Schaum's Outline of Advanced Calculus, Third Edition

SCHAUM’S outlines

Advanced Calculus

Third Edition

Robert Wrede, Ph.D.

Professor Emeritus of Mathematics

San Jose State University

Murray R. Spiegel, Ph.D.

Former Professor and Chairman of Mathematics

Rensselaer Polytechnic Institute

Hartford Graduate Center

Schaum’s Outline Series

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Preface to the

Third Edition

The many problems and solutions provided by the late Professor Spiegel remain invaluable to students as they seek to master the intricacies of the calculus and related fields of mathematics. These remain an integral part of this manuscript. In this third edition, clarifications have been provided. In addition, the continuation of the interrelationships and the significance of concepts, begun in the second edition, have been extended.

Preface to the

Second Edition

A key ingredient in learning mathematics is problem solving. This is the strength, and no doubt the reason for the longevity of Professor Spiegel’s advanced calculus. His collection of solved and unsolved problems remains a part of this second edition.

Advanced calculus is not a single theory. However, the various sub-theories, including vector analysis, infinite series, and special functions, have in common a dependency on the fundamental notions of the calculus. An important objective of this second edition has been to modernize terminology and concepts, so that the interrelationships become clearer. For example, in keeping with present usage functions of a real variable are automatically single valued; differentials are defined as linear functions, and the universal character of vector notation and theory are given greater emphasis. Further explanations have been included and, on occasion, the appropriate terminology to support them.

The order of chapters is modestly rearranged to provide what may be a more logical structure.

A brief introduction is provided for most chapters. Occasionally, a historical note is included; however, for the most part the purpose of the introductions is to orient the reader to the content of the chapters.

I thank the staff of McGraw-Hill. Former editor, Glenn Mott, suggested that I take on the project. Peter McCurdy guided me in the process. Barbara Gilson, Jennifer Chong, and Elizabeth Shannon made valuable contributions to the finished product. Joanne Slike and Maureen Walker accomplished the very difficult task of combining the old with the new and, in the process, corrected my errors. The reviewer, Glenn Ledder, was especially helpful in the choice of material and with comments on various topics.

ROBERT C. WREDE

Contents

Chapter 1 NUMBERS

Sets.

Real Numbers.

Decimal Representation of Real Numbers.

Geometric Representation of Real Numbers.

Operations with Real Numbers.

Inequalities.

Absolute Value of Real Numbers.

Exponents and Roots.

Logarithms.

Axiomatic Foundations of the Real Number System.

Point Sets, Intervals.

Countability.

Neighborhoods.

Limit Points.

Bounds.

Bolzano-Weierstrass Theorem.

Algebraic and Transcendental Numbers.

The Complex Number System.

Polar Form of Complex Numbers.

Mathematical Induction.

Chapter 2 SEQUENCES

Definition of a Sequence.

Limit of a Sequence.

Theorems on Limits of Sequences.

Infinity.

Bounded, Monotonic Sequences.

Least Upper Bound and Greatest Lower Bound of a Sequence.

Limit Superior, Limit Inferior.

Nested Intervals.

Cauchy’s Convergence Criterion.

Infinite Series.

Chapter 3 FUNCTIONS, LIMITS, AND CONTINUITY

Functions.

Graph of a Function.

Bounded Functions.

Montonic Functions.

Inverse Functions, Principal Values.

Maxima and Minima.

Types of Functions.

Transcendental Functions.

Limits of Functions.

Right- and Left-Hand Limits.

Theorems on Limits.

Infinity.

Special Limits.

Continuity.

Right- and Left-Hand Continuity.

Continuity in an Interval.

Theorems on Continuity.

Piecewise Continuity.

Uniform Continuity.

Chapter 4 DERIVATIVES

The Concept and Definition of a Derivative.

Right- and Left-Hand Derivatives.

Differentiability in an Interval.

Piecewise Differentiability.

Differentials.

The Differentiation of Composite Functions.

Implicit Differentiation.

Rules for Differentiation.

Derivatives of Elementary Functions.

