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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Calculate the following integral $\int_{|z|=1}\frac{z^m}{(z-a)^n}dz$

Given $n,m\in\mathbb{N},|a|\neq1$ Calculate the following integral $\int_{|z|=1}\frac{z^m}{(z-a)^n}dz$ I thought maybe using Cauchy's integral formula and I'm not sure what happens when $a$ is ...
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Calculate integral along a path using Cauchy´s Integral Theorem

I have the 1-form $w = \frac{dz}{z^2+1}$ defined in $U = \mathbb{C}$ \ {i, -i} and two paths: $\alpha_{r}$ : [$r$, $-r$] $\rightarrow \mathbb{R}$ $\hspace{1,7cm}t \rightarrow t$ $\beta_{r}$ :...
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Relation between the set U of a 1-form w and Cauchy´s Integral Theorem

I have a 1-form $\omega = \frac{az+b}{z^2+1}dz$ on $U = \mathbb{C}$ \ {i,-i} and I have to find for wich $a,b \in \mathbb{C}$ $\omega$ is exact on $U$. So I want to see if the integral along a loop ...
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What happens when integrating a function of which poles appear on the branch cut

I have a complicated function to integrate from $-\infty$ to $\infty$. $$ I = \int_{-\infty}^{\infty}\frac{(2k^2 - \Omega^2)(I_0^2(\Omega) + I_2(\Omega)^2) - \Omega^2 I_0(\Omega) I_2(\Omega)}{\sqrt{...
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Proof of Cauchy integral theorem on any simply connected set

I am trying to find a reference for a proof of Cauchy integral theorem, ie the fact that given a simply connected open subset $U$ of the complex plane, a rectifiable loop $\gamma$ contained in $U$ and ...
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Unusual Application of Cauchy's Integral Formula

So, I've got the following function: $f(x) = sin(\frac{1}{z-2}) + \frac{1}{z^4+1}$ I'm integrating over a circle centered at the origin with radius between 1 and 2, so some singularities are inside ...
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Complex integration $\int_{C(0,1)}(2z-3)\cos5zdz$

The following complex integration question has me stuck... $$\int_{C(0,1)}(2z-3)\cos5zdz$$ So I wondered is there a way of applying a theorem to quickly solve this or should I look at integration by ...
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Evaluate the given integral along the indicated closed contour $\oint_C\frac{sinz}{(z^2)+(\pi)^2}dz; |z-2i|=2$

Evaluate the given integral along the indicated closed contour $\oint_C\frac{sinz}{(z^2)+(\pi)^2}dz; |z-2i|=2$ This should be solve by using Cauchy's formula but i couldnt find.( z=$\pi$i is in C) ...
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$\oint_C\frac{1+e^z}{z}dz$ with |z|=1 evaluate the given integral along the indicated closed contour.

Consider $\oint_C\frac{1+e^z}{z}dz$ with |z|=1. evaluate the given integral along the indicated closed contour. I think this can be solve by using Cauchy's formula. But i'm not sure. If I take$ f(...
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If $f$ is a nonconstant entire function such that $|f(z)|\geq M|z|^n$ for $|z|\geq R$, then $f$ is a polynomial of degree atleast $n$.

I have a question in my assignment : If $f$ is a nonconstant entire function such that $|f(z)|\geq M|z|^n$ for $|z|\geq R$ for some $n\in\mathbb N$ and some $M$ and $R$ in $(0,\infty)$ show ...
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If f is analytic on the simple closed contour C, then $\oint_C(f' (z))/((z-z_0 )dz= \oint_C f(z)/(z-z_0 )^2 dz$ [duplicate]

If f is analytic within and on the simple closed contour C and z_0 is a point within C,then $$\oint_C\frac{f' (z)}{z-z_0 }dz= \oint_C \frac{f(z)}{(z-z_0 )^2} dz$$ Is this statement true or false? ...
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Complex analysis: Cauchy Intergral exercise

I got the answer being $10\pi i\cos(5i)$ by using the Cauchy integral formula for derivatives, with $n=1$, and $g(z)= \sin(5z)$, making the derivative by $5\cos(5z)$. Is it wrong?
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Prove that a complex polynomial has $|p(z)| ≥ 1$ with $z ∈ \mathbb{C}$

I'm doing a course on complex analysis and while studying for my exam I found this problem: Consider $p(z) = z^n + a_{n-1}z^{n−1} + · · · + a_0$ with $n ≥ 1$. I have to show that exist $z ∈ ∂\mathbb{...
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Finding an equality of entire funcitons using Local Cauchy properties.

