_contents The proof uses ...
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When does 𝑅𝑒∫𝑓(𝑧)𝑑𝑧=∫𝑅𝑒(𝑓(𝑧))𝑑𝑧? [closed]

When does $Re(\int_Cf(z)dz) = \int_c Re(f(z)dz)$? I genuinely have no idea where to even begin with this. I thought about letting $f$ be a function, such as $f = \frac{1}{z}$ but don't know how that ...
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Finding partial fractions of $\frac{z^3+2z^2-2z}{(z-2)(z^2+2}$ and/or using Cauchy's formula to solve

I am trying to find the inverse z-transform of \begin{equation} x(z)=\frac{z^3+2z^2-2z}{(z-2)(z^2+2)} \end{equation} and for this we need to get partial fractions. I have tried multiple approaches, ...
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7 votes
0 answers
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Cauchy's integral formula and essential singularities

Let $f$ be holomorphic at $z_0\in\mathbb C$. I would like to compute the integral $$\oint_{\gamma_{z_0}} f(z)\, e^{\frac{1}{z-z_0}}dz,$$ where $\gamma_{z_0}$ is a small circle around $z_0$. By ...
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Intuition about residues of holomorphic functions

Does the concept of residue go beyond the identification with the coefficient of the term $z^{-1}$ of the Laurent series of a holomorphic function? Wouldn't the name for this algebraic entity give a ...
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2 votes
2 answers
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How do I evaluate $\frac{1}{2 \pi i} \oint_{C} \frac{z^{2} d z}{z^{2}+4}$

How do I evaluate the following integral when where $C$ is the square with vertices at $\pm2, \pm2+4i$, $$\frac{1}{2 \pi i} \oint_{C} \frac{z^{2} d z}{z^{2}+4}$$ Using Cauchy integral: $\frac{z^{2}}{z^...
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1 vote
1 answer
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Proving the generalized Cauchy integral theorem using Green's theorem

I was looking for proofs of the generalized Cauchy integral theorem: Theorem (Generalized Cauchy integral theorem): Let $\Omega\subset\mathbb{C}$ be an open set and $\gamma:[a,b]\to\Omega$ be a ...
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Identifying an inverse z-transform using Cauchy's theorem

I have the z-transform of a sequence, and I want to find the sequence. The transform is: \begin{equation} V(z)=\frac{z}{2z^2-7z-4} \end{equation} Instead of using tables of inverse z-transforms, I ...
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1 vote
0 answers
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$\zeta'(s)\ll (\log t)^2$ using Cauchy's formula, proof verification

For $s=\sigma+it$, $t\geq 8$, $1-\frac{1}{2}(\log t)^{-1}\leq \sigma\leq 2$, \begin{equation} \zeta(s)\ll \log t \text{ and } \zeta'(s)\ll (\log t)^2 \end{equation} The first part can easily been ...
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1 answer
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If $P$ is a polynomial and $max_{|z|=1}P(z)\leq1$ then $P(z)\leq|z|^n$ when $|z|$>1 [duplicate]

I've seen many questions that regard a monic polynomial, where the main trick is to use Cauchy's integral formula to deduce about $P(z)$. Well, this question is concerned about a general $P(z)$. If $P(...
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2 votes
1 answer
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Cauchy's integral with sin used in it [closed]

$$ \int_{0}^{2 \pi} \frac{1}{5-3 \sin \theta} d \theta $$ hello, i think there are multiple ways to solve this question, but i need to use Cauchy's. would you help me with it? thanks
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-1 votes
1 answer
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Cauchy theorem in a general domain

Let $f$ be an analytic function in $D$ and continuous on $\partial D$, where $D\subseteq\mathbb{C}$ is an open set, bounded by a Jordan curve. Is it true that $f\left(z\right)=\frac{1}{2\pi i}\int\...
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5 votes
1 answer
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Ahlfors page 123: Compute $\int_{|z|=2}z^n(1-z)^mdz$. What happens when $n,m<0$? Residue Theory? Cauchy's Theorem?

This is a question from Ahlfors, page $123$, number $1b$: Compute $\int_{|z|=2}z^n(1-z)^mdz$. He doesn't specify anything about $n$ and $m$ on the page, so I am not sure if they are natural numbers ...
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1 answer
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What is the Cauchy's integral formula for this? [closed]

$$\oint_{c : |z -\pi i| = 1} \frac{e^{iz}}{z(z-\pi i)^2} dz $$ I tried to solve it and the answer is $-\frac{1}{\pi} + i$ How can I make sure that the answer is right ?
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1 vote
1 answer
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Laplace transform and Cauchy integral formula

Let $f : \Bbb R_+ \to \Bbb C$ be a bounded and continuous function such that its Laplace transform is holomorphic on all $z \in \Bbb C$ such that $Re(z)>0$ and has a meromorphic extension on $\Bbb ...
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integrals with cauchy theorem

I have complex analysis exam next week and one of the main question is to solve some integrals with the Cauchy formula which is $$f(z)=\frac{1}{2\pi i}\int \frac{f(w)}{w-z}dx$$ and I can solve a big ...
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3 answers
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Integrating f(z)=$z/(16-z^2)(z+i)$ over a circle $|z|=5$

i know the approach of how to solve the question as i need to trifurcate f(z) such that it becomes something like $(A/4-z) + (B/4+z) + C/(z+i) $ and then simply apply cauchy integral formula but i am ...
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1 answer
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Find the complex integral of $f(z) = \frac{z^3e^{1/z}}{1+z^3}$ on the circle contour $|z|=3$ [duplicate]

If $f(z) = \frac{z^3e^{1/z}}{1+z^3}$ and $C:=\{z \in \mathbb{c} :|z|=3\}$ be in a positive sense (anti-clockwise), then $$\int_{C}f(z)dz = $$ (a) $2\pi i$ (b) 0 (c) $-2\pi i$ (d) $-3\pi i$ Attempt: ...
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