Chapter 4 Contour Integration

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 $n.$