| 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} $ |
| 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).\, $ |