Higher-Order Derivatives.

Mean Value Theorems.

L’Hospital’s Rules.

Applications.

Chapter 5 INTEGRALS

Introduction of the Definite Integral.

Measure Zero.

Properties of Definite Integrals.

Mean Value Theorems for Integrals.

Connecting Integral and Differential Calculus.

The Fundamental Theorem of the Calculus.

Generalization of the Limits of Integration.

Change of Variable of Integration.

Integrals of Elementary Functions.

Special Methods of Integration.

Improper Integrals.

Numerical Methods for Evaluating Definite Integrals.

Applications.

Arc Length.

Area.

Volumes of Revolution.

Chapter 6 PARTIAL DERIVATIVES

Functions of Two or More Variables.

Neighborhoods.

Regions.

Limits.

Iterated Limits.

Continuity.

Uniform Continuity.

Partial Derivatives.

Higher-Order Partial Derivatives.

Differentials.

Theorems on Differentials.

Differentiation of Composite Functions.

Euler’s Theorem on Homogeneous Functions.

Implicit Functions.

Jacobians.

Partial Derivatives Using Jacobians.

Theorems on Jacobians.

Transformations.

Curvilinear Coordinates.

Mean Value Theorems.

Chapter 7 VECTORS

Vectors.

Geometric Properties of Vectors.

Algebraic Properties of Vectors.

Linear Independence and Linear Dependence of a Set of Vectors.

Unit Vectors.

Rectangular (Orthogonal) Unit Vectors.

Components of a Vector.

Dot, Scalar, or Inner Product.

Cross or Vector Product.

Triple Products.

Axiomatic Approach To Vector Analysis.

Vector Functions.

Limits, Continuity, and Derivatives of Vector Functions.

Geometric Interpretation of a Vector Derivative.

Gradient, Divergence, and Curl.

Formulas Involving ∇.

Vector Interpretation of Jacobians and Orthogonal Curvilinear Coordinates.

Gradient, Divergence, Curl, and Laplacian in Orthogonal Curvilinear Coordinates.

Special Curvilinear Coordinates.

Chapter 8 APPLICATIONS OF PARTIAL DERIVATIVES

Applications To Geometry.

Directional Derivatives.

Differentiation Under the Integral Sign.

Integration Under the Integral Sign.

Maxima and Minima.

Method of Lagrange Multipliers for Maxima and Minima.

Applications To Errors.

Chapter 9 MULTIPLE INTEGRALS

Double Integrals.

Iterated Integrals.

Triple Integrals.

Transformations of Multiple Integrals.

The Differential Element of Area in Polar Coordinates, Differential Elements of Area in Cylindrical and Spherical Coordinates.

Chapter 10 LINE INTEGRALS, SURFACE INTEGRALS, AND INTEGRAL THEOREMS

Line Integrals.

Evaluation of Line Integrals for Plane Curves.

Properties of Line Integrals Expressed for Plane Curves.

Simple Closed Curves, Simply and Multiply Connected Regions.

Green’s Theorem in the Plane.

Conditions for a Line Integral To Be Independent of the Path.

Surface Integrals.

The Divergence Theorem.

Stokes’s Theorem.

Chapter 11 INFINITE SERIES

Definitions of Infinite Series and Their Convergence and Divergence.

Fundamental Facts Concerning Infinite Series.

Special Series.

Tests for Convergence and Divergence of Series of Constants.

Theorems on Absolutely Convergent Series.

Infinite Sequences and Series of Functions, Uniform Convergence.

Special Tests for Uniform Convergence of Series.

Theorems on Uniformly Convergent Series.

Power Series.

Theorems on Power Series.

Operations with Power Series.

Expansion of Functions in Power Series.

Taylor’s Theorem.

Some Important Power Series.

Special Topics.

Taylor’s Theorem (For Two Variables).

Chapter 12 IMPROPER INTEGRALS

Definition of an Improper Integral.

Improper Integrals of the First Kind (Unbounded Intervals).

Convergence or Divergence of Improper Integrals of the First Kind.

Special Improper Integers of the First Kind.

