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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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Wirtinger derivatives and Cauchy integral

Suppose a complex valued function $f$ is of class $C^1$ defined on the disk $|z-z_0|<R,$ and let $C_r$ denote the circle $|z-z_0|=r$ with $0<r<R$. Prove that $$\lim_{r \to 0}\frac{1}{2\pi i ...
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Proving $g'(z)=\frac{1}{2\pi{}i}\int_{C}\frac{g(u)du}{(u-z)^2}$ for $g(z)$ holomorphic in and on contour $C$, and $z$ in $C$'s interior

Prove that if $g(z)$ is holomorphic everywhere inside and on a simple closed contour $C$, taken in a positive sense, and $z$ is any point interior to $C$, then $$g'(z)=\dfrac{1}{2\pi{}i}\int_{C}\...
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Residue theorem and a two-dimensional integral: not working?

Consider the following integral: \begin{align} \iint_{\mathbb{R}^2}d t\,dT\, \frac{e^{-i(t-T)}e^{-t^2}e^{-{T}^2}}{(t-T-i\epsilon)^2}\,. \end{align} The $i\epsilon$ prescription simply tells me that if ...
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Cauchy integral with positive lower limit of integration

I’m trying to solve an integral of the following form: $\int ^b _a \frac{1}{(\omega^2+r_1)(\omega^2+r_2)}d\omega$ Where $r_1$ and $r_2$ are complex conjugates. I used the Cauchy theorem to solve ...
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Evaluate $\frac{1}{2\pi i}\int_\gamma\frac{f(z)}{(z-z_1)(z-z_2)}-\frac{f(z)}{(z-z_0)^2}$

In Marsden's Complex Analysis, section 2.4, the main theorem is Cauchy's integral formula (C.I.F) and there appears this problem: Let $f$ be analytic inside and on $\gamma: |z-z_0|=R$. Prove that ...
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Evaluate the integral $\int_C f(z)dz$ where $C$ is the unit circle centered at the origin with $f(z) = e^{iz}/(z-a)$, $0<a<1$?

Evaluate the integral $$\int_C f(x)dx \,$$ where C is the unit circle centered at C the origin with f(z) = $$\frac {e^{iz}}{z-a}\,$$ for 0 < a < 1. I'm in complex and we are using the idea of ...
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A question about Cauchy Integral Theorem.

I am having some trouble understanding the definition of Cauchy Integral from my textbook. My problem is $\partial D$. I understand $\partial D$ is the boundary of $D$. Why would you take the ...
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Evaluate the complex integral $\oint\frac{4i(z^2+4)}{z(z^2-16)}\sin\Big(\frac{5\pi}{z^2+4}\Big)dz$

I have to evaluate the following contour integral, along the positively oriented contour $C:|z-3|=2$ $$\oint\frac{4i(z^2+4)}{z(z^2-16)}\sin{\Bigg(\frac{5\pi}{z^2+4}\Bigg)}dz$$ Attempt: since $z(z^...
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Analytic functions and their relation to harmonic functions

So we have the Cauchy Integral formula that states that $$ f(z_0)=\frac{1}{2\pi i}\oint_{C_r}\frac{f(z)}{z-z_0}dz\dots(1)$$ I've managed to show that (1) can be equivalently expressed (for a ...
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How does the fact that every integral around a toy contour vanishes imply that the function is holomorphic?

I know that if a function is holomorphic in the enclosed domain, then it follows that the integral around the contour vanishes. However, my question is rather, how does the other direction follow? If ...
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Complex Integration - Cauchy's Formula

Wanted to check if I got the right answer/ idea for this question: $$\int_{|z|=1} \frac{\sin(z)}{z}\mathrm dz$$ Attempt: The region of the curve is the unit circle so there is a singularity at the ...
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Integral $\int_{-\infty}^{\infty}\frac{1}{e^{\frac{x-\mu}{T}}+1}\frac{\gamma}{(x-x_{0})^{2}+\frac{\gamma^{2}}{4}}dx$

