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\centerline{\bf {\it Class 8} (Friday January 30)}
\begin{description}
\item[SUMMARY] Continuity and Differentiability of Complex Functions
\vspace{-12pt}
\item[CURRENT READING] Brown \& Curchill, pages 40-45
\vspace{-12pt}
\item[NEXT READING] Brown \& Curchill, pages 45-57
\end{description}
 
{\large\bf Continuity}\\
A complex function $f(z)$ is {\bf continuous} at a point $\z0$ if {\it all three} of the following statements are true
\begin{itemize}
\item[1:] ${\displaystyle \lim_{z \rightarrow \z0}} f(z)$ exists
\item[2:] $f(\z0)$ exists
\item[3:] ${\displaystyle \lim_{z \rightarrow \z0}} f(z) = f(\z0)$
\end{itemize}

Consider the function below:
$$f(z) = 
\left\{
\begin{array}{cr}
{\displaystyle \frac{z^2+4}{z-2i} },& \qquad z \neq 2i\\
 & \\
3 + 4i, & \qquad  z=2i
\end{array}
\right.
$$
Answer the following questions
\begin{enumerate}
\begin{enumerate}
\item What is the value of ${\displaystyle \lim_{z \rightarrow 2i}} f(z)$?
\item Is $f(z)$ continuous at $z=2i$?
\item Is $f(z)$ continuous at points $z \neq 2i$?
\end{enumerate}
\end{enumerate}

We say that the function $f(z)$ defined above has a {\bf removable singularity} at $z=2i.$

Write down the definition of $f(z)$ which has had the singularity removed.
\vspace*{1in}

{\bf More Aspects of Continuity}\\
As with real functions of a real variable, {\bf sums}, {\bf differences}, {\bf products} and {\bf compositions} of continuous functions are continuous.\\

When $f(z)$ continuous  $\Longleftrightarrow u(x,y)$ and $v(x,y)$ continuous\\

When $f(z)$ continouus in a region $R$, then $|f(z)|$ is also continuous in the region $R$ and if $R$ is a {\it bounded} and {\it closed} set then there exists a positive number $M$ so that $|f(z)| \le M \forall z \ni R$. 

\newpage
{\large\bf Derivative}\\
Let $f$ be defined in a neigborhood around $\z0$. The {\bf derivative} of $f$
at $\z0$, denoted by $f'(\z0)$, is defined by
$$f'(\z0) = \lim_{z \rightarrow \z0} \frac{f(z) - f(\z0)}{z-\z0}$$
provided the above limit exists. The function $f$ is said to be {\it differentiable} at $\z0$.

Consider $f(z) = z^2$. Write down the expression $\frac{ \Delta w}{\Delta z} = \frac{f(z+\Delta z) - f(z)}{\Delta z}$\\

\vspace*{1in}
The derivative ${\displaystyle \frac{dw}{dz}} = f'(z)$ is defined as $f'(z) = {\displaystyle \lim_{\Delta z \rightarrow 0} \frac{\Delta w}{\Delta z}}$

Evaluate this limit for our function $f(z) = z^2$.\\[.75in]


Write down $f'(z)$\\[12pt]

Write down the real and imaginary parts of the function $f(z) = z^2$\\[12pt]

Write down the real and imaginary parts of the function $f'(z)$ See any patterns?\\[24pt]


{\large\bf Rules of Differentiation}\\
The standard rules of differentiating function that you learned for real functions basically apply to complex functions. Scilicet:

$$ \frac{d}{d z}(c) = 0 \qquad \frac{d}{dz}(z) =1 \qquad  \qquad \frac{d}{dz}(z^n) = n z^{n-1} \qquad \frac{d}{dz}(e^z) = e^z$$

{\bf Linearity}
$$\frac{d}{dz}[cf(z)+g(z)] = cf'(z) + g'(z) \qquad c \mbox{ constant}$$
{\bf Product Rule}
$$\frac{d}{dz}[f(z)g(z)] = f'(z)g(z) + f(z)g'(z)$$
{\bf Quotient} Rule
$$\frac{d}{dz} \left[\frac{f(z)}{g(z)} \right] = \frac{f'(z)g(z) - f(z)g'(z)}{(g(z))^2}$$

{\bf Aspects of Differentiation}\\
One of the most important aspects to remember about differentiability and continuity is:
\begin{center}
{\bf \Large DIFFERENTIABILITY $\Rightarrow$ CONTINUITY}\\[12pt]
 
CONTINUITY {\bf \Large  DOES NOT IMPLY} DIFFERENTIABILITY.
\end{center}

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