Complex Analysis

Math 312 Spring 1998  MWF 10:30am - 11:25am
Buckmire  Fowler 112

Worksheet �2 (Wednesday 01/14/98)

SUMMARY Graphical Representations of Complex Numbers and Inequalities

READING Churchill & Brown, pages 6-11

Consider 2 complex numbers z_{1} = 3 + 0.5i and z_{2} = 1 + 2i Draw an Argand diagram depicting these two complex numbers in the complex plane. What physical quantity do |z_{1}| and |z_{2}| represent in the diagram?

Then draw in vectors that represent the complex numbers z_{1} + z_{2} and z_{1} − z_{2} Indicate what the value of |z_{1} + z_{2}| is. If I had two points at (3,0.5) and (1,2) what would the distance between these two points be?

\underline{Exercise} Consider the equation |z − 2 + i| = 2. What curve does this equation represent in the complex plane?

Consider the equation 2 = Re ( � z − i) What curve does this equation represent in the complex plane?

Sketch the set of points which solve these equations on the grid provided.

Complex Inequalities

There are a number of interesting inequalities that one can prove using complex variables which have signi�cance in other arenas. The most famous of these is the Triangle Inequality

Triangle Inequality

|z_{1} + z_{2}| ≤ |z_{1}| + |z_{2}|  (1)

Can you express a geometric or vector interpretation of the Triangle Inequality?

This expression (1) is also true in a more general form known as the Cauchy-Schwarz Inequality Cauchy-Schwarz Inequality

|z_{1} + z_{2} + z_{3} + … + z_{n}| ≤ |z_{1}| + |z_{2}| + |z_{3}| + … + |z_{n}|  (2)
|∑_{k=1}^{n}z_{k}| ≤ ∑_{k=1}^{n}|z_{k}|  (3)

The Cauchy-Schwarz Inequality can be proved by mathematical induction.

Some Other Inequalities For the inequalities below, try to come up with a geometric interpretation and prove that they are true.

Re z ≤ | Re z| ≤ |z| and Im z ≤ | Im z| ≤ |z|
|z_{1} + z_{2}| ≥ ||z_{1}| − |z_{2}||
|z_{1} − z_{2}| ≤ |z_{1}| + |z_{2}|
|z_{1} − z_{2}| ≥ ||z_{1}| − |z_{2}||