Convergence Tests for Improper Integrals of the First Kind.

Improper Integrals of the Second Kind.

Cauchy Principal Value.

Special Improper Integrals of the Second Kind.

Convergence Tests for Improper Integrals of the Second Kind.

Improper Integrals of the Third Kind.

Improper Integrals Containing a Parameter, Uniform Convergence.

Special Tests for Uniform Convergence of Integrals.

Theorems on Uniformly Convergent Integrals.

Evaluation of Definite Integrals.

Laplace Transforms.

Linearity.

Convergence.

Application.

Improper Multiple Integrals.

Chapter 13 FOURIER SERIES

Periodic Functions.

Fourier Series.

Orthogonality Conditions for the Sine and Cosine Functions.

Dirichlet Conditions.

Odd and Even Functions.

Half Range Fourier Sine or Cosine Series.

Parseval’s Identity.

Differentiation and Integration of Fourier Series.

Complex Notation for Fourier Series.

Boundary-Value Problems.

Orthogonal Functions.

Chapter 14 FOURIER INTEGRALS

The Fourier Integral.

Equivalent Forms of Fourier’s Integral Theorem.

Fourier Transforms.

Chapter 15 GAMMA AND BETA FUNCTIONS

The Gamma Function.

Table of Values and Graph of the Gamma Function.

The Beta Function.

Dirichlet Integrals.

Chapter 16 FUNCTIONS OF A COMPLEX VARIABLE

Functions.

Limits and Continuity.

Derivatives.

Cauchy-Riemann Equations.

Integrals.

Cauchy’s Theorem.

Cauchy’s Integral Formulas.

Taylor’s Series.

Singular Points.

Poles.

Laurent’s Series.

Branches and Branch Points.

Residues.

Residue Theorem.

Evaluation of Definite Integrals.

INDEX

CHAPTER 1

Numbers

Mathematics has its own language, with numbers as the alphabet. The language is given structure with the aid of connective symbols, rules of operation, and a rigorous mode of thought (logic). These concepts, which previously were explored in elementary mathematics courses such as geometry, algebra, and calculus, are reviewed in the following paragraphs.

Sets

Fundamental in mathematics is the concept of a set, class, or collection of objects having specified characteristics. For example, we speak of the set of all university professors, the set of all letters A, B, C, D,. . ., Z of the English alphabet, and so on. The individual objects of the set are called members or elements. Any part of a set is called a subset of the given set, e.g., A, B, C is a subset of A, B, C, D, . . ., Z. The set consisting of no elements is called the empty set or null set.

Real Numbers

The number system is foundational to the modern scientific and technological world. It is based on the symbols 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. Thus, it is called a base ten system. (There is the implication that there are other systems. One of these, which is of major importance, is the base two system.) The symbols were introduced by the Hindus, who had developed decimal representation and the arithmetic of positive numbers by 600 A.D. In the eighth century, the House of Wisdom (library) had been established in Baghdad, and it was there that the Hindu arithmetic and much of the mathematics of the Greeks were translated into Arabic. From there, this arithmetic gradually spread to the later-developing western civilization.

The flexibility of the Hindu-Arabic number system lies in the multiple uses of the numbers. They may be used to signify: (a) order—the runner finished fifth; (b) quantity—there are six apples in the barrel; (c) construction—2 and 3 may be used to form any of 23, 32, .23 or .32; (d) place—0 is used to establish place, as is illustrated by 607, 0603, and .007.

Finally, note that the significance of the base ten terminology is enhanced by the following examples:

The collection of numbers created from the basic set is called the real number system. Significant subsets of them are listed as follows. For the purposes of this text, it is assumed that the reader is familiar with these numbers and the fundamental arithmetic operations.

1.   Natural numbers 1, 2, 3, 4, . . ., also called positive integers, are used in counting members of a set. The symbols varied with the times; e.g., the Romans used I, II, III, IV, . . . . The sum a + b and product a · b or ab of any two natural numbers a and b is also a natural number. This is often expressed by saying that the set of natural numbers is closed under the operations of addition and multiplication, or satisfies the closure property with respect to these operations.

