Problems and Methods in Analysis

Foreword

Volume 1


1. Infinite Sequences

§ 1. General Remarks

§ 2. Theorems and Properties

Exercise 1


2. Number Series

§ 1. General Remarks

§ 2. Theorems and Properties


2.1 Necessary Condition for Convergence


2.2 Comparison Test


2.3 The Geometric Series


2.4 The Harmonic Series


2.5 The Harmonic Series of Order α


2.6 Two Groups of Series

§ 3. Series of Non-Negative Terms


3.1 Criterion for Convergence


3.2 Criterion for Divergence


3.3 D'Alembert's Ratio Test


3.4 Cauchy's Test

§ 4. Alternating Series


4.1 Leibniz's Test


4.2 Absolute Convergence

§ 5. Other Series

Exercise 2


3. Functions and their Derivatives

§ 1. Definition of a Function

§ 2. Limit of a Function

§ 3. Continuity


3.1 Properties of Continuous Functions


3.2 Examples of Continuous Functions

Exercise 3

§ 4. First Order Derivatives of Functions of One Variable


4.1 Definition


4.2 Definition


4.3 Geometrical Interpretation


4.4 Differentiability of Continuous Functions


4.5 Properties of First Derivatives


4.6 Function of a Function


4.7 Inverse Functions


4.8 A List of First Derivatives

Exercise 4

§ 5. Derivatives of Higher Order


5.1 Definition


5.2 Derivatives of Order n

Exercise 5

§ 6. Derivatives of a Function Given by Parametric Equations


6.1 First Derivatives


6.2 Second Derivatives

Exercise 6


4. Partial Differentiation

§ 1. Continuity of a Function of Two Variables

§ 2. First Order Derivatives

§ 3. Second and Higher Order Derivatives


3.1 Introduction


3.2 Theorem


3.3 Definition


3.4 Function of a Function


3.5 Functions of Several Variables

Exercise 7

§ 4. Derivatives of Implicit Functions


4.1 Definition


4.2 The Existence of Implicit Functions

Exercise 8


5. Algebra

§ 1. Complex Numbers


1.1 Definitions


1.2 Trigonometric Interpretation


1.3 De Moivre's Theorem

Exercise 9

§ 2. The Solution of Algebraic Equations


2.1 General Properties of Algebraic Equations

Exercise 10


2.2 Cubic Equations

Exercise 11


2.3 Quartic Equations

Exercise 12


6. Curve Tracing

§1. Maxima and Minima


1.1 Increasing and Decreasing Functions


1.2 Definition of Maxima and Minima


1.3 Stationary Points


1.4 Turning Points


1.5 Determination of the Nature of Turning Points

§ 2. Concavity


2.1 Definition


2.2 Points of Inflexion

§ 3. Asymptotes


3.1 Definition


3.2 Asymptotes Parallel to the Coordinate Axes


3.3 Oblique Asymptotes

§4. Curve Tracing


4.1 Procedure


4.2 General Notes

§ 5. Implicit Functions


5.1 Definition


5.2 Conditions for a Multiple Point


5.3 Tangents at Multiple Points


5.4 Two Rules for Finding Asymptotes


5.5 Technique

§ 6. Functions of the type yq = xp


6.1 Symmetry


6.2 p/q > 1


6.3 p/q < 1

Exercise 13


7. Power Series

§ 1. General Remarks


1.1 Definition


1.2 Radius of Convergence


1.3 Theorems

Exercise 14

§ 2. Taylor's Theorem


2.1 Definition


2.2 The Taylor Series of a Function


2.3 The Maclaurin Series of a Function


2.4 The Integration of Any Series

Exercise 15


8. Limiting Values of Indeterminate Forms

§ 1. The Indeterminate Symbols 0/0, ∞/∞


1.1 The rule of 1'Hôpital


1.2 Generalization of Rule

§ 2. The Indeterminate Symbols 0.∞, ∞ - ∞, 1∞, ∞°, 0°


2.1 Theorem (0.∞)


2.2 Theorem (∞ - ∞)


2.3 Theorem (1∞, ∞°, 0°)

Exercise 16


9. Approximation to Roots of Equations

§ 1. A Graphical Method

§ 2. Routh's Method

§ 3. The Methods of Chords

§ 4. The Method of Tangents (Newton's Method)

