# 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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### Residue theorem and a two-dimensional integral: not working?

Consider the following integral: \begin{align} \iint_{\mathbb{R}^2}d t\,dT\, \frac{e^{-i(t-T)}e^{-t^2}e^{-{T}^2}}{(t-T-i\epsilon)^2}\,. \end{align} The $i\epsilon$ prescription simply tells me that if ...
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### Cauchy integral with positive lower limit of integration

I’m trying to solve an integral of the following form: $\int ^b _a \frac{1}{(\omega^2+r_1)(\omega^2+r_2)}d\omega$ Where $r_1$ and $r_2$ are complex conjugates. I used the Cauchy theorem to solve ...
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### Evaluate $\frac{1}{2\pi i}\int_\gamma\frac{f(z)}{(z-z_1)(z-z_2)}-\frac{f(z)}{(z-z_0)^2}$

In Marsden's Complex Analysis, section 2.4, the main theorem is Cauchy's integral formula (C.I.F) and there appears this problem: Let $f$ be analytic inside and on $\gamma: |z-z_0|=R$. Prove that ...
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### Evaluate the integral $\int_C f(z)dz$ where $C$ is the unit circle centered at the origin with $f(z) = e^{iz}/(z-a)$, $0<a<1$?

Evaluate the integral $$\int_C f(x)dx \,$$ where C is the unit circle centered at C the origin with f(z) = $$\frac {e^{iz}}{z-a}\,$$ for 0 < a < 1. I'm in complex and we are using the idea of ...
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### A question about Cauchy Integral Theorem.

I am having some trouble understanding the definition of Cauchy Integral from my textbook. My problem is $\partial D$. I understand $\partial D$ is the boundary of $D$. Why would you take the ...
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### For what simple closed curves does the equation hold?

$\int_{\gamma} \frac{dz}{z^{2}+z+1} = 0$. I know that by Cauchy's Theorem, if $f$ is analytic everywhere on and inside a simple closed curve, then $\int_{\gamma} f(z)dz = 0$. So I thought that the ...
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### Using Cauchy’s Theorem/ Cauchy’s Integral Formula for Higher Derivatives or otherwise, evaluate, with justification, the following integrals:

$$\int \frac{5 \cos(\pi z)}{(z+3i)(z-7i)}dz$$ i) where γ is the circle centre 0 and radius 4; ii)γ is the circle centre 0 and radius 10 For part i) I have calculated using CIF that because z=-3i is ...
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### If $f$ is an entire function with $|\,f(z)|\le|\operatorname{Re}(z)|$, then $\,f\equiv 0$.

If $f$ is an entire function so that $|\,f(z)|\le|\operatorname{Re}(z)|$ for all $\mathbb{C}$, then $f\equiv0$ on $\mathbb{C}$. There are various ways showing the above property. For example, using ...
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### Undestand the proof of Cauchy Integral Formula using homotopy of curves.

https://math.berkeley.edu/~vvdatar/m185f16/notes/Lecture-16_CIF_Analyticity.pdf I'm trying to understand the proof of Theorem 0.1. Only the following part is missing: "Consider the contour in the ...
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### Evaluating the integral $\int_c \frac{e^z}{z(z-4)^3}dz$
I would like to evaluate \begin{align} \int_c \frac{e^z}{z(z-4)^3}dz \end{align} where $C$ is the square centered at the origin with sides length 6 units and let $C_1$ be the unit circle centered at ...
### $\int_0^∞ \dfrac{x^a}{(x^2 + 1)^2} dx$ for $-1<a<3$
I tried to evaluate $A=\int_0^∞ \dfrac{x^a}{(x^2 + 1)^2} dx$ for $-1<a<3$, several times. I followed lengthy examples of the book which supposes real positive axis as branch cut. Every time I am ...