Added uniform structures for completely regular spaces. Added calculus lemma.

src/dg/complex/derivative.tex

@@ -1,6 +1,28 @@
 \section{Complex Differentiability}
 \label{section:complex-derivative}
 
+\begin{lemma}
+\label{lemma:cauchy-circle}
+    Let $E$ be a separated locally convex space over $\complex$, $U \subset \complex$, and $f \in C^1(U; E)$. For any $a \in U$ and $r > 0$ such that $\overline{B(a, r)} \subset U$, let
+    \[
+    \gamma: [0, 2\pi] \to U \quad t \mapsto a + re^{it}
+    \]
+
+    then for any $z \in B(a, r)$, 
+    \[
+    f(z) = \frac{1}{2\pi i}\int_\gamma \frac{f(w)}{w - z}dw
+    \]
+\end{lemma}
+\begin{proof}
+    Assume without loss of generality that $a = 0$ and $r = 1$, then by the \hyperref[change of variables formula]{theorem:rs-change-of-variables}, 
+    \[
+    \frac{1}{2\pi i}\int_\gamma \frac{f(w)}{w - z}dw = \frac{1}{2\pi} \frac{f(e^{it})e^{it}}{e^{it} - z}dt
+    \]
+\end{proof}
+
+
+
+
 \begin{definition}[Complex Analytic]
 \label{definition:complex-analytic}
     Let $E$ be a separated locally convex space over $\complex$, $U \subset \complex$, and $f: U \to E$, then the following are equivalent: