1. D. R. Wilton ECE Dept. ECE 6382 Introduction to the Theory of Complex Variables 8/24/10

2. Some Applications of Complex Variables

3. Complex Arithmetic and Algebra A complex number z may be thought of simply as an ordered pair of real numbers ( x,y) with rules for addition, multiplication, etc. e z x y r Argand diagram z

4. Complex Arithmetic and Algebra, Cont’d x y z1z1 z2z2 x1x1 y1y1 x2x2 y2y2 z 1 + z 2 -z 2 z 1 - z 2

5. Complex Arithmetic and Algebra, Cont’d z x y r z r z*

6. Euler’s Formula

7. Application to Trigonometric Identities

8. Application to Trigonometric Identities, Cont’d

9. DeMoivre’s Theorem z x y r z

10. Roots of a Complex Number

11. Roots of a Complex Number,cont’d z x y u v w Re Im Cube root of unity