Complex Numbers what does the function f(z) = 1/z do to the region S which is all points outside...


Complex Numbers

What does the function {eq}f(z) = 1/z {/eq} do to the region {eq}S {/eq} which is all points outside of the circle of radius 4 centered at the origin?

Writing the Polar and Exponential Form of a Complex Number


Consider a complex number of the form :

$$z = x + y \ i \\ $$

1. Its modulus (distance from the origin) is given by :

$$r = \sqrt{x^2 + y^2 } \\ $$

2. Its argument is given by the formula :

$$\tan{(arg(z))} = \frac{y}{x} \\ $$

Where, arg(z) is expressed in radians. We need to pick a suitable value of arg(z) such that :

$$-\pi \leq arg(z) \ \leq \pi \\ $$

3. Its polar form is given by

$$z = r\angle \theta \\ $$

{eq}\theta = arg(z) \\ {/eq}

4. Its exponential form is given by :

$$z = r e^{i\theta} \\ $$

{eq}\theta = arg(z) \\ {/eq}

Answer and Explanation:

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The equation of a circle in complex plane centered at origin and of radius 4 is given as :

$$|Z| = 4 \\ $$

Therefore, all the points...

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Learn more about this topic:

Complex Numbers in Polar Form: Process & Examples


Chapter 24 / Lesson 2

After watching this video lesson, you will be able to convert complex numbers from rectangular form to polar form easily by following the formulas you will see here. You will also learn how to find the power of a complex number.

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