2.   Negative integers and zero, denoted by –1, –2, –3, . . ., and 0, respectively, arose to permit solutions of equations such as x + b = a, where a and b are any natural numbers. This leads to the operation of subtraction, or inverse of addition, and we write x = a – b.

The set of positive and negative integers and zero is called the set of integers.

3.   Rational numbers or fractions such as , . . . arose to permit solutions of equations such as bx = a for all integers a and b, where b ≠ 0. This leads to the operation of division, or inverse of multiplication, and we write x = a/b or a ÷ b, where a is the numerator and b the denominator.

The set of integers is a subset of the rational numbers, since integers correspond to rational numbers where b = 1.

4.   Irrational numbers such as and π are numbers which are not rational; i.e., they cannot be expressed as a/b (called the quotient of a and b), where a and b are integers and b ≠ 0.

The set of rational and irrational numbers is called the set of real numbers.

Decimal Representation of Real Numbers

Any real number can be expressed in decimal form, e.g., 17/10 = 1.7, 9/100 = 0.09, 1/6 = 0.16666. . . . In the case of a rational number, the decimal expansion either terminates or if it does not terminate, one or a group of digits in the expansion will ultimately repeat, as, for example, in . . . . In the case of an irrational number such as . . . no such repetition can occur. We can always consider a decimal expansion as unending; e.g., 1.375 is the same as 1.37500000 . . . or 1.3749999 . . . To indicate recurring decimals we sometimes place dots over the repeating cycle of digits, e.g., .

It is possible to design number systems with fewer or more digits; e.g., the binary system uses only two digits, 0 and 1 (see Problems 1.32 and 1.33).

Geometric Representation of Real Numbers

The geometric representation of real numbers as points on a line, called the real axis, as in Figure 1.1, is also well known to the student. For each real number there corresponds one and only one point on the line, and, conversely, there is a one-to-one (see Figure 1.1) correspondence between the set of real numbers and the set of points on the line. Because of this we often use point and number interchangeably.

Figure 1.1

While this correlation of points and numbers is automatically assumed in the elementary study of mathematics, it is actually an axiom of the subject (the Cantor Dedekind axiom) and, in that sense, has deep meaning.

The set of real numbers to the right of 0 is called the set of positive numbers, the set to the left of 0 is the set of negative numbers, while 0 itself is neither positive nor negative.

(Both the horizontal position of the line and the placement of positive and negative numbers to the right and left, respectively, are conventions.)

Between any two rational numbers (or irrational numbers) on the line there are infinitely many rational (and irrational) numbers. This leads us to call the set of rational (or irrational) numbers an everywhere dense set.

Operations with Real Numbers

If a, b, c belong to the set R of real numbers, then:

0 is called the identity with respect to addition; 1 is called the identity with respect to multiplication.

8.    For any a there is a number x in R such that x + a = 0.

x is called the inverse of a with respect to addition and is denoted by –a.

9.    For any a ≠ 0 there is a number x in R such that ax = 1.

x is called the inverse of a with respect to multiplication and is denoted by a–1 or 1/a.

Convention: For convenience, operations called subtraction and division are defined by a – b = a + (–b) and , respectively.

These enable us to operate according to the usual rules of algebra. In general, any set, such as R, whose members satisfy the preceding is called a field.

Inequalities

If a – b is a nonnegative number, we say that a is greater than or equal to b or b is less than or equal to a, and write, respectively, a ≧ b or b ≦ a. If there is no possibility that a = b, we write a > b or b < a. Geometrically, a > b if the point on the real axis corresponding to a lies to the right of the point corresponding to b.

Properties of Inequalities

If a, b, and c are any given real numbers, then:

5.    If a > b and c < 0, then ac < bc

EXAMPLES. 3 < 5 or 5 > 3; –2 < –1 or –1 > –2; x ≦ 3 means that x is a real number which may be 3 or less than 3.

Absolute Value of Real Numbers

The absolute value of a real number a, denoted by |a|, is defined as a if a > 0, –a if a < 0, and 0 if a = 0.

