Added more complex analysis.

src/dg/complex/derivative.tex

@@ -234,7 +234,7 @@
         \[
         f(z_0) = \frac{1}{2\pi i} \int_\gamma \frac{f(z)}{z - z_0}dz
         \]
-        \item (\textbf{Analyticity}) For each $z_0 \in U$, there exists $r > 0$ and $\seq{a_n} \subset E$ such that for each $z \in B(z_0, r)$, 
+        \item (\textbf{Analyticity}) For each $z_0 \in U$ and $r > 0$ with $\ol{B(0, r)} \subset U$, there exists $\seq{a_n} \subset E$ such that for each $z \in B(z_0, r/2)$, 
         \[
         f(z) = \sum_{n = 0}^\infty a_n(z - z_0)^n
         \]
@@ -257,7 +257,7 @@
     
     Let
     \[
-    g(z) = \sum_{k = 0}^\infty \frac{1}{k!} D^kf(z_0)(z - z_0)^n
+    g(z) = \sum_{k = 0}^\infty \frac{1}{k!} D^kf(z_0)(z - z_0)^k
     \]
 
     then by \hyperref[Cauchy's Estimate]{corollary:cauchy-estimate}, for any $k \in \natz$ and continuous seminorm $[\cdot]_E: E \to [0, \infty)$, 
@@ -281,3 +281,4 @@
 \end{proof}
 
 
+