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pages/Math416/Math416_L9.md

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+# Math416 Lecture 9
+
+## Review
+
+### Power Series
+
+Let $f(\zeta)=\sum_{n=0}^{\infty}a_n(\zeta-\zeta_0)^n$ be a power series.
+
+#### Radius of Convergence
+
+The radius of convergence of a power series is
+
+$$
+R=\frac{1}{\limsup_{n\to\infty}|a_n|^{1/n}}.
+$$
+
+## New Material on Power Series
+
+### Derivative of Power Series
+
+Let $f(\zeta)=\sum_{n=0}^{\infty}a_n(\zeta-\zeta_0)^n$ be a power series.
+
+Let $g(\zeta)=\sum_{n=0}^{\infty}na_n(\zeta-\zeta_0)^{n-1}$ be another power series.
+
+Then $g$ is holomorphic on $D(\zeta_0,R)$ and $g'(\zeta)=f(\zeta)$ for all $\zeta\in D(\zeta_0,R)$. and $f'(\zeta)=g(\zeta)$.
+
+Proof:
+
+Note radius of convergence of $g$ is also $R$.
+
+$\limsup_{n\to\infty}|na_n|^{1/(n-1)}=\limsup_{n\to\infty}|a_n|^{1/n}$.
+
+Let $\zeta\in D(\zeta_0,R)$.
+
+let $|\zeta-\zeta_0|<\rho<R$.
+
+Without loss of generality, assume $\zeta_0=0$. Let $|w|<\rho$.
+
+$$
+\begin{aligned}
+\frac{f(\zeta)-f(w)}{\zeta-w}-g(\zeta)&=\sum_{n=0}^{\infty}\left[\frac{1}{\zeta-w}\left(a_n(\zeta^n-w^n)\right)-na_n\zeta^{n-1}\right] \\
+&=\sum_{n=0}^{\infty}a_n\left[\frac{\zeta^n-w^n}{\zeta-w}-n\zeta^{n-1}\right]
+\end{aligned}
+$$
+
+Notice that
+
+$$
+\begin{aligned}
+\frac{\zeta^n-w^n}{\zeta-w}&=\sum_{k=0}^{n-1}\zeta^{n-1-k}w^k \\
+&=\zeta^{n-1}+\zeta^{n-2}w+\cdots+w^{n-1}
+\end{aligned}
+$$
+
+Since
+
+$$
+|w^k-\zeta^k|=\left|(w-\zeta)\left(\sum_{j=0}^{k-1}w^{k-1-j}\zeta^j\right)\right|\leq|w-\zeta|k\rho^{k-1}
+$$
+
+$$
+\begin{aligned}
+\frac{\zeta^n-w^n}{\zeta-w}-n\zeta^{n-1}&=(\zeta^{n-1}-\zeta^{n-1})+(\zeta^{n-2}w-\zeta^{n-1})+\cdots+(\zeta w^{n-1}-\zeta^{n-1}) \\
+&=\zeta^{n-2}(w-\zeta)+\zeta^{n-3}(w^2-\zeta^2)+\cdots+\zeta^0(w^{n-1}-\zeta^{n-1}) \\
+&=\sum_{k=0}^{n-1}\zeta^{n-1-k}(w^k-\zeta^k)\\
+&\leq\sum_{k=0}^{n-1}\zeta^{n-1-k}|w-\zeta|k\rho^{k-1} \\
+&\leq|w-\zeta|\sum_{k=0}^{n-1}k\rho^{k-1} \\
+\end{aligned}
+$$
+
+Apply absolute value,
+
+$$
+\begin{aligned}
+\left|\frac{f(\zeta)-f(w)}{\zeta-w}-g(\zeta)\right|&\leq\sum_{n=0}^{\infty}|a_n||w-\zeta|\left[\sum_{k=1}^{n-1}\rho^{n-1-k}k\rho^{k-1}\right] \\
+&=|w-\zeta|\sum_{n=0}^{\infty}|a_n|\left[\sum_{k=1}^{n-1}\rho^{n-2}k\right] \\
+&=|w-\zeta|\sum_{n=0}^{\infty}|a_n|\frac{n(n-1)}{2}\rho^{n-2} \\
+\end{aligned}
+$$
+
+Using Cauchy-Hadamard theorem, the radius of convergence of $\sum_{n=0}^{\infty}\frac{ n(n-1)}{2}|a_n|\zeta^{n-2}$ is at least
+
+$$
+1/\limsup_{n\to\infty}\left[\frac{n(n-1)}{2}|a_n|\right]^{1/(n-1)}=R.
+$$
+
+Therefore,
+
+$$
+|w-\zeta|\sum_{n=0}^{\infty}|a_n|\frac{n(n-1)}{2}\rho^{n-2} \leq C|w-\zeta|
+$$
+
+where $C$ is dependent on $\rho$.
+
+So $lim_{w\to\zeta}\left|\frac{f(\zeta)-f(w)}{\zeta-w}-g(\zeta)\right|=0$. as desired.
+