Properties of Absolute Value

EXAMPLES.

The distance between any two points (real numbers) a and b on the real axis is |a – b| = |b – a|.

Exponents and Roots

The product a · a . . . a of a real number a by itself p times is denoted by ap, where p is called the exponent and a is called the base. The following rules hold:

These and extensions to any real numbers are possible so long as division by zero is excluded. In particular, by using 2, with p = q and p = 0, respectively, we are led to the definitions a⁰ = 1, a–q = 1/aq.

If ap = N, where p is a positive integer, we call a a pth root of N, written . There may be more than one real pth root of N. For example, since 2² = 4 and (–2)² = 4, there are two real square roots of 4—namely, 2 and –2. For square roots it is customary to define as positive; thus, and then .

If p and q are positive integers, we define .

Logarithms

If ap = N, p is called the logarithm of N to the base a, written p = loga N. If a and N are positive and a ≠ 1, there is only one real value for p. The following rules hold:

1.    loga MN = loga M + loga N

3.    loga Mr = r loga M

In practice, two bases are used: base a = 10, and the natural base a = e = 2.71828. . . . The logarithmic systems associated with these bases are called common and natural, respectively. The common logarithm system is signified by log N; i.e., the subscript 10 is not used. For natural logarithms, the usual notation is ln N.

Common logarithms (base 10) traditionally have been used for computation. Their application replaces multiplication with addition and powers with multiplication. In the age of calculators and computers, this process is outmoded; however, common logarithms remain useful in theory and application. For example, the Richter scale used to measure the intensity of earthquakes is a logarithmic scale. Natural logarithms were introduced to simplify formulas in calculus, and they remain effective for this purpose.

Axiomatic Foundations of the Real Number System

The number system can be built up logically, starting from a basic set of axioms or self-evident truths, usually taken from experience, such as statements 1 through 9 on Page 3.

If we assume as given the natural numbers and the operations of addition and multiplication (although it is possible to start even further back, with the concept of sets), we find that statements 1 through 6, with R as the set of natural numbers, hold, while 7 through 9 do not hold.

Taking 7 and 8 as additional requirements, we introduce the numbers –1, –2, –3, . . ., and 0. Then, by taking 9, we introduce the rational numbers.

Operations with these newly obtained numbers can be defined by adopting axioms 1 through 6, where R is now the set of integers. These lead to proofs of statements such as (–2)(–3) = 6, –(–4) = 4, (0)(5) = 0, and so on, which are usually taken for granted in elementary mathematics.

We can also introduce the concept of order or inequality for integers, and, from these inequalities, for rational numbers. For example, if a, b, c, d are positive integers, we define a/b > c/d if and only if ad > bc, with similar extensions to negative integers.

Once we have the set of rational numbers and the rules of inequality concerning them, we can order them geometrically as points on the real axis, as already indicated. We can then show that there are points on the line which do not represent rational numbers (such as , π, etc.). These irrational numbers can be defined in various ways, one of which uses the idea of Dedekind cuts (see Problem 1.34). From this we can show that the usual rules of algebra apply to irrational numbers and that no further real numbers are possible.

Point Sets, Intervals

A set of points (real numbers) located on the real axis is called a one-dimensional point set.

The set of points x such that a ≦ x ≧ b is called a closed interval and is denoted by [a, b]. The set a < x < b is called an open interval, denoted by (a, b). The sets a < x ≦ b and a ≦ x < b, denoted by (a, b] and [a, b), respectively, are called half-open or half-closed intervals.

The symbol x, which can represent any number or point of a set, is called a variable. The given numbers a or b are called constants.

Letters were introduced to construct algebraic formulas around 1600. Not long thereafter, the philosopher-mathematician Rene Descartes suggested that the letters at the end of the alphabet be used to represent variables and those at the beginning to represent constants. This was such a good idea that it remains the custom.

EXAMPLE. The set of all x such that |x| < 4, i.e., –4 < x < 4, is represented by (–4, 4), an open interval.

