# 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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### Complex integrals that look like they agree, differ by sign (according to Mathematica)

Consider the integral $$\int_0^\infty \frac{dz}{1-z^2 +i0^+},$$ I would assume it to agree with the integral $$\int_0^\infty \frac{dz}{(1-z+i0^+)(1+z+i0^+)}.$$ However, according to Mathematica the ...
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### Proof of zeta functional equation in Edwards

I'm trying to understand the first proof of the zeta functional equation given in Riemann's Zeta Function by H M Edwards. Referring to the excerpt from page 13 below, I'm stuck on how he derives the ...
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### Application of Cauchy integral formula / residue theorem for evaluation of real-valued integrals

I try to evaluate integrals of the form $$I(x) = \int_{0}^{x}\frac{f(t)}{t^{2}}\,\mathrm{d}t$$ with $f(t)$ being a differentiable and real-valued function of the real-...
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### Integral over the real line of a function with a second-order pole $\int_{-\infty}^\infty \frac{e^{-(A\omega+iB)^2+C}}{\omega^2} d\omega$

I am trying to solve an integral of the form $$\int_{-\infty}^\infty \frac{e^{-(A\omega+iB)^2+C}}{\omega^2} d\omega,$$ where $A,B,C\in \mathbb{R}$, $A>0$. Attempt 1: ...
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### Show that there does not exist any holomorphic function on the open unit disk and continuous on the closed unit disk with the given property. [duplicate]

Let $\mathbb D : = \left \{z \in \mathbb C\ :\ \left \lvert z \right \rvert < 1 \right \}.$ Prove that there is no continuous function $f : \overline {\mathbb D} \longrightarrow \mathbb C$ such ...
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### Cycles around compacta and the global Cauchy theorem

Recall the global Cauchy theorem/formula: Let $U$ be open and $\Gamma\subset U$ a cycle. If $\Gamma$ is homologous to zero in $U$, then for all $f:U\to\Bbb C$ holomorphic and $w\in U\setminus\Gamma$ ...
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### Would the following not be a correct proof of Cauchy's Integral Theorem?

I'm somewhat confused as to how the general version of Cauchy's Theorem does not follow (almost) immediately from its version in a disk. At least Ahlfors as well as Stewart & Tall prove the ...
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### Cauchy's integral theorem for rectifiable simple closed curve. [closed]

Γ is a rectifiable simple closed curve and Ω is its interior. f is holomorphic on Ω and continuous on Ω∪Γ. Prove the integral of f along Γ is 0. Since we already know the Cauchy's integral theorem for ...
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### Given f holomorphic on $B_1(0)$ and $max_{z \in C_r(0)} |f(z)| \to 0$ as $r \to 1$. Show f is identically 0.

I have tried to use cauchy integral formula and the deformation theorem but that got nowhere: I got $f(z_0) = max_{z \in C_r'(z_0)}|f(z)|$ where $r' > 0$ such that $B_r'(z_0) \subset B_1(0)$. Note: ...
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### How is Ahlfors applying Cauchy's Integral Formula when deriving Taylor's Theorem?

From Ahlfors' Complex Analysis (by 'analytic' he means 'complex-differentiable' or 'holomorphic'; he does not mean that the function can be expanded as a power series, as that is what he sets out to ...
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### Cauchy Integral Formula and the Dirichlet Problem

Does the Cauchy Integral Formula by itself proves that the Dirichlet Problem is solvable for any simply connected region? I was looking for the proof of the Poisson Integral Formula and one uses the ...
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### Evaluate the integral $\int_{\gamma} \frac{z^2+1}{(z+1)(z+4)}dz$

Evaluate the integral $$\int_{\gamma} \frac{z^2+1}{(z+1)(z+4)}dz$$ if $\gamma = \beta + [4 \pi ,0]$ and $\beta(t) = te^{it}$ for $0 \le t \le 4 \pi$. My attempt By Cauchy's integral formula. \begin{...
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