§ 5. A Combined Method

Exercise 17

Answers

Index

Volume 2


10. Indefinite Integrals

§ 1. Introduction

§ 2. Some Standard Forms

§ 3. Some General Properties of Indefinite Integrals


3.1. The Addition of Integrals


3.2. Multiplication by a Constant


3.3. Integration by Parts


3.4. Substitution


3.5. Two Important Results

Exercise 18

§ 4. The Integration of Rational Functions


4.1 Introduction


4.2 Integrals of the Type ∫ mx+n/ax2+bx+c dx


4.3 Practical Methods

Exercise 19

§ 5. The Integration of Irrational Functions


5.1 Integrals Containing Roots of Linear Expressions

Exercise 20


5.2 Functions of the Type 1/√ax2+bx+c


5.3 Functions of the Type Ax+B/√ax2+bx+c


5.4 Two Pairs of Related Functions


5.5 The Method of Undetermined Multipliers


5.6 Functions of the Type 1/(ax2+b) √(px2+q)

Exercise 21

§ 6. The Integration of Trigonometric Functions


6.1 Some Standard Integrals


6.2 Integrals of Linear Functions


6.3 Integrals of Squared Functions


6.4 Integrals of Products


6.5 Reduction Formulae


6.6 The Transformation of General Trigonometric Functions Into Rational Functions

Exercise 22

§ 7. The Integration of Inverse Circular Functions

Exercise 23

§ 8. The Integration of Logarithmic and Exponential Function

Exercise 24


11. Definite Integrals

§ 1. General Remarks


1.1 Definition


1.2 Geometric Interpretation of a Definite Integral

§ 2. Properties of Definite Integrals


2.1 The Law of Addition


2.2 Multiplication by a Constant


2.3 Sum of Integrals


2.4 A Mean Value Theorem


2.5 A Function Defined as a Definite Integral


2.6 The Relationship Between Definite and Indefinite Integrals


2.7 Integration by Parts


2.8 Integration by Substitution


2.9 Dummy Symbols

Exercise 25

§ 3. Applications of Definite Integrals


3.1 The Calculation of Areas Given Parametric or Polar Equations of Curves

Exercise 26


3.2 The Calculation of Lengths of Arc

Exercise 27


3.3 Volumes of Revolution and Surface Area

Exercise 28


3.4 Moment of Inertia and Centre of Gravity

Exercise 29

§4. Irregular Integrals


4.1 Integrals of Functions Not Defined at a Finite Number of Points


4.2 Integrals of Unbounded Functions


4.3 Integrals of Infinite Interval


4.4 Geometric Interpretation

Exercise 30


12. Maxima and Minima of a Function of Two Variables

§ 1. Definition

§2. Theorem

Exercise 31


13. Tangents and Normals. Curvature

§ 1. Tangents and Normals


1.1 Equation of Tangent and Normal

Exercise 32

§ 2. Curvature and Radius of Curvature


2.1 Definition of Curvature


2.2 Definition of Radius of Curvature


2.3 Formulae for Curvature and Radius of Curvature


2.4 Sign Convention

Exercise 33

§ 3. Evolute and Involute


3.1 Coordinates of the Centre of Curvature


3.2 Definition


3.3 Two Properties of the Evolute

Exercise 34


14. Fourier Series

§ 1. General Ideas


1 1 The Euler—Fourier Formulae


1.2 The Fourier Series


1.3 Dirichlet's Conditions


1.4 Definition


1.5 Odd and Even Functions


1.6 Functions of Period 2l


1.7 Functions Defined in the Interval [a, b]

Exercise 35

Answers

Index