Added appropriate form of Taylor's formula.

src/dg/complex/derivative.tex

@@ -221,7 +221,7 @@
 \end{proof}
 
 
-\begin{definition}[Complex Analytic]
+\begin{definition}[Holomorphic]
 \label{definition:complex-analytic}
     Let $E$ be a complete separated locally convex space over $\complex$, $U \subset \complex$ be open, and $f \in C(U; E)$, then the following are equivalent:
     \begin{enumerate}
@@ -234,16 +234,16 @@
         \[
         f(z_0) = \frac{1}{2\pi i} \int_\gamma \frac{f(z)}{z - z_0}dz
         \]
-        \item (\textbf{Analyticity}) For each $z_0 \in U$ and $r > 0$ such that $\ol{B(z_0, r)} \subset U$, there exists $\seq{a_n} \subset E$ such that $f$ may be expressed as a power series
+        \item (\textbf{Analyticity}) For each $z_0 \in U$, there exists $r > 0$ and $\seq{a_n} \subset E$ such that for each $z \in B(z_0, r)$, 
         \[
         f(z) = \sum_{n = 0}^\infty a_n(z - z_0)^n
         \]
 
-        with radius of convergence at least $r$.
+        where the radius of convergence of the series is at least $r$. 
         \item (\textbf{Weak Holomorphy}) For each $\phi \in E^*$, $\phi \circ f$ satisfies the above. 
     \end{enumerate}
 
-    If the above holds, then $f$ is \textbf{complex analytic}. 
+    If the above holds, then $f$ is \textbf{holomorphic/complex analytic} on $U$. 
 \end{definition}
 \begin{proof}
     (1) $\Leftrightarrow$ (2): \autoref{lemma:complex-analytic}. 
@@ -267,7 +267,7 @@
 
     Thus $[D^kf(z_0)/k!]_E \le C/r^k$ for all $k \in \natz$, and the radius of convergence of $g$ is at least $r$. 
     
-    Let $z \in B(z_0, r/2)$, $s = |z - z_0|$, and $n \in \natp$, then by \hyperref[Taylor's Formula]{theorem:taylor-lagrange} and \hyperref[Cauchy's Estimate]{corollary:cauchy-estimate}, 
+    Let $z \in B(z_0, r/2)$, $s = |z - z_0|$, and $n \in \natp$, then by \hyperref[Taylor's Formula]{theorem:taylor-integral} and \hyperref[Cauchy's Estimate]{corollary:cauchy-estimate}, 
     \begin{align*}
         \braks{f(z) - \sum_{k = 0}^n \frac{1}{k!} D^kf(z_0)(z - z_0)^n}_E &\le s^{n+1} \cdot \sup_{z' \in \ol{B(z_0, s)}} [D^{n+1}f(z')]_E \\
         &\le \frac{Cs^{n+1}}{(r-s)^{n+1}}