$\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 11, 9/16/2022.
This page is for Section 1 only
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ACMS 20550: Applied Mathematics Method I
Instructor: Bei Hu, b1hu@nd.edu, Hurley 174A
Complex trig function $e^z$
$\fcolorbox{white}{yellow}{$\sin z =\frac{e^{iz}-e^{-iz}}{2i} $}$
$\fcolorbox{white}{yellow}{$\cos z =\frac{e^{iz}+e^{-iz}}2 $}$
$ \sin^2 z + \cos^2 z =1, \m \frac{d}{dz} \sin z = \cos z, \m \frac{d}{dz} \cos z = - \sin z$.
Applications: From $\int e^{(a+bi)x} dx \m $ to $ \m\int e^{ax} \cos bx dx $.
Applications: Using exponential functions to evaluate trig integral, e.g, $\int \cos 2x\cdot \cos x dx $.
Hyperbolic function
$\fcolorbox{white}{yellow}{$\sinh z =\frac{e^{z}-e^{-z}}{2} $}$
$\fcolorbox{white}{yellow}{$\cosh z =\frac{e^{z}+e^{-z}}{2} $}$
$ \cosh^2 z -\sinh^2 z =1, \m \frac{d}{dz} \sinh z = \cosh z, \m \frac{d}{dz} \cosh z = - \sinh z$.
$\fcolorbox{white}{yellow}{$\tanh z =\frac{\sinh z}{\cosh z}, \m \coth z =\frac{1}{\tanh z}, \m \sech z = \frac1{\cosh z}, \m \csch z =\frac1{\sinh z} $}$
Log functions
Step 1: Write $ z= re^{i\theta} = r e^{i(\theta+2k\pi)}$, Step 2: $\ln z = \ln (r e^{i(\theta+2k\pi)}) = \Ln \; r + i (\theta+ 2k\pi ), \m k=0,\pm1,\pm2, \cdots$
Don't forget the $2k\pi i$