# 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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### Show $\frac{2}{\pi} \int_{0}^{2\pi} f(e^{i\theta})\cos^2(\frac{\theta}{2})d\theta = 2f(0)+f'(0)$ when $f\in A(|z|<1+\varepsilon)$. [duplicate]

Saw this question in a test, but got stuck, here is my attempt: \begin{align} \frac{2}{\pi} \int_{0}^{2\pi} f(e^{i\theta})\cos^2\bigg(\frac{\theta}{2}\bigg)\,d\theta &= \frac{2}{\pi} \int_{0}^{2\...
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### Does the analytic continuation of $\zeta(s)$ for $s\in\mathbb{C} \ \backslash \ \{1\}$ require Cauchy's integral formula?

I've seen Cauchy's integral formula used in proofs for the analytic continuation of the Riemann zeta function to the entire complex plane ($s\neq1$). My question is whether a counterexample can be ...
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### Cauchy residue formula question

The Cauchy formula can be used to find the $n$th derivative of an analytical function $f(z)$ as $$\frac{d^nf}{dz^n}\bigg|_{z=z_0}=\frac{n!}{2\pi i }\oint \frac{f(z)}{(z-z_0)^{n+1}}dz$$ which happens ...
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### how to show this integral is $\in \mathbb{R}$ (attempt added)

Let $f: \Omega \rightarrow \mathbb{C}$ holomorphic such that $\Omega \subseteq \mathbb{C}$. Let $z_0 \in \Omega$ such that $\overline{D_{R}}\left(z_{0}\right) \subseteq \Omega$. Given that for every ...
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### Derivative question and nested Cauchy formula

Consider $F(z)=\sum_i a_iz^i$ to be a formal power series with coefficients $a_i$. It is known that the coefficients of the series can be recovered from the $n$th terms of the associated Taylor series ...
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### Find $\int_{|z|=1} f(z) d z$, when $f(z)=(z \sin z) /(z+2)+\bar{z}$

Find $\int_{|z|=1} f(z) d z$, when $f(z)=(z \sin z) /(z+2)+\bar{z}$ $f(z)$ looks like analytic function. If it is analytic, then by Cauchy-Theorem, integration will be zero. I tried to show Cauchy-...
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### A condition of linearity for a holomorphic function [duplicate]

Consider the following exercise from the book Complex Analysis Stein and Shakarchi I already proved that $\varphi(z)=z+a_nz^n+O(z^{n+1})$ and $\varphi_k(z)=z+ka_nz^n+O(z^{n+1})$, but I do not see how ...
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### holomorphic function on unit disk having zero of order $n$ at $0$

Assume that $f: \mathbb{D} \rightarrow \mathbb{D}$ is holomorphic function with a zero at $0$ of order $n$, where $n \geq 2$. What can we say about $f^{(n)}(0)$? and what can we say about $f^{(n)}(0)$...
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### two disjoint compact sets, holomorphic function there exists a decomposition $f=f_1+f_2$

Let $D_1$ and $D_2$ be two compact sets in $\Bbb C$, $D_1\cap D_2=\emptyset$, and $f\colon \Bbb C\setminus(D_1\cup D_2)\to\Bbb C$ be a holomorphic function. Show that there exist two holomorphic ...
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### What techniques can be used to find fused global-local estimators of fractional derivatives?

The history of fractional calculus is a long one. A question on this topic appeared earlier today. This is a nice example of a minimal version of a global fractional derivative in a Fourier sense. But ...
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### Numbers of the solution in a Cauchy problem

I am looking for an example for a real first order ODE which has $0$, $\infty$ or exactly $1$ solution depending on the initial $x(t_{0})$ value. The solution doesn't necessary has to have the domain ...
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### Cauchy integral calculation outside the region

I have an integral $\oint \frac{2z}{z^2-9}dz$. It's said that the integral taken around the circle |z|=2 Here, the roots of z are $3,-3$ so it's outside the region. Does it mean the integral is 0?
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### Sharpness of bound on derivative of bounded holomorphic function on a strip

Let $U$ be the open set of all $z \in \mathbb C$ such that $-1< \text{Im } z <1$. Let $f: U \to \mathbb C$ be a bounded homomorphic function and let $A :=\sup_{z\in U}|f(z)|$. Then using ...
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### Verify integral using Residue Theorem

Verify the following equation: $$\int_{0}^{2\pi}{\log{\left(\sin^{2}{2\theta}\right)}\,\mathrm{d}\theta} = 4\int_{0}^{\pi}{\log{\left(\sin{\theta}\right)}\,\mathrm{d}\theta} = -4\pi \log{2}$$ I am ...
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### Understanding the Cauchy Integral Formula

This is the statement of the Cauchy Integral Formual. I have seen its proof, and I have used this formula many times to compute integrals. However, I feel like I do not understand this formula ...
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### Can Cauchy's integral formula be used on the boundary point?

Can Cauchy's integral formula be used if the point a is on the boundary of the circle ? The theorem is given below.
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### Prove that $f(z)=g(z)$ on $D$ considering Taylor expansions

Let $f$ and $g$ be differentiable on the strip $D = \{z \in \mathbb{C}: -2 < \operatorname{Im}z <2$. Suppose that $f(z) =g(z)$ for all $z$ such that $|z|<0.01$. By considering Taylor ...