I want to solve the integral \begin{align} I=\int_{-\infty}^{\infty}\frac{1}{e^{\frac{x-\mu}{T}}+1}\frac{\gamma}{(x-x_{0})^{2}+\frac{\gamma^{2}}{4}}dx \end{align} where $T,\mu,\gamma,x_{0}\in\mathbb{...
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Integrate $\int_{|z|=1}z^{3}e^{1/z}dz$ - verification

I integrate over a circular path centered at 0 with radius 1 $\int_{|z|=1}z^{3}e^{1/z}dz=\int_{|z|=1}z^{3}\sum\limits_{n=0}^{\infty}\frac{1}{n!z^{n}}dz=\int_{|z|=1}\sum\limits_{n=0}^{\infty}\frac{1}{...
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Integrate $\int_{|z|=1/2}\frac{e^{1-z}}{z^{3}(1-z)}dz$ verification

I integrate over the edge of a circle $K$ with radius 1/2 $\int_{|z|=1/2}\frac{e^{1-z}}{z^{3}(1-z)}dz=\int_{|z|=1/2}-\frac{e^{1-z}}{z^{3}}\frac{1}{(z-1)}dz$ By the Cauchy Integral form $f(w)=\frac{...
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For what simple closed curves does the equation hold?

$\int_{\gamma} \frac{dz}{z^{2}+z+1} = 0$. I know that by Cauchy's Theorem, if $f$ is analytic everywhere on and inside a simple closed curve, then $\int_{\gamma} f(z)dz = 0$. So I thought that the ...
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Integral involving a Gaussian and a rational function.

Let $a \ge 0$ and $b \ge 0$ be real numbers. By generalizing the approach from Evaluating $\int_{\mathbb{R}}\frac{\exp(-x^2)}{1+x^2}\,\mathrm{d}x$ . we have derived the following results. Let $n\ge ...
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Contour integral vanishes

I need some help regarding why the contour integral $$\int_{\gamma} \frac{f(w)}{w-\frac{1}{\bar{z}}} \mathrm dw$$ is equal to zero, where $\gamma$ is the unit circle and $f$ is holomorphic on the unit ...
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Recommendations for a scoring formula for sorting a location based social network's post feed

I'm building a location based social networking app. The idea is to allow users to post and interact with people nearby. The main feed of the app would show posts based on their proximity (d = how far ...
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Is it possible to evaulate $I = \frac{2i}{\pi} \int_\gamma \ln(z) dz$ explicitly even though $\ln(z)$ isn't holomorphic at $z=0$?

Is it possible to calculate the following integral explicitly: $$ I = \frac{2i}{\pi} \int_\gamma \ln(z) dz, $$ where $\gamma$ can be a disk at the origin of $\mathbb{C}$? Unfortunately $\ln$ isn't ...
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Questions About the Proof of Cauchy–Pompeiu Integral Formula.

I am studying function theory in several complex variables and the book I am using is "Tasty Bits of Several Complex Variables" by Jiří Lebl: https://www.jirka.org/scv/scv.pdf. At the moment I am ...
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Solving a complex integral $\oint_L\frac{e^{1/(z-a)}}zdz$ using Cauchy's formula

I am practicing complex integration using Cauchy's formula, and I ran into a problem. The following integral: $$\oint_{L}{\frac{e^{\frac{1}{z-a}}}{z}}dz$$ where $$L=\{z\in\mathbb{C}:|z|=r\}$$ for ...
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Integral with two different answers using real and complex analysis

The integral is$$\int_0^{2\pi}\frac{\mathrm dθ}{2-\cosθ}.$$Just to skip time, the answer of the indefinite integral is $\dfrac2{\sqrt{3}}\tan^{-1}\left(\sqrt3\tan\left(\dfracθ2\right)\right)$. ...
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Real integrals with two poles in the complex plane

I'm studying the Cauchy Integral Theorem / Formula, but realised I have a misunderstanding. Consider an integral over the function $f: \mathbb{R} \to \mathbb{C}$ $$ I = \int^\infty_{-\infty} f(x) \, ...
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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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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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$L_f(z) = \frac 1 {2 \pi i}\int_{ \mathbb{T} } \frac{ \zeta+z}{ \zeta ( \zeta -z)} f( \zeta ) d\zeta$