+#### Corollary of power series
+
+If $f(\zeta)=\sum_{n=0}^{\infty}a_n(\zeta-\zeta_0)^n$ in $D(\zeta_0,R)$, then $a_0=f(\zeta_0), a_1=f'(\zeta_0)/1!, a_2=f''(\zeta_0)/2!$, etc.
+
+#### Definition (Analytic)
+
+A function $h$ on an open set $U\subset\mathbb{C}$ is called analytic if for every $\zeta\in U$, $\exists \epsilon>0$ such that on $D(\zeta,\epsilon)\subset U$, $h$ can be represented as a power series $\sum_{n=0}^{\infty}a_n(\zeta-\zeta_0)^n$.
+
+#### Theorem (Analytic implies holomorphic)
+
+If $f$ is analytic on $U$, then $f$ is holomorphic on $U$.
+
+$\sum_{n=0}^{\infty}\frac{1}{n!}f^{(n)}(\zeta)^n$
+
+Radius of convergence is $\infty$.
+
+So $f(0)=1=ce^0=c$
+
+$\sum_{n=0}^{\infty}\frac{1}{n}\zeta^n$
+
+Radius of convergence is $1$.
+
+$f'=\sum_{n=1}^{\infty}\zeta^{n-1}=\frac{1}{1-\zeta}$ (Geometric series)
+
+So $g(\zeta)=c+\log(\frac{1}{1-\zeta})=c+2\pi k i=\log(\frac{1}{1-\zeta})+2\pi k i$
+
+#### Cauchy Product of power series
+
+Let $f(\zeta)=\sum_{n=0}^{\infty}a_n\zeta^n$ and $g(\zeta)=\sum_{n=0}^{\infty}b_n\zeta^n$ be two power series.
+
+Then $f(\zeta)g(\zeta)=\sum_{n=0}^{\infty}=\sum_{n=0}^{\infty}c_n\zeta^n=\sum_{n=0}^{\infty}\sum_{k=0}^{n}a_kb_{n-k}\zeta^n$
+
+#### Theorem of radius of convergence of Cauchy product
+
+Let $f(\zeta)=\sum_{n=0}^{\infty}a_n\zeta^n$ and $g(\zeta)=\sum_{n=0}^{\infty}b_n\zeta^n$ be two power series.
+
+Then the radius of convergence of $f(\zeta)g(\zeta)$ is at least $\min(R_f,R_g)$.
+
+Without loss of generality, assume $\zeta_0=0$.
+
+$$
+\begin{aligned}
+\left(\sum_{j=0}^{N}a_j\zeta^j\right)\left(\sum_{k=0}^{N}b_k\zeta^k\right)-\sum_{l=0}^{N}c_l\zeta^l&=\sum_{j=0}^{N}\sum_{k=N-j}^{N}a_jb_k\zeta^{j+k}\\
+&\leq\sum_{N/2\leq\max(j,k)\leq N}|a_j||b_k||\zeta^{j+k}|\\
+&\leq\left(\sum_{j=N/2}^{N}|a_j||\zeta^j|\right)\left(\sum_{k=0}^{\infty}|b_k||\zeta^k|\right)+\left(\sum_{j=0}^{\infty}|a_j||\zeta^j|\right)\left(\sum_{k=N/2}^{\infty}|b_k||\zeta^k|\right)\\
+\end{aligned}
+$$
+
+Since $\sum_{j=0}^{\infty}|a_j||\zeta^j|$ and $\sum_{k=0}^{\infty}|b_k||\zeta^k|$ are convergent, and $\sum_{j=N/2}^{N}|a_j||\zeta^j|$ and $\sum_{k=N/2}^{\infty}|b_k||\zeta^k|$ converges to zero.
+
+So $\left|\left(\sum_{j=0}^{N}a_j\zeta^j\right)\left(\sum_{k=0}^{N}b_k\zeta^k\right)-\sum_{l=0}^{N}c_l\zeta^l\right|\leq\left(\sum_{j=N/2}^{N}|a_j||\zeta^j|\right)\left(\sum_{k=0}^{\infty}|b_k||\zeta^k|\right)+\left(\sum_{j=0}^{\infty}|a_j||\zeta^j|\right)\left(\sum_{k=N/2}^{\infty}|b_k||\zeta^k|\right)\to 0$ as $N\to\infty$.
+
+So $\sum_{n=0}^{\infty}c_n\zeta^n$ converges to $f(\zeta)g(\zeta)$ on $D(0,R_fR_g)$.

pages/Math416/_meta.js

@@ -11,4 +11,5 @@ export default {
     Math416_L6: "Complex Variables (Lecture 6)",
     Math416_L7: "Complex Variables (Lecture 7)",
     Math416_L8: "Complex Variables (Lecture 8)",
+    Math416_L9: "Complex Variables (Lecture 9)"
 }