Question

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

Answer 1

Answer on Question #58320 – Math – Complex Analysis

Question

The polar form of the complex number

$$z = x + iy$$

is given by

Solution

The algebraic form of the complex number $z$ is $z = x + iy$.

The polar form of the complex number $z$ is given by

$$z = A(\cos\varphi + \sin\varphi).$$

Relationship between the polar and algebraic forms of a complex number:

$$A = \sqrt{x^2 + y^2},$$ $$\varphi = \begin{cases} \arccos \dfrac{x}{\sqrt{x^2 + y^2}}, & y \geq 0, \sqrt{x^2 + y^2 > 0}, \\ -\arccos \dfrac{x}{\sqrt{x^2 + y^2}}, & y < 0, \sqrt{x^2 + y^2} > 0, \\ \text{undefined}, & \sqrt{x^2 + y^2} = 0; \end{cases}$$

or

$$\varphi = \begin{cases} \arctan\dfrac{y}{x}, & x > 0, \\ +\dfrac{\pi}{2}, & x = 0, y > 0, \\ -\dfrac{\pi}{2}, & x = 0, \quad y < 0, \\ \arctan\dfrac{y}{x} + \pi, x < 0, y \geq 0, \\ \arctan\dfrac{y}{x} - \pi, x < 0, y < 0. \end{cases}$$

Relationship between the algebraic and polar forms of a complex number:

$$x = A\cos\varphi, y = A\sin\varphi.$$

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