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\centerline{{\it  Class 27} {\bf (Wednesday April 1)}}
 
\begin{description}
\item[SUMMARY] Theorems Following Cauchy's Integral Formulas
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\item[CURRENT READING] Brown \& Curchill pages 130-135
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\item[NEXT READING] Brown \& Curchill pages 180-196
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\centerline{\bf Implications of Cauchy's Integral Formula}

We will take some time to see that we understand how the following theorems are derived from the CIF.


{\bf Cauchy's Inequality}\\
Let $f$ be analytic inside and on a circle $|z-z_0|=R$, taken in the positive sense, and denoted by $C_R$, then the derivatives of $f$ as $z_0$ satisfy
$$|f^{n}(z_0)| \le \frac{n! M}{R^n} \qquad \qquad (n=1,2,3,\ldots)$$
where $|f(z)| \le M$ for all $z$ on $C_R$.\\
\underline{\bf Proof}: (Use the generalized CIF to try and bound $f^{(n)}(z_0)$)
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{\bf Liouville's Theorem}\\
If $f$ is {\it entire} and {\it bounded} in the complex plane, then $f(z)$ is constant throughout the plane.\\

%{\bf Proof:}\\
\underline{\bf Proof}: (Use Cauchy's Inequality when $n=1$ and assume the radius has to be big enough to include the entire plane, e.g. ${R \rightarrow \infty}$)
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%{\bf Morera's Theorem} \\
% If $f(z)$ is continuous in a simply-connected region $R$ and if $\oint_C f(z) dz = 0$ around {\it every} simple closed curve $C$ in $R$, then $f(z)$ is analytic in $R$.\\
%(NOTE: Morera's Theorem is the converse of the Cauchy-Goursat theorem.)\\
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{\bf Fundamental Theorem of Algebra}\\
Every polynomial equation $P(z) = a_0 + a_1 z + a_2 z^2 + \cdots  + a_n z^n = 0$ with degree $n \ge 1$ and $a_n \neq 0$ has atleast one root.\\
\underline{\bf Proof}: The proof is by contradiction and uses Liouville's Theorem.
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{\bf Gauss' mean value theorem}\\
If $f(z)$ is analytic inside and on a circle $C$ with center $z_0$ and radius $r$ then $f(z_0)$ is the mean of the values of $f(z)$ on $C$, namely
$${\displaystyle f(z_0) = \frac{1}{2 \pi} \int_0^{2 \pi} f(z_0 + r e^{i \theta}) d\theta }$$
\underline{\bf Proof}: This is obvious by using the CIF and a parametrization for $|z-z_0|=r$ and interpreting the results.
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{\bf Maximum modulus theorem}\\
If $f(z)$ is analytic inside and on a simple closed curve $C$ and is not identically equal to a constant, then the maximum value of $|f(z)|$ occurs on $C$.\\
\underline{\bf Proof}:
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{\bf Minimum modulus theorem}\\
If $f(z)$ is analytic inside and on a simple closed curve $C$ and $f(z) \neq 0$ inside $C$, then the minimum value of $|f(z)|$ occurs on $C$.\\
\underline{\bf Proof}: Use the maximum modulus theorem with $g(z) = 1/f(z)$.\\

\underline{\bf Cauchy's Residue Theorem}\\
If $f$ is analytic on a simple (positively oriented) closed contour $C$ and everywhere inside $C$ except a finite number of points $z_1, z_2, \cdots z_n$ inside $C$, then
$${\displaystyle \oint_{C} f(z) \, dz = 2 \pi i \sum_{k=1}^n \mbox{\bf Res}(z_k)}$$

Before we talk about what a {\bf residue} of a function $f(z)$ at $z_k$, denoted by Res$(f,z_k)$ or Res$(z_k)$ we will need to have a discussion about {\bf poles},{\bf zeroes} and {\bf singularities}.
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