Complex Number Formula

Complex number Formula

A complex number is one of the form of a + ib, where a and b are real number and $i = \sqrt{-1}$ . A is called real part of the complex number and b is called imaginary part of the complex number.

1.De moivre’s Theorem

De Moivre’s Theorem is a relatively simple formula for calculating powers of complex numbers.

If “Z” is a complex number and:

$Z = r( \cos \theta + i \sin \theta)$ as expressed in Polar form of complex numbers.

where,

$Z = a + ib \\ \text{ and } a = Re(Z) = r \cos \theta \\ \text{ and } b = Im(Z) = r \sin \theta$

Then De Moivre’s Theorem states that:

The complex Number “Z” raised to an integer power “n” is given by:

$Z^n = r^n[ \cos ( n \times \theta ) + i \sin (n \times \theta)]$

Where ,

$\theta = \arg(z) = \arctan(\frac{b}{a}) \text{ and }r = |Z| = \sqrt{a^2+b^2}$

2.Square root of complex number
$x^2 = \dfrac{\sqrt{a^2 + b^2} + a}{2}$ and $y^2 = \dfrac{\sqrt{a^2 + b^2}- a}{2}$

For $(x + iy)^2 = a + ib$ .

3.Cube root of unity.
Let For $z^3 = 1$ the requrd cube root of unity are
$1, \dfrac{-1 + \sqrt{3i}}{2} \& \dfrac{-1- \sqrt{3i}}{2}$ or $1, \omega, \omega^2$ $(\omega$ = omega)

Note:

i. $1 + \omega + \omega^2 = 0 \\[3mm]$

ii. $\omega.\omega^2 = 1$

iii. $\omega^3n = 1$ for any integer n.

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