The set x > a can also be represented by a < x < ∞. Such a set is called an infinite or unbounded interval. Similarly, –∞ < x < ∞ represents all real numbers x.

Countability

A set is called countable or denumerable if its elements can be placed in 1-1 correspondence with the natural numbers.

EXAMPLE. The even natural numbers 2, 4, 6, 8, . . . is a countable set because of the 1-1 correspondence shown.

A set is infinite if it can be placed in 1-1 correspondence with a subset of itself. An infinite set which is countable is called countable infinite.

The set of rational numbers is countable infinite, while the set of irrational numbers or all real numbers is noncountably infinite (see Problems 1.17 through 1.20).

The number of elements in a set is called its cardinal number. A set which is countably infinite is assigned the cardinal number ℵ0 (the Hebrew letter aleph-null). The set of real numbers (or any sets which can be placed into 1-1 correspondence with this set) is given the cardinal number C, called the cardinality of the contimuum.

Neighborhoods

The set of all points x such that |x – a| < δ, where δ > 0, is called a δ neighborhood of the point a. The set of all points x such that 0 < |x – a| < δ, in which x = a is excluded, is called a deleted δ neighborhood of a or an open ball of radius δ about a.

Limit Points

A limit point, point of accumulation, or cluster point of a set of numbers is a number l such that every deleted δ neighborhood of l contains members of the set; that is, no matter how small the radius of a ball about l, there are points of the set within it. In other words, for any δ > 0, however small, we can always find a member x of the set which is not equal to l but which is such that |x – l| < δ. By considering smaller and smaller values of δ, we see that there must be infinitely many such values of x.

A finite set cannot have a limit point. An infinite set may or may not have a limit point. Thus, the natural numbers have no limit point, while the set of rational numbers has infinitely many limit points.

A set containing all its limit points is called a closed set. The set of rational numbers is not a closed set, since, for example, the limit point is not a member of the set (Problem 1.5). However, the set of all real numbers x such that 0 ≦ x ≦ 1 is a closed set.

Bounds

If for all numbers x of a set there is a number M such that x ≦ M, the set is bounded above and M is called an upper bound. Similarly if x ≧ m, the set is bounded below and m is called a lower bound. If for all x we have m ≦ x ≦ M, the set is called bounded.

If M is a number such that no member of the set is greater than M but there is at least one member which exceeds M – ε for every ε > 0, then M is called the least upper bound (l.u.b.) of the set. Similarly, if no member of the set is smaller than + ε for every ε > 0, then is called the greatest lower bound (g.l.b.) of the set.

Bolzano-Weierstrass Theorem

The Bolzano-Weierstrass theorem states that every bounded infinite set has at least one limit point. A proof of this is given in Problem 2.23.

Algebraic and Transcendental Numbers

A number x which is a solution to the polynomial equation

where a0 ≠ 0, a1, a2, . . ., an are integers and n is a positive integer, called the degree of the equation, is called an algebraic number. A number which cannot be expressed as a solution of any polynomial equation with integer coefficients is called a transcendental number.

EXAMPLES. and which are solutions of 3x – 2 = 0 and x² – 2 = 0, respectively, are algebraic numbers.

The numbers π and e can be shown to be transcendental numbers. Mathematicians have yet to determine whether some numbers such as eπ or e + π are algebraic or not.

The set of algebraic numbers is a countably infinite set (see Problem 1.23), but the set of transcendental numbers is noncountably infinite.

The Complex Number System

Equations such as x² + 1 = 0 have no solution within the real number system. Because these equations were found to have a meaningful place in the mathematical structures being built, various mathematicians of the late nineteenth and early twentieth centuries developed an extended system of numbers in which there were solutions. The new system became known as the complex number system. It includes the real number system as a subset.

We can consider a complex number as having the form a + bi, where a and b are real numbers called the real and imaginary parts, and is called the imaginary unit. Two complex numbers a + bi and c + di are equal if and only if a = c and b = d. We can consider real numbers as a subset of the set of complex numbers with b = 0. The complex number 0 + 0i corresponds to the real number 0.

