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A First Course in Electrical and Computer Engineering . The LaTeX source files for this collection were created using an optical character recognition technology, and because of this process there may be more errors than usual. Please contact us if you discover any errors.
${e}^{j\theta}$ , cos
θ , and sin
θ
${e}^{j\theta}=\underset{n\to \infty}{lim}{(1+j\frac{\theta}{n})}^{n}=\sum n=0\infty \frac{1}{n!}{\left(j\theta \right)}^{n}=cos\theta +jsin\theta cos\theta =\sum n=0\infty \frac{{(-1)}^{n}}{\left(2n\right)!}{\theta}^{2n};sin\theta =\sum n=0\infty \frac{{(-1)}^{n}}{(2n+1)!}{\theta}^{2n+1}$
$cos\theta =\sum n=0\infty \frac{{(-1)}^{n}}{\left(2n\right)!}{\theta}^{2n};sin\theta =\sum n=0\infty \frac{{(-1)}^{n}}{(2n+1)!}{\theta}^{2n+1}$
Trigonometric identities
$${sin}^{2}\theta +{cos}^{2}\theta =1$$
$sin(\theta +\phi )=sin\theta cos\phi +cos\theta sin\phi $
$cos(\theta +\phi )=cos\theta cos\phi -sin\theta sin\phi $
$sin(\theta -\phi )=sin\theta cos\phi -cos\theta sin\phi $
$cos(\theta -\phi )=cos\theta cos\phi +sin\theta sin\phi $
Euler's equations
${e}^{j\theta}=cos\theta +jsin\theta $
$$sin\theta =\frac{{e}^{j\theta}-{e}^{-j\theta}}{2j}$$
$cos\theta =\frac{{e}^{j\theta}+{e}^{-j\theta}}{2}$
De moivre's identity
$(cos\theta +jsin{\theta )}^{n}=cosn\theta +jsinn\theta $
Binomial expansion
$${(x+y)}^{N}=\sum n=0NNn{x}^{n}{y}^{N-n};Nn=\frac{N!}{(N-n)!n!}$$
$${2}^{N}=\sum n=0NNn$$
Geometric sums
$$\sum k=0\infty a{z}^{k}=\frac{a}{1-z}\left|z\right|<1$$
$$\sum k=0N-1a{z}^{k}=\frac{a(1-{z}^{N})}{1-z}z\ne 1$$
Taylor's series
$$f\left(x\right)=\sum k=0\infty {f}^{\left(k\right)}\left(a\right)\frac{{(x-a)}^{k}}{k!}$$
$$(\text{Maclaurin's Series if}a=0)$$