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Power Series Expansions

These can easily be evaluated using the Taylor series discussed in the last section, expanded around the origin $ z = 0$ , and are an alternative way of seeing that $ z = e^{i\theta}$ . In the case of exponential and trig functions, the expansions converge for all $ z$ , not just small ones (although they of course converge faster for small ones).


$\displaystyle e^{x}$ $\displaystyle =$ $\displaystyle 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \ldots$ (77)
$\displaystyle \cos(x)$ $\displaystyle =$ $\displaystyle 1 - \frac{x^2}{2!} + \frac{x^4}{4!} + \ldots$ (78)
$\displaystyle \sin(x)$ $\displaystyle =$ $\displaystyle x - \frac{x^3}{3!} + \frac{x^5}{5!} + \ldots$ (79)

Depending on where you start, these can be used to prove the relations above. They are most useful for getting expansions for small values of their parameters. For small x (to leading order):
$\displaystyle e^{x}$ $\displaystyle \approx$ $\displaystyle 1 + x$ (80)
$\displaystyle \cos(x)$ $\displaystyle \approx$ $\displaystyle 1 - \frac{x^2}{2!}$ (81)
$\displaystyle \sin(x)$ $\displaystyle \approx$ $\displaystyle x$ (82)
$\displaystyle \tan(x)$ $\displaystyle \approx$ $\displaystyle x$ (83)

We will use these fairly often in this course, so learn them.


next up previous contents
Next: An Important Relation Up: Complex Numbers and Harmonic Previous: Trigonometric and Exponential Relations   Contents
Robert G. Brown 2011-04-19