Complex Number Identities and Formulas
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| 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 | $ \text{if}\ z=a+ib,\ \text{for}\ a,\ b \in {\mathbb R},\ \text{then} \ \bar{z}=a-ib $ |
| (info) magnitude of a complex number | $ \| z \| = \sqrt{ 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},\ \text{for}\ a,b\in {\mathbb R} $ |
| (info) magnitude of a complex number | $ \| r e^{i \theta} \| = r,\ \text{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) \ $ |
| multiplication in polar form | $ \left( r_1 (\cos \theta_1 + i \sin \theta_1) \right) \left( r_2 (\cos \theta_2 + i \sin \theta_2) \right)= r_1 r_2 \left( \cos (\theta_1+\theta_2)+i \sin (\theta_1-\theta_2) \right)\ $ |
| division | $ \frac{a+ib} {c+id}=\frac{ac+bd} {c^2+d^2}+ i \frac{bc-ad} {c^2+d^2} \ $ |
| division in polar form | $ \frac{ r_1 (\cos \theta_1 + i \sin \theta_1)}{ r_2 (\cos \theta_2 + i \sin \theta_2) }= \frac{r_1}{ r_2} \left( \cos (\theta_1-\theta_2)+i \sin (\theta_1+\theta_2) \right)\ $ |
| 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).\, $ |
| Root of a complex number | $ \left( r (\cos x+i\sin x) \right)^{\frac{1}{n}}=r^{\frac{1}{n}} \cos\left(\frac{x+2 k \pi}{n}\right) +i\sin\left(\frac{x+2 k \pi}{n} \right), k=0,1,\ldots, n-1.\, $ |
