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| align="right" style="padding-right: 1em;" | electrical engineers' imaginary number || <math>j=\sqrt{-1}\ </math> | | align="right" style="padding-right: 1em;" | electrical engineers' imaginary number || <math>j=\sqrt{-1}\ </math> | ||
|- | |- | ||
| − | | align="right" style="padding-right: 1em;" | [[more_on_complex_conjugate|(more)]] conjugate of a complex number || if <math>z=a+ | + | | align="right" style="padding-right: 1em;" | [[more_on_complex_conjugate|(more)]] conjugate of a complex number || if <math>z=a+ib</math>, for <math>a,b\in {\mathbb R}</math>, then <math> \bar{z}=a-ib </math> |
|- | |- | ||
| align="right" style="padding-right: 1em;" | [[more_on_complex_magnitude|(more)]] magnitude of a complex number || <math>\| z \| = z \bar{z} </math> | | align="right" style="padding-right: 1em;" | [[more_on_complex_magnitude|(more)]] magnitude of a complex number || <math>\| z \| = z \bar{z} </math> | ||
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| align="right" style="padding-right: 1em;" | [[more_on_complex_magnitude|(more)]] magnitude of a complex number || <math> \| z \| = \sqrt{\left(Re(z)\right)^2+\left(Im(z)\right)^2}</math> | | align="right" style="padding-right: 1em;" | [[more_on_complex_magnitude|(more)]] magnitude of a complex number || <math> \| z \| = \sqrt{\left(Re(z)\right)^2+\left(Im(z)\right)^2}</math> | ||
|- | |- | ||
| − | | align="right" style="padding-right: 1em;" | [[more_on_complex_magnitude|(more)]] magnitude of a complex number || <math>\| a+ | + | | align="right" style="padding-right: 1em;" | [[more_on_complex_magnitude|(more)]] magnitude of a complex number || <math>\| a+ib \| = \sqrt{a^2+b^2} </math>, for <math>a,b\in {\mathbb R}</math> |
|- | |- | ||
| align="right" style="padding-right: 1em;" | [[more_on_complex_magnitude|(more)]] magnitude of a complex number || <math>\| r e^{i \theta} \| = r </math>, for <math>r,\theta\in {\mathbb R}</math> | | align="right" style="padding-right: 1em;" | [[more_on_complex_magnitude|(more)]] magnitude of a complex number || <math>\| r e^{i \theta} \| = r </math>, for <math>r,\theta\in {\mathbb R}</math> | ||
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| align="right" style="padding-right: 1em;" | Cosine function in terms of complex exponentials|| <math>\cos\theta=\frac{e^{i\theta}+e^{-i\theta}}{2}</math> | | align="right" style="padding-right: 1em;" | Cosine function in terms of complex exponentials|| <math>\cos\theta=\frac{e^{i\theta}+e^{-i\theta}}{2}</math> | ||
|- | |- | ||
| − | | align="right" style="padding-right: 1em;" | Sine function in terms of complex exponentials||<math>\sin\theta=\frac{e^{i\theta}-e^{-i\theta}}{ | + | | align="right" style="padding-right: 1em;" | Sine function in terms of complex exponentials||<math>\sin\theta=\frac{e^{i\theta}-e^{-i\theta}}{2i}</math> |
|- | |- | ||
|} | |} | ||
---- | ---- | ||
[[ MegaCollectiveTableTrial1|Back to Collective Table]] | [[ MegaCollectiveTableTrial1|Back to Collective Table]] | ||
Revision as of 09:33, 2 November 2009
| Complex Number Identities and Formulas | |
|---|---|
| Basic Definitions | |
| imaginary number | $ i=\sqrt{-1} \ $ |
| electrical engineers' imaginary number | $ j=\sqrt{-1}\ $ |
| (more) conjugate of a complex number | if $ z=a+ib $, for $ a,b\in {\mathbb R} $, then $ \bar{z}=a-ib $ |
| (more) magnitude of a complex number | $ \| z \| = z \bar{z} $ |
| (more) magnitude of a complex number | $ \| z \| = \sqrt{\left(Re(z)\right)^2+\left(Im(z)\right)^2} $ |
| (more) magnitude of a complex number | $ \| a+ib \| = \sqrt{a^2+b^2} $, for $ a,b\in {\mathbb R} $ |
| (more) magnitude of a complex number | $ \| r e^{i \theta} \| = r $, for $ r,\theta\in {\mathbb R} $ |
| Euler's Formula and Related Equalities | |
| Euler's formula | $ e^{iw_0t}=\cos w_0t+i\sin w_0t \ $ |
| A really cute formula | $ e^{i\pi}=-1 \ $ |
| Cosine function in terms of complex exponentials | $ \cos\theta=\frac{e^{i\theta}+e^{-i\theta}}{2} $ |
| Sine function in terms of complex exponentials | $ \sin\theta=\frac{e^{i\theta}-e^{-i\theta}}{2i} $ |
