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TABLE OF CONTENTS
Preface
0.1 Advice to Teachers
0.2 Acknowledgments
0.3 Warnings and Disclaimers
0.4 Suggested Use
0.5 About the Title
I Algebra 1
1 Sets and Functions 2
1.1 Sets
1.2 Single Valued Functions
1.3 Inverses and Multi-Valued Functions
1.4 Transforming Equations
1.5 Exercises
2 Vectors 22
2.1 Vectors
2.1.1 Scalars and Vectors
2.1.2 The Kronecker Delta and Einstein Summation Convention
2.1.3 The Dot and Cross Product
2.2 Sets of Vectors in n Dimensions

II Calculus
3 Differential Calculus
3.1 Limits of Functions
3.2 Continuous Functions
3.3 The Derivative
3.4 Implicit Differentiation
3.5 Maxima and Minima
3.6 Mean Value Theorems
3.6.1 Application: Using Taylor’s Theorem to Approximate Functions
3.6.2 Application: Finite Difference Schemes
3.7 L’Hospital’s Rule
3.8 Exercises
3.8.1 Limits of Functions
3.8.2 Continuous Functions
3.8.3 The Derivative
3.8.4 Implicit Differentiation
3.8.5 Maxima and Minima
3.8.6 Mean Value Theorems
3.8.7 L’Hospital’s Rule

4 Integral Calculus
4.1 The Indefinite Integral
4.2 The Definite Integral
4.2.1 Definition
4.2.2 Properties
4.3 The Fundamental Theorem of Integral Calculus
4.4 Techniques of Integration
4.4.1 Partial Fractions
4.5 Improper Integrals
4.6 Exercises
4.6.1 The Indefinite Integral
4.6.2 The Definite Integral
4.6.3 The Fundamental Theorem of Integration
4.6.4 Techniques of Integration
4.6.5 Improper Integrals
4.7 Hints
4.8 Solutions
4.9 Quiz
4.10 Quiz Solutions

5 Vector Calculus 154
5.1 Vector Functions
5.2 Gradient, Divergence and Curl
5.3 Exercises

III Functions of a Complex Variable 179
6 Complex Numbers 180
6.1 Complex Numbers
6.2 The Complex Plane
6.3 Polar Form
6.4 Arithmetic and Vectors
6.5 Integer Exponents
6.6 Rational Exponents
6.7 Exercises

7 Functions of a Complex Variable 239
7.1 Curves and Regions
7.2 The Point at Infinity and the Stereographic Projection
7.3 A Gentle Introduction to Branch Points
7.4 Cartesian and Modulus-Argument Form
7.5 Graphing Functions of a Complex Variable
7.6 Trigonometric Functions
7.7 Inverse Trigonometric Functions
7.8 Riemann Surfaces
7.9 Branch Points
7.10 Exercises

8 Analytic Functions
8.1 Complex Derivatives
8.2 Cauchy-Riemann Equations
8.3 Harmonic Functions
8.4 Singularities
8.4.1 Categorization of Singularities
8.4.2 Isolated and Non-Isolated Singularities
8.5 Application: Potential Flow
8.6 Exercises

9 Analytic Continuation
9.1 Analytic Continuation
9.2 Analytic Continuation of Sums
9.3 Analytic Functions Defined in Terms of Real Variables
9.3.1 Polar Coordinates
9.3.2 Analytic Functions Defined in Terms of Their Real or Imaginary Parts
9.4 Exercises

10 Contour Integration and the Cauchy-Goursat Theorem 462
10.1 Line Integrals
10.2 Contour Integrals
10.2.1 Maximum Modulus Integral Bound
10.3 The Cauchy-Goursat Theorem
10.4 Contour Deformation
10.5 Morera’s Theorem
10.6 Indefinite Integrals
10.7 Fundamental Theorem of Calculus via Primitives
10.7.1 Line Integrals and Primitives
10.7.2 Contour Integrals
10.8 Fundamental Theorem of Calculus via Complex Calculus

