1. ECE 6382 Notes 1 Introduction to Complex Variables Fall 2017
David R. Jackson
Notes 1
Introduction to Complex Variables
Notes are adapted from D. R. Wilton, Dept. of ECE

2. Some Applications of Complex Variables
Phasor-domain analysis in physics and engineering
Laplace and Fourier transforms
Evaluation of integrals
Asymptotics (method of steepest descent)
Conformal Mapping (solution of Laplace’s equation)
Radiation physics (branch cuts, poles)

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. z x y r
Argand diagram
Note: In Euler's formula, the angle  must be in radians.
Note: Usually we will use i to denote the square-root of -1.
However, we will often switch to using j when we are doing an engineering example.

4. Complex Arithmetic and Algebra (cont.)y z1 z2 x1 y1 x2 y2
z1+ z2
-z2
z1- z2
Division is kind of messy in rectangular coordinates!

5. Complex Arithmetic and Algebra (cont.)
Geometrical interpretation of addition and subtraction of complex numbers:
Geometrically, this works the same way and adding and subtracting two-dimensional vectors. x y z1 z2 x1 y1 x2 y2
z1+ z2
-z2
z1- z2
Note: We can multiply and divide complex numbers. We cannot divide two-dimensional vectors. We can multiply two-dimensional vectors in different ways (dot product and cross product).

6. Complex Arithmetic and Algebra (cont.)z x y r z*

7. Euler’s Formula

8. Application to Trigonometric Identities

9. Application to Trigonometric Identities (cont.)

10. DeMoivre’s Theorem

11. Roots of a Complex Number

12. Roots of a Complex Number (cont.)z x y u v w Re Im
Cube root
of unity