The absolute value or modulus of a + bi is defined as . The complex conjugate of a + bi is defined as a – bi. The complex conjugate of the complex number z is often indicated by .

The set of complex numbers obeys rules 1 through 9 on Pages 3, and thus constitutes a field. In performing operations with complex numbers, we can operate as in the algebra of real numbers, replacing i² by –1 when it occurs. Inequalities for complex numbers are not defined.

From the point of view of an axiomatic foundation of complex numbers, it is desirable to treat a complex number as an ordered pair (a, b) of real numbers a and b subject to certain operational rules which turn out to be equivalent to the aforementioned rules. For example, we define (a, b) + (c, d) = (a + c, b + d), (a, b) (c, d) = (ac – bd, ad + bc), m(a, b) = (ma, mb), and so on. We then find that (a, b) = a(1, 0) + b(0, 1) and we associate this with a + bi, where i is the symbol for (0, 1).

Polar Form of Complex Numbers

If real scales are chosen on two mutually perpendicular axes X′ OX and Y′ OY (the x and y axes), as in Figure 1.2, we can locate any point in the plane determined by these lines by the ordered pair of numbers (x, y) called rectangular coordinates of the point. Examples of the location of such points are indicated by P, Q, R, S, and T in Figure 1.2.

Figure 1.2

Figure 1.3

Since a complex number x + iy can be considered as an ordered pair (x, y), we can represent such numbers by points in an xy plane called the complex plane or Argand diagram. Referring to Figure 1.3, we see that x = ρ cos ϕ, y = ρ sin ϕ, where and ϕ, called the amplitude or argument, is the angle which line OP makes with the positive x axis OX. It follows that

called the polar form of the complex number, where ρ and ϕ are called polar coordinates. It is sometimes convenient to write cis ϕ instead of cos ϕ + i sin ϕ.

If z1 = x1 + iyi = ρ1 (cos ϕ1 + i sin ϕ1) and z2 = x 2 + i y2 = ρ2(cosϕ2 + i sin ϕ2) and by using the addition formulas for sine and cosine, we can show that

where n is any real number. Equation (5) is sometimes called De Moivre’s theorem. We can use this to determine roots of complex numbers. For example, if n is a positive integer,

from which it follows that there are in general n different values of z¹/n. In Chapter 11 we will show that eiϕ = cos ϕ + i sin ϕ where e = 2.71828. . . . This is called Euler’s formula.

Mathematical Induction

The principle of mathematical induction is an important property of the positive integers. It is especially useful in proving statements involving all positive integers when it is known, for example, that the statements are valid for n = 1, 2, 3 but it is suspected or conjectured that they hold for all positive integers. The method of proof consists of the following steps:

1.    Prove the statement for n = 1 (or some other positive integer).

2.    Assume the statement is true for n = k, where k is any positive integer.

3.    From the assumption in 2, prove that the statement must be true for n = k + 1. This is part of the proof establishing the induction and may be difficult or impossible.

4.    Since the statement is true for n = 1 (from Step 1) it must (from Step 3) be true for n = 1 + 1 = 2 and from this for n = 2 + 1 = 3, and so on, and so must be true for all positive integers. (This assumption, which provides the link for the truth of a statement for a finite number of cases to the truth of that statement for the infinite set, is called the axiom of mathematical induction.)

SOLVED PROBLEMS

Operations with numbers

1.1.       If x = 4, y = 15, z = –3, , evaluate (a) x + (y + z), (b) (x + y) + z, (c) p(qr), (d) (pq)r, (e) x(p + q).

(a)   x + (y + z) = 4 + [15 + (– 3)] = 4 + 12 = 16

(b)   (x + y) + z = (4 + 15) + (– 3) = 19 – 3 = 16

The fact that (a) and (b) are equal illustrates the associative law of addition.

The fact that (c) and (d) are equal illustrates the associative law of multiplication.

Another method:

using the distributive law.

