The functions e^x and e^jθ: introduction

Notes to teachers and students

It is essential to write out, term-by-term, every sequence and sum in this chapter. This demystifies the seemingly mysterious notation. The example on compound interest shows the value of limiting arguments in everydaylife and gives $e^x$ some real meaning. The function $e^{j\theta }$ , covered in the section "The Function of e ^jθ and the Unit Circle and "Numerical Experiment (Approximating e ^jθ , must be understood by all students before proceeding to "Phasors" . The Euler and De Moivre identities provide every tool that students need toderive trigonometric formulas. The properties of roots of unity are invaluable for the study of phasors in "Phasors" .

The MATLAB programs in this chapter are used to illustrate sequences and series and to explore approximations to $sin\theta$ and $cos\theta$ . The numerical experiment in "Numerical Experiment (Approximating e ^jθ illustrates, geometrically and algebraically, how approximations to $e^{j\theta }$ converge.

“Second-Order Differential and Difference Equations” is a little demanding for freshmen, but we give it a once-over-lightly to illustratethe power of quadratic equations and the functions e ^x and $e^{j\theta }$ . This section also gives a sneak preview of more advanced courses in circuits and systems.

Introduction

It is probably not too strong a statement to say that the function $e^x$ is the most important function in engineering and applied science. In this chapter we study the function $e^x$ and extend its definition to the function $e^{j\theta }$ . This study clarifies our definition of $e^{j\theta }$ from "Complex Numbers" and leads us to an investigation of sequences and series. We use the function $e^{j\theta }$ to derive the Euler and De Moivre identities and to produce a number of importanttrigonometric identities. We define the complex roots of unity and study their partial sums. The results of this chapter will be used in "Phasors" when we study the phasor representation of sinusoidal signals.