Complex Exercises

Michael Fowler Link to Lecture PDF

1. Show where in the complex plane are: 1, i, and write all these numbers in the form

2. State the rule for multiplying two complex numbers of the form , and from that figure out the inverse of a complex number: that is, express as

3. Find how to invert a number in the other notation: if find a, b in terms of x, y.

Hint: it helps to multiply

4. Show on a diagram where in the complex plane is a cube root of -1, we’ll call it How many cube roots does -1 have? Show all possibilities on the diagram. Next, what about cube roots of 1? Show them on another figure. (Note: is commonly used for a cube root of -1. we also use it, of course, for angular frequency. Take care not to confuse the two.)

5. Draw a complex number z as a vector (pointing from the origin to z), then draw on the same diagram as vectors iz, z/i, (being the cube root of -1.)

6. Using , from , deduce the formulas for

7. Suppose the point z moves in the complex plane is such a way that at time t , where A is real and . Where is z at t = 0? Where at t = 1 second? Where at t = 0.5 seconds? Where at t = 0.25 seconds? Describe how z moves as time progresses.

How would your answer change if A were pure imaginary instead of real?

8. Consider again , . Differentiate both sides to find an expression for the velocity as the point moves along its path. How does the velocity vector relate to the position vector? Next, find by differentiating again the acceleration vector, and comment on its direction.

9. State briefly how z behaves in time if for real . How would this behavior change if had a small imaginary part, , where is small? Sketch how z would move in the complex plane, both for positive and negative.

10. Consider the quadratic equation . For b = 1, both roots equal 1. Sketch (in the complex plane) how the larger root moves as b varies from 1.2 down through 1 to 0.8.

When you’ve done that, do the same for the other root, preferably in a different color.