update notations and fix typos

pages/Math416/index.md

@@ -1,3 +1,15 @@
 # Math 416
 
-Complex variables.
+Complex variables. This is a course that explores the theory and applications of complex analysis as extension of Real analysis.
+
+The course is taught by Professor.
+John E. McCarthy <mccarthy@math.wustl.edu>
+
+Some interesting fact is that he cover the lecture terribly quick. At least for me. I need to preview and review the lecture after the course ended. The only thing that I can take granted of is that many theorem in real analysis still holds in the complex. By elegant definition designing, we build a wonderful math with complex variables and extended theorems, which is more helpful when solving questions that cannot be solved in real numbers.
+
+McCarthy like to write $\zeta$ for $z$ and his writing for $\zeta$ is almost identical with $z$, I decided to use the traditional notation system I've learned to avoid confusion in my notes.
+
+I will use $B_r(z_0)$ to denote a disk in $\mathbb{C}$ such that $B_r(z_0) = \{ z \in \mathbb{C} : |z - z_0| < r \}$
+
+I will use $z$ to replace the strange notation of $\zeta$. If that makes sense.
+