# Doc/Elementary Analysis

## Elementary Analysis

by Kenneth E. Iverson

published by APL Press, 1976
(preliminary edition entitled *Elementary Functions* published by IBM, 1974)

## Contents

- 1.1 Notation
- 1.2 Expressions, Equations, and Proofs
- 1.3 Formal Function Definition
- 1.4 Use of an APL Computer
- 1.5 Polynomials
- 1.6 Approximation of a Polynomial
- 1.7 Sum, Product, and Quotient of Polynomials

**Chapter 2 Sum Formulas**

- 2.1 Graphs and the Properties of Functions
- 2.2 Power Functions
- 2.3 The Sum of Functions
- 2.4 Multiplication by a Constant
- 2.5 Function Tables
- 2.6 Maps

**Chapter 3 The Derivative and Anti-Derivative**

- 3.1 Reading Graphs
- 3.2 The Tangent Slope or Derivative
- 3.3 Derivatives of Various Functions
- 3.4 Derivative of a Polynomial
- 3.5 Some Applications of Derivatives
- 3.6 Anti-Derivatives
- 3.7 Anti-Derivative of a Polynomial
- 3.8 Applications of Anti-Derivatives
- 3.9 Taylor's Theorem: Sum Formulas From Derivatives

**Chapter 4 The Exponential Family**

- 4.1 Growth and Decay Functions
- 4.2 Graphs and Tables of the Exponential Function
- 4.3 Sum Formula for the Exponential
- 4.4 Applications of the Exponential
- 4.5 Parity of Functions
- 4.6 The Hyperbolic Functions
- 4.7 Circular Functions
- 4.8 Periodicity
- 4.9 Applications of the Circular and Hyperbolic Functions
- 4.10 Notation
- 4.11 Parametric Equations

**Chapter 5 The Circular Functions**

- 5.1 Sine and Cosine
- 5.2 Trigonometry
- 5.3 Limiting Values of
`(sin s)÷s`and`(1-cos s)÷s` - 5.4 Sum Formulas
- 5.5 Derivatives
- 5.6 Polynomials for Sine and Cosine
- 5.7 General Triangles
- 5.8 Notation for Matrices
- 5.9 Coordinate Geometry
- 5.10 An Electrical Application
- 5.11 The Pythagorean Functions

**Chapter 6 The Inverse and Reciprocal Functions**

- 6.1 Rules for Obtaining Derivatives of Composite Functions
- 6.2 The Derivation of Derivative Rules
- 6.3 Inverse Circular Functions
- 6.4 The Logarithm
- 6.5 A Function to Produce Derivatives

**Chapter 7 The Dyadic Logarithm**

- 7.1 The Dyadic Power Function
- 7.2 The Dyadic Logarithm
- 7.3 Properties of
`*`and`⍟` - 7.4 Tables of Logarithms
- 7.5 Applications of Base-10 Logarithms
- 7.6 The Pythagorean Functions

**Chapter 8 Complex Numbers**

- 8.1 The Number System
- 8.2 Functions on Complex Arguments
- 8.3 The Exponential Family

**Chapter 9 Conic Sections**

- 9.1 The Circle
- 9.2 The Ellipse
- 9.3 The Hyperbola
- 9.4 The Parabola

**Chapter 10 Formal Function Definition**

- 10.1 Variables in a Function Definition
- 10.2 Recursive Definition
- 10.3 Functions for Handling Polynomials
- 10.4 The Function
*DEF*

**Exercises**

**Index**

**References**

## Introduction

Elementary algebra is concerned with certain simple functions
such as addition and multiplication, with the development of
notation for their use, and with the treatment of properties
of these functions (such as commutativity) and of relations
among them. This text continues the treatment of functions,
but with more emphasis on their properties and their applications,
and introduces the so-called *elementary functions* which form
the basis for most work in the applications of mathematics.
The elementary functions include the logarithm, antilogarithm,
and the circular (also called *trigonometric*) and the
hyperbolic functions.

The present treatment of elementary functions differs from most traditional texts in three major respects. Firstly, each function introduced is defined explicitly in terms of known functions so that means are provided for the evaluation of each function introduced. This contrasts with the common definition of logarithms and trigonometric functions in terms of tables alone.

Secondly, the treatment of each function is based largely
on the *slope* of the function, and upon the related
solution of an addition formula for the function.

Thirdly, this text is based on the notation and the treatment of algebra presented in Reference 1. The text includes a summary and review of the notation; most of this review appears in this chapter and the rest is interspersed as needed. Any reader familiar with the material of algebra but not with the notation should find the text and exercises of this chapter an adequate introduction.

The simplicity and precision of the notation permits an algorithmic treatment of the material. In particular, every expression in the text can be executed directly by simply typing it on an appropriate computer terminal. The use of a computer is not essential, but it can be useful in exercises and other explorations of the functions treated. Suggestions for the use of an APL computer may be found in Section 1.4, in Chapter 10, and in Appendix C of Reference 1. Further information about the computer may be found in References 2-4.

The serious student should treat the exercises as an integral
part of the text. The point at which each group of exercises
can be attempted is indicated by a marginal note consisting
of a domino (`⌹`) followed by the number of the first exercise
in the group. The groups are separated by horizontal lines.

The inclusion of too many simple exercises may bore the quick student, but their exclusion may leave unbridgeable gaps in the experience of some. The student should therefore learn to use discretion in the doing of exercises, ranging ahead and skipping detail, but being prepared to return to do earlier exercises whenever unintelligible difficulties arise in later ones. The most serious difficulty most students find with this approach is psychological; one must learn to treat exercises as a potential source of light and delight rather than as a capriciously imposed drudgery.