I'm trying to prove that for any harmonic function $u$, we have : let $ \Omega \subset \mathbb{R}^2$ and $ \overline B(0,R) \subset \Omega $ $$ u \colon \Omega \to \mathbb R $$ $$\forall z \in ...
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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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Using Cauchy’s Theorem/ Cauchy’s Integral Formula for Higher Derivatives or otherwise, evaluate, with justification, the following integrals:

$$\int \frac{5 \cos(\pi z)}{(z+3i)(z-7i)}dz$$ i) where γ is the circle centre 0 and radius 4; ii)γ is the circle centre 0 and radius 10 For part i) I have calculated using CIF that because z=-3i is ...
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Stuck on proof using Cauchy's integral formula

I posted my attempted proof to this question here but I realized that I was wrong in taking the limit, and that the proof did not make sense. So I am still stuck on this problem let $f: \Omega \...
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Prove inequalities with Cauchy's integral theorem [closed]

Let $$f:\overline{B(0,1)}\rightarrow\mathbb{C}$$ be continuous and holomorphic on $B(0,1)$. Consider the function $$z\mapsto F(z):=f(z)\overline{f(\overline{z})}.$$ Show (i) $\int_{\gamma}F(z)dz=0$ ...
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Proof using Cauchy's Integral Formula

let $f: \Omega \rightarrow \mathbb{C}$ be analytic and $z_0 \in \mathbb{C}$. Define $$g(z) = \begin{cases} \frac{f(z)-f(z_0)}{z- z_0} & z \not = z_0 \\ f'(z_0) & z = z_0 \...
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Cauchy Integral Theorem with $f(z)=e^{z^2}$

I have $z(t)=t(1-t)e^t + \cos(2 \pi \cdot t^3)i$ with $0 \le t \le 1$ and need to evaluate the line integral of $e^{z^2}$. I know that the endpoints are $z(0)=z(1)=0+i$, so the line is a closed ...
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Proving a certain holomorphic function is polinomic

Suppose we have an holomorphic function $f$ on the open unit disk $D(0,1)$ s.t.: $$\forall r\in (0,1) \exists n\in \mathbb{N}| \max_{C(0,r)}|f|=r^n$$ prove that $f$ is polinomic. Honestly I don't ...
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Show that for a function $f$ bounded by $M$ on a disk $D_{r}$ show that $|f^{n}(z)|\leq n!M/\delta^{n}$ for $D_{r-\delta}$

Suppose that a function $f$ is analytic in the open disk $$D_{r}=\{z\in \mathbb{C}:|z|<r\}$$ where $r>0$, and there is a number $M\in\mathbb{R} $ such that $|f(z)| \leq M$ for all $z \...
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Questions about Cauchy's thm. on complex integration

I have the following integral $$I = \int_{\gamma_{1}} \frac{e^{z^2}}{(z-1)^2}dz,$$ where $$\gamma_{1} : [0,2Pi] \rightarrow \mathbb{C}, \\ \quad t \quad \mapsto \quad 2e^{i t} .$$ I want to show that ...
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Integral by Residue Theorem

I'm working through a Complex Analysis text and am working through the Residue chapter. I am not sure if I am approaching this question correctly. $$ \int_\gamma \frac{1}{(z-1)^2(z^2+1)}$$ Such that ...
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How to evaluate integrals like $ \int_C \frac{z}{e^z-1} dz$ without Residue Theorem?

I'm trying to evaluate two integrals: $$ I_1 = \int_{C(0, 1)} \frac{z}{e^z-1} dz,\quad I_2 = \int_{C(1, 3)} \frac{\sin(iz)}{e^{iz}-1} dz $$ $ C(a, r) $ is a circular contour with radius $r$ ...
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Proving $\int ^{2\pi} _{0} (\cos(t)^{2n})={2n \choose {n}}\frac{2\pi}{2^{2n}}$

using the following result $\int _{\gamma}(z+ \frac{1}{z})^{2n}\frac{dz}{z}= {2n \choose {n}}2\pi i $ Prove $\int ^{2\pi} _{0} (\cos(t)^{2n})={2n \choose {n}}\frac{2\pi}{2^{2n}}$ I cant see how ...
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Cauchy's integral formula with special contour 4

suppose $\gamma: [a,b] \rightarrow \mathbb{C} $ is a path of integration with $ \gamma(a)=0, \gamma(b)=1 \ and \ \pm i \notin\gamma([a,b]) $ Show that, $$ \int_{\gamma} \frac{1}{1+z^2} = \frac{\pi}{...
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Holomorphic integrals

