# 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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### 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 $C$,...