The question is the following: Let $a \in \mathbb{R}$ and $f$, $g$ entire functions such that $$Re(f(z)) \leq a Re(g(z))$$ for every $z \in \mathbb{C}$. prove that exists $c \in \mathbb{C}$ such that ...
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Evaluate for $n \in \mathbb N$ $\int_{C(i,2)} \frac{e^z}{(z-1)^n}$

Evaluate for $n \in \mathbb N$ $\int_{C(i,2)} \frac{e^z}{(z-1)^n}dz$ My Understanding $f(z)=\sum_{n=0} a_n(z-\xi)^n$ $\frac{f^n(\xi)}{(n)!} = a_n = \frac{1}{2\pi i } \int_C \frac{f(z)}{(z-\xi)^{n+1}...
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Complex Integral $\int_{C(0,2)} \frac{e^z}{i\pi -2z}$

Evaluate: $\int_{C(0,2)} \frac{e^z}{i \pi -2z}dz$ So using the Cauchy Integral Formula: $\int_{C} \frac{f(z)}{z-z_0} = 2\pi i f(z_0)$ If I define $f(z)= \frac{e^z}{-2}$ then the $z-z_0=\frac{-i\pi}{...
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Complex Integral $\frac{e^z}{z-1}$

I'm looking to evaluate the following integral $\int_{C(0,2)} \frac{e^z}{z-1} dz$. So if I let $f(z)=\frac{e^z}{(z-1)^1}$ then: $e^z$ is entire and the root of $z-1$ is $1$ and $1 \in D(0,2)$. But ...
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Using cauchy integral for a gamma path. Is the integration done correctly?

Given the following integral of gamma on the path for the circle $[0,2]$ and circle $[-2,2]$. Where 0 is the center and 2 is the radius as well for the 2nd circle 0,-2 is the center and 2 is the ...
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Complex Analysis. How to use cauchy intergral

Given the following integral of gamma on the path $[0,2]$ and $[-2,2]$ we have the integral $$ \int_\gamma \frac{z} {(z^2-1)(z-3)}dz$$ I set it up like $$ \int_\gamma \frac{\frac{z}{(z+1)(z-3)}}{z-...
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Why isn't $\oint_{C} f(z) = 2\pi i\, \mathrm{d}z$?

I was going over some practice problems T/F: If $C$ is the circle in $\mathbb{C}$ of radius $10$ centered at $z=2i$ with positive orientation and $$f(z) = \frac{\cos(4z)}{z}$$ then $$ \oint_{...
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a formula for $f(z)$ in a ring between two contours

Let $\gamma_1$ be the positively oriented circle $|z| = 1$, $\gamma_2$ the positively oriented circle$|z| = 2$ and $f$ be analytic on $\gamma_1$ and $\gamma_2$ and on the area between them, I need to ...
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On complex integration

The question given is to integrate: $∫z^2e^{1/z}dz$ over the unit circle I managed to get $z^2e^{1/z}$ as $z^2 + \frac{z}{1!}+ \frac{1}{2!} +....$ However I fail to understand how to get the ...
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Question in Complex Analysis

Let $f$ be a analytic in Domain $D$ which is an open disk of radius 1 centered at $z=0$ , it is given that in Domain $D$ , $|f(z)|\leq 1-|z|$ then prove that $f(z)=0$ in Domain $D$ It is asked that ...
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Problem with the Cauchy´s integration formula

