# 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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### Basic Complex Integration Question on a circle

I'm trying to find $$\int \dfrac{dz}{(z-1)^3}$$ on the circle $|z|=2$. I'm pretty new to this, so my instincts were to use Cauchy's theorem for this, namely that the integral of an analytic function ...
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### $\int _C \frac{\exp(z^2)}{z^2\left(z-1-i\right)}dz$, $C$ consists of $|z|=2$ counterclockwise and $|z|=1$ clockwise.

$\int _C \frac{\exp(z^2)}{z^2\left(z-1-i\right)}dz$, $C$ consists of $|z|=2$ counterclockwise and $|z|=1$ clockwise. Not sure where to begin and was wondering if anyone could give me a hint. I ...
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### Is it possible to calculate $\int_0^\pi\frac{dx}{2+\sin^2x}$ without residue theorem

I came across the following integral: Calculate $\int_0^\pi\frac{dx}{2+\sin^2x}$. I know that you probably can solve it using residue theorem, but since we haven't proved it yet, I tried another ...
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### Justification of interchanging limit and integral in the proof of the derivative of Cauchy integral formula

On Stein and Shakarchi's Complex Analysis book, in the course of proving that for $f$ a holomorphic (i.e. differentiable) function on an open set, and $C$ a circle whose interior is also contained in ...
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### Calculating the integral $I(w)=\int_\mathbb{R}\exp(-\pi(x+w)^2)dx$ for $w\in\mathbb{C}$

Suppose $I(w)=\int_\mathbb{R}\exp(-\pi(x+w))^2dx$ with $w\in\mathbb{C}$. There are two parts to this problem : i) I have to show that $I(w)$ converges uniformly on compact sets $K\subset\mathbb{C}$ ii)...
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### How to calculate $\int_{\partial D(0;a)} \frac{dz}{(z-a)(z-b)}$

How to calculate $\displaystyle\int_{\partial D(0;r)} \frac{dz}{(z-a)(z-b)}$ when $|a|<r<|b|$?. My first idea: I tried to separate by partial fractions, so for some $A,B$ \...
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### Complex integral of $\int_{-\pi}^{\pi}\frac{\cos\theta\,d\theta}{\csc \alpha+\cos\theta}$ [closed]

A current $I_1$ flows in a circular circuit of radius a and a current $I_2$ flows through a very long straight conductor in the same plane of the circular circuit (see the figure). From the laws of ...
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### Closed rectifiable curve with arbitary winding number.

This is an exercise from Conway's Functions of One Complex Variable(page 83, exercise 2). Give an example of a closed rectifiable curve $\gamma$ in $\mathbb C$ s.t. for any integer $k$ there is a ...
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### Cauchy's theorem with a non closed curve

I'm reading Wavelets - tools for science and technology by Jaffard, Meyer and Ryan and I have a problem with a statement of section 10.4.1. They say that there is a form of Cauchy's theorem that says :...
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### Integral of a complex function (contour integral)

This is the integral: $$\int_{\gamma}z^2\frac{f'(z)}{f(z)}dz$$ we know that: $\gamma$ is a simple closed curve; $f$ is analytical in $R$ (the region bounded by the curve) and in all the points of the ...
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### Why do holomorphic functions on a disk admit so many different representation formulae?

Consider a holomorphic function $f$ on the unit disk that extends continuously to the boundary circle, i.e. $f \in H(\mathbb{D}) \cap C(\overline{\mathbb{D}})$ with $f|_{\partial \mathbb{D}} = f_1$. ...
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### Using the Cauchy integral formula show that $\oint_{ |z|=2} \dfrac { e^zdz}{(z-1)^2(z-3)}= -\dfrac 3 2je\pi$

Using the Cauchy integral formula show that $$\oint_{ |z|=2} \dfrac { e^zdz}{(z-1)^2(z-3)}= -\frac 3 2je\pi.$$ My work is in this picture. The answer is $-\frac 3 2je\pi$. I used the singularities ...