$\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} \newcommand{\sech}{\text{sech}} \newcommand{\csch}{\text{csch}} \newcommand{\Ln}{\text{Ln}} $

Lecture 12, 9/19/2022. This page is for Section 1 only.
ACMS 20550: Applied Mathematics Method I
Instructor: Bei Hu, b1hu@nd.edu, Hurley 174A

  1. Complex roots and power
    1. $\fcolorbox{white}{yellow}{$a^b = e^{b\ln a} $} $ Don't forget the $2k\pi i$ from $\ln a$
    2. Examples: (a) Writing polar represenation $ a = r e^{i(\theta + 2k\pi)}$, (b) $\ln a = \Ln\; r + i(\theta+2 k\pi)$, (c) plug in and doing algebra
  2. Inverse trignomitry function
    1. $\fcolorbox{white}{yellow}{$z = \arccos w \m $ if $\m w =\cos z = \frac{e^{iz}+e^{-iz}}2$}$
    2. Example: Solve $\arccos w$ (with $w$ a given number)
      (a) $u = e^{iz}$;
      (b) $\cos z = \frac{e^{iz}+e^{-iz}}2 = \frac{ u+ \frac1u}2 = w$;
      (c) Solve quodratic equaiton $2 u w = u^2 +1$, or $u^2 - 2u w +1 =0$;
      (d) Solve $u$ and then $z$;