A First Course in Electrical and Computer Engineering by Louis Scharf - HTML preview

/ Home / Engineering (Academic) / A First Course in Electrical and Computer Engineering

PLEASE NOTE: This is an HTML preview only and some elements such as links or page numbers may be incorrect.
Download the book in PDF, ePub, Kindle for a complete version.

Chapter 2. The Functions e^x and e^jΘ

2.1. The Functions e^x and e^jθ: Introduction^*

This module is part of the collection, A First Course in Electrical and Computer Engineering. The LaTeX source files for this collection were created using an optical character recognition technology, and because of this process there may be more errors than usual. Please contact us if you discover any errors.

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 everyday life and gives e^x some real meaning. The function e^jθ, 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 to derive 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 θ and cos θ . The numerical experiment in "Numerical Experiment (Approximating e^jθ illustrates, geometrically and algebraically, how approximations to e^jθ converge.

“Second-Order Differential and Difference Equations” is a little demanding for freshmen, but we give it a once-over-lightly to illustrate the power of quadratic equations and the functions e^x and e^jθ. 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^jθ. This study clarifies our definition of e^jθ from "Complex Numbers" and leads us to an investigation of sequences and series. We use the function e^jθ to derive the Euler and De Moivre identities and to produce a number of important trigonometric 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.

2.2. The Functions e^x and e^jθ: The Function e^x^*

Many of you know the number e as the base of the natural logarithm, which has the value 2.718281828459045. . . . What you may not know is that this number is actually defined as the limit of a sequence of approximating numbers. That is,

(2.1)

(2.2)

This means, simply, that the sequence of numbers , . . . , gets arbitrarily close to 2.718281828459045. . . . But why should such a sequence of numbers be so important? In the next several paragraphs we answer this question.

Exercise 1.

(MATLAB) Write a MATLAB program to evaluate the expression for n=1,2,4,8,16,32,64 to show that f_n≈e for large n.

Derivatives and the Number e. The number arises in the study of derivatives in the following way. Consider the function

(2.3)

and ask yourself when the derivative of f(x) equals f(x). The function f(x) is plotted in Figure 2.1 for a>1. The slope of the function at point x is

(2.4)

Figure 2.1.

The Function f(x)=a^x

If there is a special value for a such that

(2.5)

then would equal f(x). We call this value of a the special (or exceptional) number e and write

(2.6)

The number e would then be e=f(1). Let's write our condition that converges to 1 as

(2.7)

or as

(2.8)e≅(1+Δx)^1/Δx.

Our definition of amounts to defining and allowing n→∞ in order to make Δx→0. With this definition for e, it is clear that the function e^x is defined to be (e)^x :

(2.9)

By letting we can write this definition in the more familiar form

(2.10)

This is our fundamental definition for the function e^x. When evaluated at x=1, it produces the definition of e given in Equation 2.1.

The derivative of e^x is, of course,

(2.11)

This means that Taylor's theorem^[3] may be used to find another characterization for e^x :

(2.12)

When this series expansion for e^x is evaluated at x=1, it produces the following series for e:

(2.13)

In this formula, n! is the product n(n–1)(n–2)⋯( 2 ) 1 . Read n! as " n factorial.”

Exercise 2.

(MATLAB) Write a MATLAB program to evaluate the sum

(2.14)

for N=1,2,4,8,16,32,64 to show that S_N≅e for large N . Compare S₆₄ with f₆₄ from Exercise 1.. Which approximation do you prefer?

Compound Interest and the Function . There is an example from your everyday life that shows even more dramatically how the function e^x arises. Suppose you invest V₀ dollars in a savings account that offers 100 x % annual interest. (When x=0.01, this is 1%; when x=0.10, this is 10% interest.) If interest is compounded only once per year, you have the simple interest formula for V₁, the value of your savings account after 1 compound (in this case, 1 year):

V₁=(1+x)V₀. This result is illustrated in the block diagram of Figure 2.2(a). In this diagram, your input fortune V₀ is processed by the “interest block” to produce your output fortune V₁. If interest is compounded monthly, then the annual interest is divided into 12 equal parts and applied 12 times. The compounding formula for V₁₂, the value of your savings after 12 compounds (also 1 year) is

(2.15)

This result is illustrated in Figure 2.2b. Can you read the block diagram? The general formula for the value of an account that is compounded n times per year is

(2.16)

V_n is the value of your account after n compounds in a year, when the annual interest rate is 100x%.

(a)

(b)

Figure 2.2.

Block Diagram for Interest Computations; (a) Simple Annual Interest, and (b) Monthly Compounding

Exercise 3.

Verify in Equation 2.16 that a recursion is at work that terminates at V_n. That is, show that for i=0,1,...,n–1 produces the result .

Bankers have discovered the (apparent) appeal of infinite, or continuous, compounding:

(2.17)

We know that this is just

(2.18)V_∞=e^xV₀.

So, when deciding between 100x₁ % interest compounded daily and 100x₂% interest compounded continuously, we need only compare

(2.19)

We suggest that daily compounding is about as good as continuous compounding. What do you think? How about monthly compounding?

Exercise 4.

(MATLAB) Write a MATLAB program to compute and plot simple interest, monthly interest, daily interest, and continuous interest versus interest rate 100x. Use the curves to develop a strategy for saving money.

2.3. The Functions e^x and e^jθ: The Function e^jθ and the Unit Circle^*

Let's try to extend our definitions of the function e^x to the argument x = j Θ . Then e^jΘ is the function

(2.20)

The complex number is illustrated in Figure 2.3. The radius to the point is and the angle is This means that the n^th power of has radius and angle (Recall our study of powers of z.) Therefore the complex number may be written as