In this second short week, we will discuss briefly descriptions and properties of sets in the complex plane and then move on to concentrate on complex-valued functions of a complex variable. We'll begin by examining the complex squaring function.

We will not meet in the lab this week. Both class sessions will be in the classroom.

Assignment for Friday, January 22.

Read Section 17.2. Put most of your effort into pages 891-894. Look over the rest of the section. We will come back to these topological definitions as we need them.

1. Finish the worksheet we started in class on Wednesday.

2. Write out and turn in solutions to the following exercises in Section 17.2. Think about these both geometrically (interpreting the magnitude as a distance) and algebraically -- let z = x + iy and solve the resulting simultaneous equations or inequalities. Make a sketch for each solution set.

• 1, 2, 3, 5, 6,
• 9, 10, 11, 17

Assignment for Monday, January 25.

1. Finish the worksheet started in class on Friday. The question is repeated here

   Let D be the closed unit disk with center at 0, i.e., D = {z: | z | is less than or equal to 1}. For each of the following differentiable functions f, find the maximum value of | f(z) | for z in D and locate each value of z where the maximum is assumed.

   (a) f(z) = z² + i

   (b) 1/(2 + z)

2. Find all complex numbers z such that z³ = 1.
3. Find all complex numbers z such that z⁴ = 1.

4. Find all complex numbers z such that z² = 3 + 4i.

5. Find all complex numbers z such that z³ = 3 + 4i.

6. Find the real and imaginary parts of f if
   f(z) = (3 + 4z)²

7. Find the real and imaginary parts of f if
   f(z) = 1/(2 - iz)

Last modified: January 21, 1999