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\centerline{ Class 10 {\bf (Wednesday February 4)}}
\begin{description}
\item[SUMMARY] Analyticity and the Cauchy-Riemann Equations
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\item[CURRENT READING] Brown \& Curchill, pages 45-57
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\item[NEXT READING] Brown \& Curchill, pages 48-50, 55-57, 59-64

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\underline{\sc Groupwork}\\
Given $g(z) = z^2 + z + i$ and ${\displaystyle f(z) = \frac{1}{z}}$\\
$g'(z) =$
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$f'(z) =$
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$[g(z)f(z)]' =$
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${\displaystyle [g(z)/f(z)]'} =$
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{\bf Cauchy-Riemann Equations}\\
We shall derive the Cauchy-Riemann equations by looking at the definition of the derivative of a function $f(z) = u(x,y) + i v(x,y)$ at the point $z_0.$

\begin{eqnarray}
f'(\z0) & =  & {\displaystyle \lim_{\Delta z \rightarrow 0} \frac{f(z_0 + \Delta z) - f(\z0)}{\Delta z}} \nonumber \\
 & = &{\displaystyle \lim_{(\Delta x, \Delta y) \rightarrow (0,0)} \frac{u(x_0+\Delta x,y_0+\Delta y) + i v(x_0+\Delta x,y_0+\Delta y) - u(x_0,y_0)-iv(x_0,y_0)}{\Delta x + i \Delta y}}  \nonumber
\end{eqnarray}


We shall do this limit twice, once letting $\Delta z \rightarrow 0$ horizontally and the other time letting $\Delta z \rightarrow 0$ vertically

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This shows us that the existence of $f'(z_0)$ implies the Cauchy-Riemann equations are satisfied at this point (and at every point in a  neighborhood of $z_0$).\\ This is true since if $f'(z_0)$ exists then $f$ is {\bf analytic} ``at'' this point.
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\begin{center}
{\Large ANALYTICITY $\Rightarrow$ C.R.E.}
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To make satisfying the CRE a {\it sufficient} condition one needs the added condition that the first derivatives of $u$ and $v$ are continuous. If both these conditions are true and $f$ is defined on an open set, then $f$ is analytic on the open set.
$$f'(\z0) = u_x(\x0,\y0) + i v_x (\x0,\y0) = -i (u_y(\x0,\y0) + i v_y(\x0,\y0))$$
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{\Large ANALYTICITY $\Longleftrightarrow$ C.R.E. + Continuity of $u_x,u_y,v_x,v_y$}
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\underline{\bf Example}\\
Show that $f(z) = \zc$ is {\bf not analytic} anywhere in the complex plane.
You can do this in two ways:\\
{\bf 1:}\\
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{\bf 2:}\\
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\underline{\sc Groupwork}\\
\underline{\bf Example}\\
Show that the function $f(z) = 1/z$ is analytic on the set $z \neq 0$. 
To do that you will have to answer the
following questions:
\begin{itemize}
\item What is its domain of definition? Is this an open set?
\item What are its component functions? Are there partial derivatives continuous?
\item Do they satisfy the CRE?
\item Is it analytic? On what set?
\end{itemize}
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