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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Help with contour integral for a reproducing kernel
If we assume $f$ is analytic and this integral makes sense on the unit disc $U$, then I'm trying to show this is a weighted Bergman kernel, but I'm stuck here:
$$
\frac{4}{\pi}\int_{U}\frac{f\left(...
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Cauchy Theorem Integration Issue
I need to compute the integral $\int_{0}^{2\pi}\frac{1}{1 + \sin^2(x)}dx = 4.4429$ utilizing the Cauchy theorem. However, I'm encountering a discrepancy where my result is consistently half of the ...
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Evaluating an Integral Using Cauchy's Theorem and Root-Finding
I am tasked with evaluating the following integral utilizing the Cauchy theorem:
$$\int_{0}^{2\pi} \frac{1}{(2 - \cos t)^{2}}dt$$
I have initiated my calculation as follows:
$$\begin{align*}
\int_{0}^{...
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Calculation of an Integral Using the Cauchy Formula
I need assistance with calculating the following integral using the Cauchy formula. I have been encountering incorrect results and would greatly appreciate your help in identifying the mistake in my ...
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Suitable Contour to find the value of the integral: $\int_{0}^{\infty} \frac{x}{x^3+1} dx$
I have found the three points of singularity of the functions $$f(z)=\frac{z}{z^3+1}$$ as
$z_1=e^{iπ/3}$, $z_2=e^{iπ}$, $z_3=e^{i5π/3}$
But what contour should I take to have the real axis from $0$ to ...
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How to show this equality from the Cauchy integral formula
In a paper https://projecteuclid.org/journals/brazilian-journal-of-probability-and-statistics/volume-36/issue-2/Limit-theorems-for-quasi-arithmetic-means-of-random-variables-with/10.1214/22-BJPS531....
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For $u$ harmonic and $f = u+iv$ holomorphic, show that $f(z) = \frac{1}{\pi i} \oint_{|\zeta|=r} \frac{u(\zeta)}{\zeta - z} d\zeta - \overline{f(0)}$
Here's a question from a previous complex analysis qualifying exam that I'm honestly just stumped on:
Let $u$ be a harmonic function on the unit disc $D = \{z: |z|<1\}$, which is the real part of ...
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Evaluating the integral $\oint \coth(\frac{1}{2}z) dz$ taken over the contour $C$ and $|z-\frac{\pi}{2}i|^{2} = 1$
I would highly appreciate your valuable feedback on the consistency of my demonstration:
The integral $\oint \coth\left( \frac{1}{2} \cdot z \right)\, \operatorname{d}z$ is taken over the contour $C$,...
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Exercise of Pseudodifferential operators problem
Let $\varepsilon > 0$, $\Omega = \lbrace \zeta \in \mathbb{C}^n ; |Im\zeta| < \varepsilon |Re\zeta| \rbrace$ and $a(\zeta)$ a holomorphic function on $\Omega$ satisfying an estimate $|a(\zeta)| \...
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Cauchy integral and taylor serie
I have a question about my attempt to demonstrate the following exercise:
exercise:
Let $f$ be analytic in a domain $\Omega$ and for $a \in \Omega$ let
$$
f_n = \sum_{k=1}^n f^{(k)}(a)(z-a)^k, z \in \...
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How to evaluate $\int_{0}^{\frac{\pi}{2}} \frac{d\theta}{2+\sin(\theta)}$ using Cauchy's integral theorem even tho it isn't from 0 to 2pi?
Evaluate using Cauchy's integral theorem
$$\int_{0}^{\frac{\pi}{2}} \frac{d\theta}{2+\sin(\theta)}.$$
I understand the idea is to use $\sin(\theta) = \frac{z-z^{-1}}{2i}$ and $z = e^{i\theta}$ to ...
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Solving a real integral using Cauchy theorem
How do I compute the residues of the complex function in order to evaluate the following real integral using Cauchy's theorem?
$$
I = \int_{-\infty}^{\infty} dE \left[\frac{E^2f(E)}{(E-E_n-i\eta)^2(E-...
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Complex analysis, Jameson Exercise 2.1.1
Exercise: Let $f(x + iy) = x + y$, and for $z$ in $\mathbb{C}$, define $F(z)=\int _{[0 \to z]}f$. At which points is $F$ differentiable?
Is my answer correct? Here's my answer:
$$F(z+h)-F(z)=\int _{[0 ...
