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Theorems of the month

Cauchy's integral theorem

Given the neighbourhood relations [math]B \subseteq {A}^{2}[/math] and [math]D \subseteq [a, b][/math] for some simply connected [math]h[/math]-set [math]A \subseteq {}^{\omega}\mathbb{C}[/math], infinitesimal [math]h[/math], a holomorphic function [math]f: A \rightarrow {}^{\omega}\mathbb{C}[/math] and a closed path [math]\gamma: [a, b[\rightarrow \partial A[/math], choosing [math]\curvearrowright B \gamma(t) = \gamma(\curvearrowright D t)[/math] for [math]t \in [a, b[[/math], we have that

[math]\int\limits_{\gamma }{f(z)dBz}=0.[/math]

Proof: By the Cauchy-Riemann partial differential equations and Green's theorem, with [math]x := \text{Re} \, z, y := \text{Im} \, z, u := \text{Re} \, f, v := \text{Im} \, f[/math] and [math]{A}^{-} := \{z \in A : z + h + ih \in A\}[/math], we have that

[math]\int\limits_{\gamma }{f(z)dBz}=\int\limits_{\gamma }{\left( u+iv \right)\left( dBx+idBy \right)}=\int\limits_{z\in {{A}^{-}}}{\left( i\left( \frac{\partial Bu}{\partial Bx}-\frac{\partial Bv}{\partial By} \right)-\left( \frac{\partial Bv}{\partial Bx}+\frac{\partial Bu}{\partial By} \right) \right)dB(x,y)}=0.\square[/math]

Fundamental theorem of algebra

For every non-constant polynomial [math]p \in {}^{(\omega)}\mathbb{C}[/math], there exists some [math]z \in {}^{(\omega)}\mathbb{C}[/math] such that [math]p(z) = 0[/math].

Indirect proof: By performing an affine substitution of variables, we can reduce to the case [math]1/p(0) \ne \mathcal{O}(\text{d0})[/math]. Suppose that [math]p(z) \ne 0[/math] for all [math]z \in {}^{(\omega)}\mathbb{C}[/math].

Since [math]f(z) := 1/p(z)[/math] is holomorphic, we have that [math]f(1/\text{d0}) = \mathcal{O}(\text{d0})[/math]. By the mean value inequality [math]|f(0)| \le {|f|}_{\gamma}[/math] for [math]\gamma = \partial\mathbb{B}_{r}(0)[/math] and arbitrary [math]r \in {}^{(\omega)}\mathbb{R}_{>0}[/math], and hence [math]f(0) = \mathcal{O}(\text{d0})[/math], which is a contradiction.[math]\square[/math]

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