11 Cauchy’s Integral Formula 493
11.1 Cauchy’s Integral Formula
11.2 The Argument Theorem
11.3 Rouche’s Theorem

12 Series and Convergence
12.1 Series of Constants
12.1.1 Definitions
12.1.2 Special Series
12.1.3 Convergence Tests
12.2 Uniform Convergence
12.2.1 Tests for Uniform Convergence
12.2.2 Uniform Convergence and Continuous Functions
12.3 Uniformly Convergent Power Series
12.4 Integration and Differentiation of Power Series
12.5 Taylor Series
12.5.1 Newton’s Binomial Formula
12.6 Laurent Series
12.7 Exercises
12.7.1 Series of Constants
12.7.2 Uniform Convergence
12.7.3 Uniformly Convergent Power Series
12.7.4 Integration and Differentiation of Power Series
12.7.5 Taylor Series
12.7.6 Laurent Series

13 The Residue Theorem
13.1 The Residue Theorem
13.2 Cauchy Principal Value for Real Integrals
13.2.1 The Cauchy Principal Value
13.3 Cauchy Principal Value for Contour Integrals
13.4 Integrals on the Real Axis
13.5 Fourier Integrals
13.6 Fourier Cosine and Sine Integrals
13.7 Contour Integration and Branch Cuts
13.8 Exploiting Symmetry
13.8.1 Wedge Contours
13.8.2 Box Contours
13.9 Definite Integrals Involving Sine and Cosine
13.10Infinite Sums

IV Ordinary Differential Equations 772
14 First Order Differential Equations 773
14.1 Notation
14.2 Example Problems
14.2.1 Growth and Decay
14.3 One Parameter Families of Functions
14.4 Integrable Forms
14.4.1 Separable Equations
14.4.2 Exact Equations
14.4.3 Homogeneous Coefficient Equations
14.5 The First Order, Linear Differential Equation
14.5.1 Homogeneous Equations
14.5.2 Inhomogeneous Equations
14.5.3 Variation of Parameters
14.6 Initial Conditions
14.6.1 Piecewise Continuous Coefficients and Inhomogeneities
14.7 Well-Posed Problems
14.8 Equations in the Complex Plane
14.8.1 Ordinary Points
14.8.2 Regular Singular Points
14.8.3 Irregular Singular Points
14.8.4 The Point at Infinity
14.9 Additional Exercises

15 First Order Linear Systems of Differential Equations 846
15.1 Introduction
15.2 Using Eigenvalues and Eigenvectors to find Homogeneous Solutions
15.3 Matrices and Jordan Canonical Form
15.4 Using the Matrix Exponential

16 Theory of Linear Ordinary Differential Equations 900
16.1 Exact Equations
16.2 Nature of Solutions
16.3 Transformation to a First Order System
16.4 The Wronskian
16.4.1 Derivative of a Determinant
16.4.2 The Wronskian of a Set of Functions
16.4.3 The Wronskian of the Solutions to a Differential Equation
16.5 Well-Posed Problems
16.6 The Fundamental Set of Solutions
16.7 Adjoint Equations
16.8 Additional Exercises

17 Techniques for Linear Differential Equations 930
17.1 Constant Coefficient Equations
17.1.1 Second Order Equations
17.1.2 Real-Valued Solutions
17.1.3 Higher Order Equations
17.2 Euler Equations
17.2.1 Real-Valued Solutions
17.3 Exact Equations
17.4 Equations Without Explicit Dependence on y
17.5 Reduction of Order
17.6 *Reduction of Order and the Adjoint Equation
17.7 Additional Exercises

18 Techniques for Nonlinear Differential Equations 984
18.1 Bernoulli Equations
18.2 Riccati Equations
18.3 Exchanging the Dependent and Independent Variables
18.4 Autonomous Equations
18.5 *Equidimensional-in-x Equations
18.6 *Equidimensional-in-y Equations
18.7 *Scale-Invariant Equations
18.8 Exercises

19 Transformations and Canonical Forms
19.1 The Constant Coefficient Equation
19.2 Normal Form
19.2.1 Second Order Equations
19.2.2 Higher Order Differential Equations
19.3 Transformations of the Independent Variable
19.3.1 Transformation to the form u” + a(x) u = 0
19.3.2 Transformation to a Constant Coefficient Equation
19.4 Integral Equations
19.4.1 Initial Value Problems
19.4.2 Boundary Value Problems

