5.10. Periodicity and Rotations in the Complex Plane

Try these first

1. 1.

   What is the period of the complex exponential function?

   Hint

   Look at how e^(ix) behaves when x increases by 2π.

2. 2.

   If z = 3e^(iπ/4), what angle does z represent?

   Hint

   Recall the angle θ in polar representation.

3. 3.

   What is the periodic nature of the complex exponential function?

   • π
   • 2π
   • 3π

   Hint

   Consider what periodicity means in terms of intervals.

4. 4.

   True or False: Rotating a complex number increases its modulus.

   • True
   • False

   Hint

   Examine what changes while rotating in the complex plane.

5. 5.

   Show how periodicity affects signal transmission in AC circuits.

   Hint

   Consider how amplitude and phase relate to periodicity.

6. 6.

   Given z = 5e^(iπ/4), calculate z multiplied by e^(iπ/2) and express in rectangular form.

   Hint

   Use conversion from polar to rectangular forms after the rotation.