I am struggling to understand how the center and radius effect a certain circular contour. e.g. $ \int _{\gamma} \frac {z^{2}+1} {e^{z}(z-1)(z+1)^{2}} dz $ can anyone explain this?
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Complicated Cauchy complex integral.

I have the function $f(z)$ that defined on the closed disc $ \bar D_3(0)$ by the integeral on the boundry $C_3(0)$: $f(z) = \int_{C_3(0)} \frac{3w^2+7w+1}{w-z}dw $ I need to find $f'(1+i)$ Caushy ...
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Integral Cauchy formula

How do I put: $\int_\gamma (z + \frac{1}{z})^{2n} \frac{dz} {z} $ In the form: $\int_\gamma \frac{f(z)}{(z-a)^{2n}} $ Initially I have gone for $f(z)=\frac {1}{z} $ but I cannot get the rest in ...
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Prove simple closed curves $f$'s exist, so $\Gamma = C-\sum_{i=1}^{k}{f_i}$ satisfies $ \int_{\Gamma}{\frac{z^3e^{1/z}}{(z^2 + z + 1)(z^2 + 1)}dz}=0$

Let $C$ be the circle $C(0,2)$ traversed one time counter-clockwise. Prove that there exist $k\in \mathbb {Z}_+$ and $C^1$ simple closed cuves $f_1, \dots ,f_k$ such that the cycle $\Gamma = C-\sum_{i=...
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If $f$ is an entire function with $|\,f(z)|\le|\operatorname{Re}(z)|$, then $\,f\equiv 0$.

If $f$ is an entire function so that $|\,f(z)|\le|\operatorname{Re}(z)|$ for all $\mathbb{C}$, then $f\equiv0$ on $\mathbb{C}$. There are various ways showing the above property. For example, using ...
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Undestand the proof of Cauchy Integral Formula using homotopy of curves.

https://math.berkeley.edu/~vvdatar/m185f16/notes/Lecture-16_CIF_Analyticity.pdf I'm trying to understand the proof of Theorem 0.1. Only the following part is missing: "Consider the contour in the ...
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21 views

Showing a Result Using Cauchy's Generalised Integral Formula

Using Cauchy's generalised integral formula, show that if $f$ is an entire function and $|f(z)|\leq |z|+1$, then $f^{(k)}(0)=0$. Hence deduce that there are complex constants $a,b$ such that $f(...
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1answer
17 views

Interating when one singularity is inside C and the other is not

I tried to evaluate $\frac{1}{2\pi i}\oint_C \frac{z^2}{z^2+4}dz$ where C is the square with vertices $\pm2$, $\pm2+4i$ thusly: $$\frac{1}{2\pi i}\oint_C \frac{z^2}{z^2+4}dz=\frac{1}{2\pi i}\oint_C \...
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88 views

How to show an entire function bounded from above is identically zero

Suppose that $f$ is entire. By writing $f$ as a Taylor Series, prove that if $\lvert f(z)\rvert<m\lvert z\rvert^\alpha$ for $m>0$ and $0<\alpha<1$ then $f$ must be identically zero in $\...
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Evaluating the integral $\int_c \frac{e^z}{z(z-4)^3}dz$

I would like to evaluate \begin{align} \int_c \frac{e^z}{z(z-4)^3}dz \end{align} where $C$ is the square centered at the origin with sides length 6 units and let $C_1$ be the unit circle centered at ...
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131 views

$\int_0^∞ \dfrac{x^a}{(x^2 + 1)^2} dx$ for $-1<a<3$

I tried to evaluate $A=\int_0^∞ \dfrac{x^a}{(x^2 + 1)^2} dx$ for $-1<a<3$, several times. I followed lengthy examples of the book which supposes real positive axis as branch cut. Every time I am ...