# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 "...
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 ...