Cauchy's Integral Formula Review

# Cauchy's Integral Formula Review

We will now review some of the recent material regarding Cauchy's integral formula and results that stem from it.

• Recall from the Cauchy's Integral Formula page that if $A \subseteq \mathbb{C}$ is open, $f : A \to \mathbb{C}$ is analytic on $A$, and $\gamma$ is a simple, closed, piecewise smooth, positively oriented curve contained in $A$ then for any $z_0$ inside of $\gamma$ we have that:
(1)
\begin{align} \quad f(z_0) = \frac{1}{2\pi i} \int_{\gamma} \frac{f(z)}{z - z_0} \: dz \end{align}
• On the Cauchy's Integral Formula for Derivatives page we then proved a more general result of Cauchy's integral formula. If $A \subseteq \mathbb{C}$ is open, $f : A \to \mathbb{C}$ is analytic on $A$, and $\gamma$ is a simple, closed, piecewise smooth, positively oriented curve contained in $A$ then for any $z_0$ inside of $\gamma$ and for all $k = 0, 1, 2, ...$ we have that:
(2)
\begin{align} \quad f^{(k)}(z_0) = \frac{k!}{2\pi i} \int_{\gamma} \frac{f(z)}{(z - z_0)^{k+1}} \: dz \end{align}
• As a nice consequence to this theorem we saw that if $A \subseteq \mathbb{C}$ is open and if $f : A \to \mathbb{C}$ is analytic on $A$ then all of the higher order partial derivatives of $f$ exist on $A$ (with the formula for $f^{(k)}(z_0)$ given above for all $z_0 \in A$ and for a circle $\gamma$ in $A$ with $z_0$ on the inside of $\gamma$.
• On the Cauchy's Derivative Inequalities page we looked at a nice result which gives us bounds for the values of the higher order derivatives of $f$. We saw that if $A \subseteq \mathbb{C}$ is open, $f : A \to \mathbb{C}$ is analytic, and for $z_0 \in A$ we have that $r > 0$ is such that $D(z_0, r) \subseteq A$ and there exists an $M > 0$ such that $\mid f(z) \mid \leq M$ on the circle $C_{z_0, r}$ given by $\mid z - z_0 \mid = r$ then for each $k = 0, 1, 2, ...$ we have that:
(3)
\begin{align} \quad \mid f^{(k)}(z_0) \mid \leq \frac{k! M}{r^k} \end{align}
• On the Liouville's Theorem page we used Cauchy's derivative inequalities in order to prove a very nice theorem called Liouville's theorem which states that if $f : \mathbb{C} \to \mathbb{C}$ is analytic on all of $\mathbb{C}$ and if there exists an $M > 0$ such that $\mid f(z) \mid \leq M$ for all $z \in \mathbb{C}$ then $f$ is a constant.
• Then on the Morera's Analyticity Theorem we looked at a nice criterion for a function to be analytic on a set by using Cauchy's integral formula for derivatives. Recall that if $A \subseteq \mathbb{C}$ is open, $f : A \to \mathbb{C}$ is continuous, and $\displaystyle{\int_{\gamma} f(z) \: dz = 0}$ for every closed piecewise smooth then $f$ is analytic on $A$.
• On The Maximum-Modulus Theorem page we then proved a remarkable result which said that if $A \subseteq \mathbb{C}$ is open and connected and if $f : A \to \mathbb{C}$ is analytic on $A$ then either:
* $f$ is constant on $A$.
* $\mid f(z) \mid$ does not attain a maximum value of $A$.