1.2.       Explain why we do not consider as numbers.

(a)   If we define a/b as that number (if it exists) such that bx = a, then 0/0 is that number x such that 0x = 0. However, this is true for all numbers. Since there is no unique number which 0/0 can represent, we consider it undefined.

(b)   As in (a), if we define 1/0 as that number x (if it exists) such that 0x = 1, we conclude that there is no such number.

Because of these facts we must look upon division by zero as meaningless.

1.3.       Simplify .

provided that the cancelled factor (x – 3) is not zero; i.e., x ≠ 3. For x = 3, the given fraction is undefined.

Rational and irrational numbers

1.4.       Prove that the square of any odd integer is odd.

Any odd integer has the form 2m + 1. Since (2m + 1)² = 4m² + 4m + 1 is 1 more than the even integer 4 + 4m = 2(2m² + 2m), the result follows.

1.5.       Prove that there is no rational number whose square is 2.

Let p / q be a rational number whose square is 2, where we assume that p / q is in lowest terms; i.e., p and q have no common integer factors except ± 1 (we sometimes call such integers relatively prime).

Then (p / q)² = 2, p² = 2q² and p² is even. From Problem 1.4, p is even, since if p were odd, p² would be odd. Thus, p = 2m.

Substituting p = 2m in p² = 2q² yields q² = 2m², so that q² is even and q is even.

Thus, p and q have the common factor 2, contradicting the original assumption that they had no common factors other than ±1. By virtue of this contradiction there can be no rational number whose square is 2.

1.6.       Show how to find rational numbers whose squares can be arbitrarily close to 2.

We restrict ourselves to positive rational numbers. Since (1)² = 1 and (2)² = 4, we are led to choose rational numbers between 1 and 2, e.g., 1.1, 1.2, 1.3, . . ., 1.9.

Since (1.4)² = 1.96 and (1.5)² = 2.25, we consider rational numbers between 1.4 and 1.5, e.g., 1.41, 1.42,. ., 1.49.

Continuing in this manner we can obtain closer and closer rational approximations; e.g., (1.414213562)²is less than 2, while (1.414213563)² is greater than 2.

1.7.       Given the equation a0xn + a1xn–1 + . . . + an = 0, where a0, a1, . . . an are integers and a0 and an ≠ 0, show that if the equation is to have a rational root p / q, then p must divide an and q must divide a0 exactly.

Since p / q is a root we have, on substituting in the given equation and multiplying by qn, the result is

or dividing by p,

Since the left side of Equation (2) is an integer, the right side must also be an integer. Then, since p and q are relatively prime, p does not divide qn exactly and so must divide an.

In a similar manner, by transposing the first term of Equation (1) and dividing by q, we can show that q must divide a0.

1.8.       Prove that cannot be a rational number.

If , then , and, squaring, x⁴ – 10x² + 1 = 0. The only possible rational roots of this equation are ± 1 by Problem 1.7, and these do not satisfy the equation. It follows that , which satisfies the equation, cannot be a rational number.

1.9.       Prove that between any two rational numbers there is another rational number.

The set of rational numbers is closed under the operations of addition and division (nonzero denominator). Therefore, is rational. The next step is to guarantee that this value is between a and b. To this purpose, assume a < b. (The proof would proceed similarly under the assumption b < a.) Then 2a < a + b; thus, a < and a + b < 2b; therefore, .

Inequalities

1.10.     For what values of x is x + 3(2 – x) ≧ 4 – x?

x + 3(2 – x) ≧ 4 – x when x + 6 – 3x ≧ 4 – x, 6 – 2x ≧ 4 – x, 6 – 4 ≧ 2x – x, and 2 ≧ x; i.e. x ≦ 2.

1.11.     For what values of x is x² – 3x – 2 < 10 – 2x?

The required inequality holds when x² – 3x – 2 – 10 + 2x < 0, x² – x – 12 < 0 or (x – 4)(x + 3) < 0. This last inequality holds only in the following cases.

Case 1:   x – 4 > 0 and x + 3 < 0; i.e., x > 4 and x < –3. This is impossible, since x cannot be both greater than 4 and less than –3.

Case 2:   x