I'm reading the Pinsky paper for the Erdos-Feller-Pollard theorem, but i'm stuck on one of the steps. First of all, he propose four conditions: 1.- $f_n \geq 0$, $\sum_1^\infty f_n =1$, $\sum_1^\infty ...
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Complex Contour Integral $\oint_{C(-i, 1)} \left(\frac 1 {(z+i)^3} - \frac 5 {z+i} +8\right)\,dz$

$$\oint_{C(-i,1)} \left(\frac {1} {(z+i)^3} - \frac {5} {z+i} +8\right)\,dz$$ The following integral is causing me lots of issues. I believe Cauchy Integral Theorem applys here and that the ...
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Cauchy Integral Formula Question $\int_{C(2,1)} (z^4-\frac 1 z)dz$

I'm looking to compute the following integral $$\int_{C(2,1)} (z^4-\frac 1 z)dz$$. My question here is does Cauchy’s Integral Formula hold or even apply here? Is $f(z)=(z^4-\frac 1 z)$ entire and ...
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Cauchy estimates and small derivatives at the center of discs

In the introductory chapter of S. G. Krantz's "Complex Analysis: The Geometric Viewpoint" (second edition), he writes the Cauchy estimate theorem in the following form: Let $F$ be a holomorphic ...
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Question about complex integration over half circle (not closed)

Here we consider a complex function $f(z) = \frac{\sin(z)}{z-\alpha}$ where $\alpha = \frac{\pi}{2}+i \log(2)$. I want to know $\lim_{R\to\infty}\int_{-R}^{R}f(z)dz$ (let us call it $I$ for ...
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Proof of subordination principle for holomorphic functions on $\mathbb{D}$

I am trying to prove a very simple theorem that uses the general idea in complex analysis that if $f:\mathbb{D}\to\mathbb{C}$ is holomorphic, then the quantity $|f'(0)|$ is somehow responsible for how ...
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48 views

Cauchy-Goursat Theorem vs Cauchy Integral Formula

What is the difference between Cauchy Goursat Theorem and Cauchy Integral formula? Given an integral where you're supposed to use one of the two I can't seem to differentiate between them. Thanks
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How to determine the uniqueness of the “Laurent representation” of a function?

Let $A$ be the annulus $\{z:r_1<|z|<r_2\}$, where $r_1$ and $r_2$ are given positive numbers. (a) Show that the Cauchy formula $$ f(z)=\frac{1}{2\pi{i}}\left(\int_{\gamma_1}+\int_{\...
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Using Cauchy Inequalities to bound coefficients of power series representation

How do I evaluate the coefficients of a function, that is analytic in the unit disk with a power series $\Sigma a_n z^n$ representation that has the following property $$|f'(z)| \leq \frac{1}{1-|z|}$$...
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Application of Cauchy's integral formula.

Let $f$ be an entire fundtion satisfying $|f^{\prime}(z)|\le 2|z|$ for any $z \in \Bbb C$. Then show that $f(z)=a+bz^2$ for some $a,b\in \Bbb C $ with $|b| \le 1$. My trial : I tried to show that $f^{...
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Estimates for Coefficients using Cauchy's Integral Formula (double integrals)

I'm trying to solve the following problem, but am having trouble finishing it. Here's the problem: Let $U,V$ be open discs centered at the origin and let $f(z,w)$ be continuous on $U\times V$ such ...
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38 views

Show that for all $n ∈ \mathbb N$ there exists a constant $C_n > 0$ such that $|f ^{(n)} (z)| ≤ C_n/(Imz)^ n $ for all $z ∈ H$

Let $H$ be the upper half plane and let $f : H → \mathbb C$ be holomorphic on $H$. Suppose that $f$ is bounded. Show that for all $n ∈ \mathbb N$ there exists a constant $C_n > 0$ such that $|f^{(n)...
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Showing that the integral of $f(z) dz$ over the upper semicircle tends to $0$

Define $f(z) = \frac{e^{iz}}{z} $ and given $R > 0$, let $γ_R$ is the upper semicircle of radius $R$ and center $0$. Show that $\int_{γ_R} f(z)dz → 0$ as $R → ∞$. (Hint: after a suitable ...
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Integrate $\int_0^\infty \frac{\sin x}{x(1+x^2)^2} dx$ with contour integral

How to integrate $$I=\int_0^\infty \frac{\sin x}{x(1+x^2)^2}dx$$ using contour integration? Since, it is even, $$I=0.5*\int_{-\infty}^{\infty}\frac{\sin x}{x(1+x^2)^2}dx$$ $$I=-0.25i(\int_{-\infty}^...
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Double integral over the unit circle and Cauchy's integral formula

