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## Theorems of the month

### Cauchy's integral theorem

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

$\int\limits_{\gamma }{f(z)dBz}=0.$

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

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

### Fundamental theorem of algebra

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

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

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