Math formulas: Complex numbers

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Definitions:

A complex number is written as $a + b\,i$ where $a$ and $b$ are real numbers an $i$, called the imaginary unit, has the property that $i^2 = -1$.

The complex numbers $z = a + b\,i$ and $\overline{z} = a - b\,i$ are called complex conjugate of each other.

Formulas:

Equality of complex numbers

$$a + b\,i = c + d\,i \Longleftrightarrow a = c ~~and~~ b = d$$

Addition of complex numbers

$$(a + b\,i) + (c + d\,i) = (a + c) + (b + d)\,i$$

Subtraction of complex numbers

$$(a + b\,i) - (c + d\,i) = (a - c) + (b - d)\,i$$

Multiplication of complex numbers

$$(a + b\,i)\cdot(c + d\,i) = (ac - bd) + (ad + bc)\,i$$

Division of complex numbers

$$\frac{a + b\,i}{c + d\,i} = \frac{a + b\,i}{c + d\,i}\cdot\frac{c - d\,i}{c - d\,i} = \frac{ac + bd}{c^2 + d^2} + \frac{bc - ad}{c^2 + d^2} \, i$$

Polar form of complex numbers

$$a + b\,i = r\cdot(\cos\theta + i\,\sin\theta)$$

Multiplication and division of complex numbers in polar form

$$\left[r_1\left(\cos\theta_1 + i \cdot \sin\theta_1\right)\right] \cdot \left[r_2\left(\cos\theta_2 + i \cdot \sin\theta_2\right)\right] = r_1 \cdot r_2 \left[ \cos\left(\theta_1+\theta_2\right) + i \cdot \sin\left(\theta_1+\theta_2\right) \right]$$

$$\frac{r_1\left(\cos\theta_1 + i\,\sin\theta_1\right)}{r_2\left(\cos\theta_2 + i\,\sin\theta_2\right)}= \frac{r_1}{r_2} \left[\cos\left(\theta_1-\theta_2\right) + i \cdot \sin\left(\theta_1-\theta_2\right)\right]$$

De Moivre's theorem

$$\left[r \left( \cos\theta + i\,\sin\theta \right) \right]^n = r^n \left( \cos ( n\theta) + i\,\sin (n\theta) \right)$$

Roots of complex numbers

$$\left[r \left( \cos\theta + i\,\sin\theta \right) \right]^{1/n} = r^{1/n} \left( \cos \frac{\theta + 2k\pi}{n} + i\,\sin \frac{\theta + 2k\pi}{n} \right) ~~ k=0,1,\dots, n-1$$