Question

The polar form of the complex number z = x + iy is given by

Answer 1

Answer on Question #58283 – Math – Complex Analysis

The polar form of the complex number

$$z = x + iy$$

is given by

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Solution:

The polar form of the complex number $z$ is

$$z = r(\cos \varphi + i \sin \varphi)$$

where

$r = |z| = \sqrt{x^2 + y^2}$ is the absolute value of the complex number $z$ and

$\varphi = \arg(z)$ is the argument of the complex number $z$:

$$\varphi = \arg(x + iy) = \begin{cases} \arctan \left(\frac{y}{x}\right) & \text{if } x > 0 \\ \pi + \arctan \left(\frac{y}{x}\right) & \text{if } x < 0 \text{ and } y \geq 0 \\ -\pi + \arctan \left(\frac{y}{x}\right) & \text{if } x < 0 \text{ and } y < 0 \\ \frac{\pi}{2} & \text{if } x = 0 \text{ and } y > 0 \\ -\frac{\pi}{2} & \text{if } x = 0 \text{ and } y < 0 \\ \text{indeterminate} & \text{if } x = 0 \text{ and } y = 0 \end{cases}$$

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Answer: $z = \sqrt{x^2 + y^2}(\cos[\arg(x + iy)] + i \sin[\arg(x + iy)])$