Complex Variable

Math 3146 Problem Set 1, due on Feb. 1, Monday

Exercise 1. Simplify the expressions

1 + i

1� i , �1 + 2i 3� i ,

into the form of a+ bi, where a and b are real numbers.

Exercise 2. Show that

|3 + z̄ + z3|  5 for all |z|  1.

Exercise 3 ([BC14, Exercise 5.3]). Show that

Re(z1 + z2)

|z3 + z4|  |z1|+ |z2|��|z3|� |z4|

�� whenever |z3| 6= |z4|.

Exercise 4. Describe the geometric meaning of each set of points below,

{z 2 C; |z � 1 + 3i| = 4}, {z 2 C; |z + 5i| < 6},

by using either words or sketching a picture.

Exercise 5. Find all the complex roots of the equations

z 2 + z + 1 = 0, z4 + 3 = 0.

Exercise 6. Express the complex numbers

�1 + i, 3 + 4i

into the polar form rei✓, where r > 0 and �⇡ < ✓  ⇡.

Note: [BC14] stands for the textbook, Complex Variables and Applications, by J. W. Brown and R. V. Churchill, McGraw-Hill Education, 2014.