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Regularity of Cauchy-like integral
I'm trying to understand whether a certain integral representation is holomorphic. Specifically, let's assume that $f$ belongs to the Schwartz space $\mathcal S(\mathbb R)$, and consider a function $g(...
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How do I evaluate this integral using Cauchy's theorem?
I have a physic paper to do and one of the question is to demonstrate this equality using Cauchy's theorem :
$$\int_{-\infty}^{\infty} e^{i\zeta^2 sgn \varphi''(k_0) } d\zeta = \sqrt{\Pi} e^{\frac{i\...
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Don't understand the answer of $\int_0^\pi \frac{\cos{n\theta}}{1 -2r\cos{\theta}+r^2} d\theta$
I don't understand the answer provided in this post, what does "to use in the numerator $e^{inθ}$" mean? Also, how to handle the upper limit $\pi$ instead of $2\pi$? Many thanks in advance!
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Is a holomorphic $f\colon U\to\mathbb{C}$ with continuous extension to $\overline{U}$ Lipschitz continuous on $\partial U$?
Let $U\subset\mathbb{C}$ be a bounded connected open subset with smooth boundary $\partial U$. Suppose that we have a holomorphic function $f\colon U\to\mathbb{C}$ that can be continuously extended to ...
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Why only one singularity is involved? $\int_{0}^{2\pi} \frac{1}{13+5\sin(\theta)}~d\theta$
I solved the integral $$\int_{0}^{2\pi} \frac{1}{13+5\sin(\theta)}~d\theta$$ with the residue theorem and Cauchy’s Integral Formula.
The following is the solution, but I am unsure why in the end we ...
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problem about derivatives of functions from normal family
Let D be a domain.
Let {fn} be a set of functions that are holomorphic on D and is normal. I’m trying to prove that
{fn’} is also a normal family.
The following is my attempt:
Note that if {Gn} is ...
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Prove that any entire conformal map satisfying $f(z + 2) = f(z) + 2$ and $f(z + iM) = f(z) + iM'$ for some real $M, M'$ is linear
Prove that any entire conformal map satisfying $f(z + 2) = f(z) + 2$ and $f(z + 2iM) = f(z) + 2iM'$ for some real $M, M'$ is linear.
I wanted to check if my approach is correct: By subtracting $f(0)$ ...
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Finding the Fourier transform with the Cauchy residue theorem or integration in the complex plane
I need to find the Fourier transform of the function $f(x) = \frac{1}{\sqrt{2\pi\sigma^2}}\exp(-\frac{(x-\mu)^2}{2\sigma^2}), x, \mu \in \mathbb{R}, \sigma>0$ by using the Cauchy residue theorem. ...
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Calculating Complex integral ${\int _{_{_{\left|z\right|=\frac{1}{2}}}}}\frac{e^{zt}}{z^2+1}dz$
I am trying to calculate the integral $${\int _{_{_{\left|z\right|=\frac{1}{2}}}}}\frac{e^{zt}}{z^2+1}dz$$ for all $t \in \mathbb{R}$
According to answers, it is zero because it is a closed curve.
I ...
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Cauchy's differentiation formula for bounded operators
Let $T$ be a bounded operator on a Banach space $X$. It's known that by the Dunford-Riesz functional calculus and the Cauchy formula we have that,
$$
f(T) = \frac{1}{2\pi i}\int_{\partial U} \frac{f(\...
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Can this given $f: S^1\to \mathbb C$ be extended to a continuous $F: \overline{\mathbb D}\to \mathbb C, F$ is holomorphic on $\mathbb D$?
Suppose that $f: \mathbb S^1\to \mathbb C$ is continuous such that $f(z)=f(\bar z)$ for all $z\in \mathbb S^1$. Can it be extended to a continuous $F: \overline{\mathbb D}\to \mathbb C$ such that $ F$ ...
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Compute integrals using Cauchy Integral Formula
The integrals:
$\int_{|z+1|=1}\frac{1}{(z+1)(z-1)^4} dz $
$\int_{|z|=2}\frac{\cos(z)}{(z+i)} dz $
idea:
As I mentioned in the title I want to use the Cauchy integral formula:
$f(a) = \frac{1}{2\pi ...
