| Line 18: | Line 18: | ||
| align="right" style="padding-right: 1em;" | Euler's formula || <math>e^{jw_0t}=cosw_0t+jsinw_0t</math> | | align="right" style="padding-right: 1em;" | Euler's formula || <math>e^{jw_0t}=cosw_0t+jsinw_0t</math> | ||
|- | |- | ||
| − | | align="right" style="padding-right: 1em;" | | + | | align="right" style="padding-right: 1em;" | Cosine function in terms of complex exponentials|| <math>cos\theta=\frac{e^{j\theta}+e^{-j\theta}}{2}</math> |
|- | |- | ||
| align="right" style="padding-right: 1em;" | [[Sine function in terms of complex exponential_ECE301Fall2008mboutin]] || {{:Sine function in terms of complex exponential_ECE301Fall2008mboutin}} | | align="right" style="padding-right: 1em;" | [[Sine function in terms of complex exponential_ECE301Fall2008mboutin]] || {{:Sine function in terms of complex exponential_ECE301Fall2008mboutin}} | ||
Revision as of 04:52, 27 October 2009
Some General Purpose Formulas and Definitions
| General Purpose Formulas | |
|---|---|
| Series | |
| Finite Geometric Series Formula | $ \sum_{k=0}^n x^k = \left\{ \begin{array}{ll} \frac{1-x^{n+1}}{1-x}&, \text{ if } x\neq 1\\ n+1 &, \text{ else}\end{array}\right. $ |
| Infinite Geometric Series Formula | $ \sum_{k=0}^n x^k = \left\{ \begin{array}{ll} \frac{1}{1-x}&, \text{ if } |x|\leq 1\\ \text{diverges} &, \text{ else }\end{array}\right. $ |
| Euler's Formula and Related Equalities | |
| Euler's formula | $ e^{jw_0t}=cosw_0t+jsinw_0t $ |
| Cosine function in terms of complex exponentials | $ cos\theta=\frac{e^{j\theta}+e^{-j\theta}}{2} $ |
| Sine function in terms of complex exponential_ECE301Fall2008mboutin | $ sin\theta=\frac{e^{j\theta}-e^{-j\theta}}{2j} $ |
| Definition of some Basic Functions (what engineers call "Signals") | |
| sinc function_ECE301Fall2008mboutin | $ sinc(\theta)=\frac{sin(\pi\theta)}{\pi\theta} $ |
