Page Description Equation
166 parametrization of the circle $\abs{z-z_0}=r,$ i.e. the circle centered at $z0$ with radius $r$ \[ z\left(t\right)=z_0+re^{it},0\le t\le2\pi \]
166 $C_r=$ circle $\abs{z-z_0}=r$ traversed once in the counter-clockwise direction \[ \int_{C_r}{\left(z-z_0\right)^n\dz} = \left\{ \begin{matrix} 0 &n\neq-1\\ 2\pi i &n=-1 \end{matrix} \right. \]
186 Deformation Invariance theorem. theorem 8. \[ \int_{\Gamma_0} f\left(z\right)\dz =\int_{\Gamma_1} f\left(z\right)\dz \]
187 Cauchy's integral theorem. theorem 9. $f$ is analytic in a simply connected domain $\C$ and $\Gamma$ is any loop (closed contour) in $D.$ \[ \int_{\Gamma} f\left(z\right)dz=0 \]
Cauchy's integral formula. Holds if $f$ is analytic inside and on the simple closed positively oriented contour $\Gamma$ and $z_0$ is any point inside $\Gamma.$ \[ f\left(z_0\right) =\frac{1}{2\pi i}\int_{\Gamma} {\frac{f\left(z\right)}{\left(z-z_0\right)}\dz} \]
211 generalized Cauchy integral formula. Holds if $f$ is analytic inside and on the simple closed positively oriented contour $\Gamma$ and $z$ is any point inside $\Gamma.$ \[ f^{\left(n\right)}\left(z\right) =\frac{n!}{2\pi i}\int_{\Gamma} {\frac{f\left(\zeta\right)} {\left(\zeta-z\right)^{n+1}}\,d\zeta}, \left(n=0,1,2,\ldots\right) \]
211 generalized Cauchy integral formula (equivalent form). Same conditions except now $z_0$ is any point inside $\Gamma.$ \[ f^{\left(n\right)}\left(z_0\right) =\frac{n!}{2\pi i} \int_{\Gamma} {\frac{f\left(z\right)}{\left(z-z_0\right)^{n+1}}\dz}, \left(n=0,1,2,\ldots\right) \]
211 generalized Cauchy integral formula (equivalent form). Same conditions as previous. \[ \frac{2\pi i f^{\left(m-1\right)}\left(z_0\right)}{\left(m-1\right)!} =\int_{\Gamma} {\frac{f\left(z\right)}{\left(z-z_0\right)^m}\dz}, \left(m=1,2,3,\ldots\right) \]
215 Cauchy estimates for the derivatives of a function $f.$ Holds if $f$ is analytic inside and on a circle $C_R$ of radius $R$ centered about $z_0,$ and $\abs{f\left(z\right)}\le M$ for all $z$ on $C_R.$ \[ \abs{f^{\left(n\right)}\left(z_0\right)} \le\frac{n!M}{R^n}, \left(n=1,2,3,\ldots\right) \]
216 mean value property \[ f\left(z_0\right) =\frac{1}{2\pi} \int_{0}^{2\pi}f\left(z_0+Re^{it}\right)\dt \]

Notes

Page Notes
182 Definition 5
184 Definition 6
187 The integral along $\Gamma$ must vanish whenever the integrand is analytic inside and on $\Gamma.$
187 Theorem 10
207 Theorem 15
209 Theorem 16 If $f$ is analytic in a domain $D,$ then all its derivatives $f',f'',\ldots,f^{(n)},\ldots$ exist and are analytic in $D.$
210 Theorem 17
210 Theorem 18. Morera's theorem.
205 By merely knowing the values of the analytic function $f$ on $\Gamma$ we can compute the integral in Cauchy's integral formula that hence all the values of $f$ inside $\Gamma.$
205 Chapter 6 is devoted to more efficient and powerful techniques for computing integrals than Cauchy's integral formula.
215 Theorem 21. Liouville's Theorem. The only bounded entire functions are the constant functions.
216 Fundamental Theorem of Algebra. Every nonconstant polynomial with complex coefficients has at least one zero.
217 Lemma 1
217 Theorem 23
218 Theorem 24
220 exercise 7 If $f$ is entire and $\abs{f\left(z\right)}\le M\abs{z}^n$ for $\abs{z}>r_0$ where $n$ is a nonnegative integer, then $f$ must be a polynomial of degree at most $n.$