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Complex numbers basic formulas

Advanced math

Algebraic form of complex number, where [tex]i^2=-1[/tex]. $${z=x+iy}$$
Trigonometric form of complex number, where [tex]r=\sqrt{x^2+y^2}[/tex], [tex]{\cos(\varphi)=\frac{x}{r}}[/tex] and [tex]\sin(\varphi)=\frac{y}{r}[/tex]. $${z=r(\cos(\varphi)+i\sin(\varphi))}$$
Euler's formula [tex]z=r \cdot e^{i\varphi}[/tex], where [tex]e^{i\varphi}=\cos(\varphi)+i\sin(\varphi)[/tex].

Multiplication of complex numbers in a trigonometric form $${z_1 \cdot z_2=r_1 \cdot r_2(\cos(\varphi_1+\varphi_2)+i\sin(\varphi_1+\varphi_2))}$$
Division of complex numbers in a trigonometric form $${\frac{z_1}{z_2}=\frac{r_1}{r_2}(\cos(\varphi_1-\varphi_2)+i\sin(\varphi_1-\varphi_2))}$$
Power of complex numbers in a trigonometric form $${z^n=r^n(\cos{n\varphi}+i\sin{n\varphi})}$$
Nth root of complex numbers in a trigonometric form $${\sqrt[n]{z}=\sqrt[n]{r} (\cos \frac{\varphi+2\pi k}{n}+i\sin \frac{\varphi+2\pi k}{n})}$$

Last time edited 2017-11-06

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