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Integrating $\frac {e^{-iz}}{z}$ over a semicircle counter-clockwise around $0$ of infinite radius $r$
How do I find $\lim_{r\to +\infty}$ ${∫_\gamma}_r$ $\frac {e^{-iz}}{z}dz$ where $\gamma_r$ is the counter-clockwise semicircle centred at 0 with radius $r$. I think it's equal to $2πi$, but how do I ...
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Complex conjugate by complex integration
By Cauchy's Theorem we have
$
f(a)=\frac{1}{2\pi i}\int_{\gamma}\frac{f(w)}{w-a}dw
$
where $\gamma$ is the path $\gamma(t)=b+re^{it},~t\in[0,2\pi],$ $a\in B_r(b)$ where $B_r(b)$ is the closed ball of ...
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Choosing a path for reverse Cauchy integral formula
We are told to compute the integral $$\int_0^{2\pi}\frac{4}{5+3\cos\theta}d\theta.$$ using Cauchy integral formula.
In the solution they take the path to be $\gamma(\theta)=e^{i\theta}$ for $0\leq\...
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Example for complex integral
In my question I use the following notations:
$r,R\in\mathbb{R};$ $\alpha,\beta\in\mathbb{C};$
$A'\left(\alpha,R\right)\dot{=}\left\{ z:z\in\mathbb{C},\left|z-\alpha\right|\leq R\right\};$
$A\left(\...
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Show that integral $\int \frac{f'(w)}{w-z} dw = \int \frac{f(w)}{(w-z)^2} dw$
We let $\gamma$ be a simple closed contour, and $f$ analytic in and on $\gamma$. Then how might we show that for any $z$ in the interior of $\gamma$, that the equation
$\int_\gamma \frac{f'(w)}{w-z}\,...
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How to show $\mathbb E[f(X)]=f(\mu +i\sigma)$ using Cauchy integral formula [closed]
Assume that $C(\mu ,\sigma )$ is the Cauchy distribution with location $\mu \in \mathbb{R}$ and scale $\sigma >0$ and the density function of $C(\mu ,\sigma)$ is $p(x;(\mu ,\sigma))=\frac{\sigma}{\...
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Prove that if $f$ is holomorphic inside and on a circle of radius $R$ centered at $z_0$, then $|f(z_0)| \geq \min_{z \in C_{R}(z_0)} |f(z)|$
Suppose $f$ is holomorphic on the closed disc of radius $R$ around some point $z_0$ and $f(z) \neq 0$, i.e. $f$ is holomorphic on the disc $D_{R}(z_0)$, show that the minimum value of $f(z)$ on the ...
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How to prove that a sum is inclosed by two upper and lower bounds (values)?
Show that this is true for all $n \ge 1$
$2\ln(2) - 1 \le \frac1{n} \sum_{k=1}^{n} \ln(1 + \frac{k}{n}) \le 2\ln(2) - 1 + \frac{\ln(2)}{n}$
I've no idea what to do here but here is how I've tried so ...
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Upper bound of a function in Hoffman and Mardsten's Complex Analysis 2.8
Question 2.5.8 in Mardsen and Hoffman's Basic Complex Analysis:
Let f be entire and let $|f(z)|<M$ for $z$ on the circle $|z|=R$, let $R$ be fixed, prove that
$|f^{(k)}(re^{i\theta})|<\frac{k!M}...
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Cauchy integral formula for a unbounded domain
Let $C_r$ be the positively oriented circle centered at origin with radius $2$. Let $f$ be a analytic function on $\{z : |z| > 1\}$ and $\lim_{z \to \infty} f(z)=0.$ I need to show that for $|z|&...
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Does the $\Gamma\subset\mathbb{C}$ curve always has to be smooth in Cauchy's integral theorem?
In Cauchy's integral theorem (see: https://en.wikipedia.org/wiki/Cauchy%27s_integral_theorem), does the $\gamma$ representation of the $\Gamma\subset\mathbb{C}$ curve always has to be smooth (i.e. at ...
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$f: D\to \mathbb C$ is holomorphic such that $diam f(D)=d$, then $|f'(0)|=d/2\implies f$ is linear. [duplicate]
$f: D\to \mathbb C$ is holomorphic such that $\sup_{u,v\in D}|f(v)-f(u)|=d$, then $|f'(0)|=d/2\implies f$ is linear.
Here, let's suppose that $D$ is a closed unit disk.