Let $f$ be analytic on the unit disc $D$ and assume $\int \int _{D} |f|^{2} dxdy$ exists. Let \begin{equation*} f(z)=\sum_{n=0}^{\infty} a_{n}z^{n} \end{equation*} Prove that \begin{equation*} ...
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28 views

Finding the Laurent Series expansion in positive and negative powers

Expand the function \begin{equation*} f(z)=\frac{z}{1+z^{3}} \end{equation*} a) in a series of positive powers and b) in a series of negative powers. In each case, specify the region in which the ...
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Uniform Limits of Analytic functions, complex analysis question

Here's the question I'm trying to answer: Let $f$ be analytic on an open set $U$, let $z_{0}\in U$ and $f'(z_{0})\neq 0$. Show \begin{equation*} \frac{2\pi i}{f'(z_{0})}=\int _{C}\frac{1}{f(z)-f(...
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48 views

Use Cauchy integral to calculate

a)$\displaystyle\int_{\partial B_2(0)}\dfrac{e^z}{(z+1)(z-3)^2}dz$ Apply the Cauchy integral with $f(z)=\dfrac{e^z}{(z-3)^2}$ at $ z= -1 $. Then: $$\int_{\partial B_2(0)}\dfrac{e^z}{(z+1)(z-3)^2}...
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48 views

Cauchy-Type integral

Greeting, I have an integral to solve, it is $$\int_{-\infty}^{+\infty} \frac{g(x)}{x-z}dx$$ with $g(x)$ a smooth, continuous, positive function, and $z=z_r + iz_i$ a complex number. I saw that this ...
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An upper bound for the modulus of the derivative of an analytic function in the unit disk (from D. Sarason's “Complex Function Theory”)

I'm having some trouble tackling the following, which appears as an exercise after the Schwarz Lemma part. Exercise VII.17.3. Let $f$ be a holomorphic map of the unit disk $D$ into itself. Prove that:...
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31 views

Evaluate $\int_{C}\frac{|z|\mathit{e}^{z}}{z^{2}}dz$.

Evaluate $\int_{C}\frac{|z|\mathit{e}^{z}}{z^{2}}dz$, where $C$ is the circumference of the circle of radius 2 around the origin. I wanted to use the Cauchy integral formula, but is $|z|$ analytic on ...
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92 views

Computing $\int_{|z|=2} z^n(1 - z)^m\ dz$ for $n,m\in\Bbb Z$ [closed]

Computing $\int_{|z|=2} z^n(1 - z)^m\ dz$ I need help for this question when $m, n$ are negative integers. Thanks.
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36 views

How to prove the Laurent series converges to the right thing?

From what I understand, the main "point" of the Laurent series is that we should be able to derive it easily (e.g. by stitching together known Taylor series), and then exploit its uniqueness to say ...
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51 views

Basic question about winding number in Complex Analysis

I have a question which is very basic. In the proof that the complex integral $$W(\gamma,a)=\frac{1}{2\pi i} \int_{\gamma} \frac{dw}{w-a}$$ is an integer, some proof start out-of-the-blue by saying "...
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46 views

residue integral for specific range of a

I know for $\int_0^{2\pi} \frac{d\theta}{1+a cos\theta} $ where $-1<a<1$, we take a c: unit circle as the contour and change the integrand into a rational function and then apply the residue ...
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18 views

complex integration on closed c

Let $c: |z-i|=1$ evaluate integral $I=\int_c e^{z^2}+z^4+\frac{z^3+4z}{2z-i}+\frac{1}{z^2+1} \,dz$ this is my answer please correct me if I m wrong. I m learning. $I=\int_c e^{z^2}dz\ +\int_cz^4dz\ +...
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Can I use the Cauchy formulas for the higher order derivatives

Consider the funtion $exp(\frac{z}{1-z})$. Prove that it's Taylor coefficients at $0$ are given by $a_0 = 1$ $a_n = \sum_{s=1}^n \frac{1}{s!} \binom{n-1}{s-1}$. Is it posible to use cauchy integral ...

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