20 The Dirac Delta Function
20.1 Derivative of the Heaviside Function
20.2 The Delta Function as a Limit
20.3 Higher Dimensions
20.4 Non-Rectangular Coordinate Systems

21 Inhomogeneous Differential Equations
21.1 Particular Solutions
21.2 Method of Undetermined Coefficients
21.3 Variation of Parameters
21.3.1 Second Order Differential Equations
21.3.2 Higher Order Differential Equations
21.4 Piecewise Continuous Coefficients and Inhomogeneities
21.5 Inhomogeneous Boundary Conditions
21.5.1 Eliminating Inhomogeneous Boundary Conditions
21.5.2 Separating Inhomogeneous Equations and Inhomogeneous Boundary Conditions
21.5.3 Existence of Solutions of Problems with Inhomogeneous Boundary Conditions
21.6 Green Functions for First Order Equations
21.7 Green Functions for Second Order Equations
21.7.1 Green Functions for Sturm-Liouville Problems
21.7.2 Initial Value Problems
21.7.3 Problems with Unmixed Boundary Conditions
21.7.4 Problems with Mixed Boundary Conditions
21.8 Green Functions for Higher Order Problems
21.9 Fredholm Alternative Theorem

22 Difference Equations
22.1 Introduction
22.2 Exact Equations
22.3 Homogeneous First Order
22.4 Inhomogeneous First Order
22.5 Homogeneous Constant Coefficient Equations
22.6 Reduction of Order

23 Series Solutions of Differential Equations 1184
23.1 Ordinary Points
23.1.1 Taylor Series Expansion for a Second Order Differential Equation
23.2 Regular Singular Points of Second Order Equations
23.2.1 Indicial Equation
23.2.2 The Case: Double Root
23.2.3 The Case: Roots Differ by an Integer
23.3 Irregular Singular Points
23.4 The Point at Infinity

24 Asymptotic Expansions
24.1 Asymptotic Relations
24.2 Leading Order Behavior of Differential Equations
24.3 Integration by Parts
24.4 Asymptotic Series
24.5 Asymptotic Expansions of Differential Equations
24.5.1 The Parabolic Cylinder Equation

25 Hilbert Spaces
25.1 Linear Spaces
25.2 Inner Products
25.3 Norms
25.4 Linear Independence
25.5 Orthogonality
25.6 Gramm-Schmidt Orthogonalization
25.7 Orthonormal Function Expansion
25.8 Sets Of Functions
25.9 Least Squares Fit to a Function and Completeness
25.10 Closure Relation
25.11 Linear Operators

26 Self Adjoint Linear Operators 1307
26.1 Adjoint Operators
26.2 Self-Adjoint Operators
26.3 Exercises
26.4 Hints
26.5 Solutions
27 Self-Adjoint Boundary Value Problems 1314
27.1 Summary of Adjoint Operators
27.2 Formally Self-Adjoint Operators
27.3 Self-Adjoint Problems
27.4 Self-Adjoint Eigenvalue Problems
27.5 Inhomogeneous Equations

28 Fourier Series 1330
28.1 An Eigenvalue Problem
28.2 Fourier Series
28.3 Least Squares Fit
28.4 Fourier Series for Functions Defined on Arbitrary Ranges
28.5 Fourier Cosine Series
28.6 Fourier Sine Series
28.7 Complex Fourier Series and Parseval’s Theorem
28.8 Behavior of Fourier Coefficients
28.9 Gibb’s Phenomenon
28.10 Integrating and Differentiating Fourier Series

29 Regular Sturm-Liouville Problems
29.1 Derivation of the Sturm-Liouville Form
29.2 Properties of Regular Sturm-Liouville Problems
29.3 Solving Differential Equations With Eigenfunction Expansions

30 Integrals and Convergence
30.1 Uniform Convergence of Integrals
30.2 The Riemann-Lebesgue Lemma
30.3 Cauchy Principal Value
30.3.1 Integrals on an Infinite Domain
30.3.2 Singular Functions

31 The Laplace Transform
31.1 The Laplace Transform
31.2 The Inverse Laplace Transform
31.3 Properties of the Laplace Transform
31.4 Constant Coefficient Differential Equations
31.5 Systems of Constant Coefficient Differential Equations
31.6 Exercises
31.7 Hints
31.8 Solutions

