$\newcommand{\dis}{\displaystyle} \newcommand{\m}{\hspace{1em}} \newcommand{\mm}{\hspace{2em}} \newcommand{\x}{\vspace*{1ex}} \newcommand{\xx}{\vspace*{2ex}} \let\limm\lim \renewcommand{\lim}{\dis\limm} \let\fracc\frac \renewcommand{\frac}{\dis\fracc} \let\summ\sum \renewcommand{\sum}{\dis\summ} \let\intt\int \renewcommand{\int}{\dis\intt} $

Lecture 9, 9/12/2022. This page is for Section 1 only.
ACMS 20550: Applied Mathematics Method I
Instructor: Bei Hu, b1hu@nd.edu, Hurley 174A

  1. Complex infinite series: $\sum_{n=1}^\infty z_n, \m z_n = x_n + i y_n$
    1. $ \fcolorbox{white}{yellow}{$\sum_{n=1}^\infty z_n = \sum_{n=1}^\infty x_n +i \sum_{n=1}^\infty y_n $ if both $\sum_{n=1}^\infty x_n$ and $\sum_{n=1}^\infty y_n$ are convergent.}$
    2. $\sum_{n=1}^\infty |z_n|$ converges if and only if both $\sum_{n=1}^\infty x_n$ and $\sum_{n=1}^\infty y_n$ are convergent absolutly.
    3. $ \fcolorbox{white}{yellow}{All tests in Chapter 1 applies to the case of absolute convergence}$.
  2. Complex power series $\sum_{n=0}^\infty a_n z^n$.
    1. Disk of convergence $|z|< R$.
      $\frac1R = \lim_{n\to\infty} \frac{|a_{n+1}|}{|a_n|} \fcolorbox{white}{yellow}{Caution: the formula fails if the limit does not exist}$
    2. e.g.,
      $e^z = 1 + z+\frac{z^2}{2!} + \frac{z^3}{3!}+ \cdots, $ all $|z|<\infty$.