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.