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! colspan="2" style="background: #eee;" | Basic Definitions | ! colspan="2" style="background: #eee;" | Basic Definitions | ||
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| − | | align="right" style="padding-right: | + | | align="right" style="padding-right: 2em;" | imaginary number || <math>i=\sqrt{-1} \ </math> |
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| − | | align="right" style="padding-right: | + | | align="right" style="padding-right: 2em;" | electrical engineers' imaginary number || <math>j=\sqrt{-1}\ </math> |
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| − | | align="right" style="padding-right: | + | | align="right" style="padding-right: 2em;" | [[more_on_complex_conjugate|(info)]] 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> |
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| − | | align="right" style="padding-right: | + | | align="right" style="padding-right: 2em;" | [[more_on_complex_magnitude|(info)]] magnitude of a complex number || <math>\| z \| = z \bar{z} </math> |
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| − | | align="right" style="padding-right: | + | | align="right" style="padding-right: 2em;" | [[more_on_complex_magnitude|(info)]] magnitude of a complex number || <math> \| z \| = \sqrt{\left(Re(z)\right)^2+\left(Im(z)\right)^2}</math> |
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| − | | align="right" style="padding-right: | + | | align="right" style="padding-right: 2em;" | [[more_on_complex_magnitude|(info)]] magnitude of a complex number || <math>\| a+ib \| = \sqrt{a^2+b^2} </math>, for <math>a,b\in {\mathbb R}</math> |
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| − | | align="right" style="padding-right: | + | | align="right" style="padding-right: 2em;" | [[more_on_complex_magnitude|(info)]] magnitude of a complex number || <math>\| r e^{i \theta} \| = r </math>, for <math>r,\theta\in {\mathbb R}</math> |
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! colspan="2" style="background: #eee;" | Complex Number Operations | ! colspan="2" style="background: #eee;" | Complex Number Operations | ||
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| − | | align="right" style="padding-right: | + | | align="right" style="padding-right: 2em;" |addition || <math>(a+ib)+(c+id)=(a+c) + i (b+d) \ </math> |
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| − | | align="right" style="padding-right: | + | | align="right" style="padding-right: 2em;" |multiplication || <math>(a+ib) (c+id)=(ac-bd) + i (ad+bc) \ </math> |
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| − | | align="right" style="padding-right: | + | | align="right" style="padding-right: 2em;" |division || <math>\frac{a+ib} {c+id}=\frac{ac+bd} {c^2+d^2}+ i \frac{bc-ad} {c^2+d^2} \ </math> |
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| − | | align="right" style="padding-right: | + | | align="right" style="padding-right: 2em;" | exponentiation || <math> i^n =\left\{ \begin{array}{ll}1,& \text{when }n\equiv 0\mod 4 \\ i,& \text{when }n\equiv 1\mod 4 \\-1,& \text{when }n\equiv 2\mod 4 \\-i,& \text{when }n\equiv 3\mod 4 |
\end{array} \right. \ </math> | \end{array} \right. \ </math> | ||
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! colspan="2" style="background: #eee;" | Euler's Formula and Related Equalities [[more_on_Eulers_formula|(info)]] | ! colspan="2" style="background: #eee;" | Euler's Formula and Related Equalities [[more_on_Eulers_formula|(info)]] | ||
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| − | | align="right" style="padding-right: | + | | align="right" style="padding-right: 2em;" | [[more_on_Eulers_formula|(info)]] Euler's formula || <math>e^{iw_0t}=\cos w_0t+i\sin w_0t \ </math> |
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| − | | align="right" style="padding-right: | + | | align="right" style="padding-right: 2em;" | A really cute formula || <math>e^{i\pi}=-1 \ </math> |
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| − | | align="right" style="padding-right: | + | | align="right" style="padding-right: 2em;" | Cosine function in terms of complex exponentials|| <math>\cos\theta=\frac{e^{i\theta}+e^{-i\theta}}{2}</math> |
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| − | | align="right" style="padding-right: | + | | align="right" style="padding-right: 2em;" | Sine function in terms of complex exponentials||<math>\sin\theta=\frac{e^{i\theta}-e^{-i\theta}}{2i}</math> |
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! colspan="2" style="background: #eee;" | Other Formulas | ! colspan="2" style="background: #eee;" | Other Formulas | ||
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| − | | align="right" style="padding-right: | + | | align="right" style="padding-right: 2em;" | De Moivre's theorem ||<math>\left(\cos x+i\sin x\right)^n=\cos\left(nx\right)+i\sin\left(nx\right).\,</math> |
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|} | |} | ||
Revision as of 17:34, 2 November 2009
| Complex Number Identities and Formulas (info) | |
|---|---|
| Basic Definitions | |
| imaginary number | $ i=\sqrt{-1} \ $ |
| electrical engineers' imaginary number | $ j=\sqrt{-1}\ $ |
| (info) conjugate of a complex number | if $ z=a+ib $, for $ a,b\in {\mathbb R} $, then $ \bar{z}=a-ib $ |
| (info) magnitude of a complex number | $ \| z \| = z \bar{z} $ |
| (info) magnitude of a complex number | $ \| z \| = \sqrt{\left(Re(z)\right)^2+\left(Im(z)\right)^2} $ |
| (info) magnitude of a complex number | $ \| a+ib \| = \sqrt{a^2+b^2} $, for $ a,b\in {\mathbb R} $ |
| (info) magnitude of a complex number | $ \| r e^{i \theta} \| = r $, for $ r,\theta\in {\mathbb R} $ |
| Complex Number Operations | |
| addition | $ (a+ib)+(c+id)=(a+c) + i (b+d) \ $ |
| multiplication | $ (a+ib) (c+id)=(ac-bd) + i (ad+bc) \ $ |
| division | $ \frac{a+ib} {c+id}=\frac{ac+bd} {c^2+d^2}+ i \frac{bc-ad} {c^2+d^2} \ $ |
| exponentiation | $ i^n =\left\{ \begin{array}{ll}1,& \text{when }n\equiv 0\mod 4 \\ i,& \text{when }n\equiv 1\mod 4 \\-1,& \text{when }n\equiv 2\mod 4 \\-i,& \text{when }n\equiv 3\mod 4 \end{array} \right. \ $ |
| Euler's Formula and Related Equalities (info) | |
| (info) 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} $ |
| Other Formulas | |
| De Moivre's theorem | $ \left(\cos x+i\sin x\right)^n=\cos\left(nx\right)+i\sin\left(nx\right).\, $ |
