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

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)

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.

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!

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).

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

Euler’s Formula

Application to Trigonometric Identities

Application to Trigonometric Identities (cont.)

DeMoivre’s Theorem

Roots of a Complex Number

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