By Cauchy's integral formula, ...
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Cauchy's theorem with and without Green's Theorem
I know two main approaches to proof of Cauchy's theorem ($\oint_{\gamma}^{}f(z)dz=0$ for any simple connected contour $\gamma$ and any function $f$ holomorphic inside $\gamma$). First is based on ...
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Complex integral on a simple closed curve
I try to translate a statement that I've read in a translated „Introduction to Complex Analysis” book...
Let me define $f$ as a continuous complex valued function on a set $D\subset\mathbb{C}$, where $...
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2
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Cauchy Integral formula with central angle different from $2\pi$
We know that if $f$ is holomorphic in an open set $A$ and continous in its closure, and $C$ is a closed curve inside $A$ then :
$f(z_0)2i\pi = \int_{C} \frac{f(z)}{z- z_0} dz $
Now suppose that $C$ ...
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Write a continuously differentiable function defined on the unit circle as difference of two holomorphic functions.
Given a $C^1$ function $g(z)$ on the unit circle, I would like to find $f^+$ holomorphic on $\mathbb D$ and continuous on $\overline{\mathbb D}$, $f^-$ holomorphic on $\mathbb C\setminus \overline{\...
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Calculate $\oint\frac{z^{2}}{z-4}dz$ over a contour C which is a circle with $\left|z\right|=1$ in anticlock-wise direction
Question:
Calculate $\oint\frac{z^{2}}{z-4}dz$ over a contour C which is a circle with $\left|z\right|=1$ in anticlock-wise direction.
My Approach:
Using the Cauchy-Integral Formula $f\left(z_{0}\...
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2
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Prove complex integral equality
Suppose $\triangle$ is the open unit disk and $\overline{\triangle}$ be it’s closure (closed unit disk). Let $f$ be holomorphic in an open set containing the set $D = \mathbb{C} - \overline{\triangle}$...
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Cauchy Integral Formula of $\int_C \frac{z^3+3}{z(z-i)}dz$
I'm trying calcualte this integral $\int_C \frac{z^3+3}{z(z-i)}dz$ when the path $C$ is something similar to
$$C=\gamma_1+\gamma_2$$,
where $\gamma_1:|z-i|=1$, $\gamma_2:|z+i|=1.$
My solutions is ...
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Is $D=\{z:|z-z_0|\le r\}$ an open or closed set? [closed]
My doubt arises from the Wikipedia page on the Cauchy integral formula, where the set $D=\{z:|z-z_0|\le r\}$ is defined as an open set.
As it includes also its boundary points $|z-z_0|=r$, shouldn't ...
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Harmonic function and complex analysis can't prove a result.
I'm trying to show that given $u:\bar D(0,R)\subset \Omega\rightarrow\mathbb R$ an harmonic function, one has that: $$\forall z\in D(0,R): u(z)=\operatorname{Re}\left(\frac{1}{2\pi i}\oint_{\partial ...
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Is $z^{3/2}$ analytic? If so why can't I use Cauchy-Goursat Theorem?
$$\oint_\Gamma z^{3/2} dz, \quad \text{where } \Gamma : |z| = 1$$
It should meet the requirements of the Theorem of Cauchy-Goursat so it should be $0$. But when I do it:
$$ \oint_0^{2\pi}e^{3i\theta/...
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1
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Verification: Complex integral computation using cauchy's integral formula
I want to know if the way I computed this integral is rigorous (and correct):
Compute: $$\oint_{\partial D(0,8)}\frac{1+\cos^2(z)}{z-\pi}dz$$
I'll be using the following corollary of the Cauchy ...
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Application of Cauchy's integral formula [duplicate]
Let $f(z)$ be analytic in a neighbourhood of $z_0$, where $f'(z_0)$ does not equal $0$. Show that
$$\int_C\frac{\mathrm{d}z}{f(z)-f(z_0)} = \frac{2\pi i}{f'(z_0)}$$
where $C$ is a small (as small as ...
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Getting Contradicting results with the Fundamental Theorem of Complex integrals and Cauchy's integral theorem
The way I prove the fundemental theorem of complex integrals is as follows:
$$\int_\gamma f'(z)dz = \int_a^b f'(\gamma(t))\gamma'(t)dt = \int_a^b g'(t)dt = g(b)-g(a)=f(z_2)-f(z_1)$$
Where in the first ...
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