32 The Fourier Transform
32.1 Derivation from a Fourier Series
32.2 The Fourier Transform
32.3 Evaluating Fourier Integrals
32.4 Properties of the Fourier Transform
2.4.1 Closure Relatio
32.4.2 Fourier Transform of a Derivative
32.4.3 Fourier Convolution Theorem
32.4.4 Parseval’s Theorem
32.4.5 Shift Property
32.4.6 Fourier Transform of x f(x)
32.5 Solving Differential Equations with the Fourier Transform
32.6 The Fourier Cosine and Sine Transform
32.7 Properties of the Fourier Cosine and Sine Transform
32.8 Solving Differential Equations with the Fourier Cosine and Sine Transforms

33 The Gamma Function
33.1 Euler’s Formula
33.2 Hankel’s Formula
33.3 Gauss’ Formula
33.4 Weierstrass’ Formula
33.5 Stirling’s Approximation

34 Bessel Functions
34.1 Bessel’s Equation
34.2 Frobeneius Series Solution about z = 0
34.2.1 Behavior at Infinity
34.3 Bessel Functions of the First Kind
34.3.1 The Bessel Function Satisfies Bessel’s Equation
34.3.2 Series Expansion of the Bessel Function
34.3.3 Bessel Functions of Non-Integer Order
34.3.4 Recursion Formulas
34.3.5 Bessel Functions of Half-Integer Order
34.4 Neumann Expansions
34.5 Bessel Functions of the Second Kind
34.6 Hankel Functions
34.7 The Modified Bessel Equation

V Partial Differential Equations 1680
35 Transforming Equations 1681

36 Classification of Partial Differential Equations 1685
36.1 Classification of Second Order Quasi-Linear Equations
36.1.1 Hyperbolic Equations
36.1.2 Parabolic equations
36.1.3 Elliptic Equations
36.2 Equilibrium Solutions

37 Separation of Variables
37.1 Eigensolutions of Homogeneous Equations
37.2 Homogeneous Equations with Homogeneous Boundary Conditions
37.3 Time-Independent Sources and Boundary Conditions
37.4 Inhomogeneous Equations with Homogeneous Boundary Conditions
37.5 Inhomogeneous Boundary Conditions
37.6 The Wave Equation
37.7 General Method

38 Finite Transforms
38.1 Exercises
38.2 Hints
38.3 Solutions
39 The Diffusion Equation
39.1 Exercises
39.2 Hints
39.3 Solutions
40 Laplace’s Equation

41 Waves
41.1 Exercises
41.2 Hints
41.3 Solutions
42 Similarity Methods 1888

43 Method of Characteristics 1897
43.1 First Order Linear Equations
43.2 First Order Quasi-Linear Equations
43.3 The Method of Characteristics and the Wave Equation
43.4 The Wave Equation for an Infinite Domain
43.5 The Wave Equation for a Semi-Infinite Domain
43.6 The Wave Equation for a Finite Domain
43.7 Envelopes of Curves
43.8 Exercises
43.9 Hints
43.10Solutions
44 Transform Methods 1918
44.1 Fourier Transform for Partial Differential Equations
44.2 The Fourier Sine Transform
44.3 Fourier Transform

45 Green Functions
45.1 Inhomogeneous Equations and Homogeneous Boundary Conditions
45.2 Homogeneous Equations and Inhomogeneous Boundary Conditions
45.3 Eigenfunction Expansions for Elliptic Equations
45.4 The Method of Images
45.5 Exercises
45.6 Hints
45.7 Solutions
46 Conformal Mapping
46.1 Exercises
46.2 Hints
46.3 Solutions
47 Non-Cartesian Coordinates 2051
47.1 Spherical Coordinates
47.2 Laplace’s Equation in a Disk
47.3 Laplace’s Equation in an Annulus

VI Calculus of Variations
48 Calculus of Variations

VII Nonlinear Differential Equations
49 Nonlinear Ordinary Differential Equations
50 Nonlinear Partial Differential Equations
50.1 Exercises
50.2 Hints
50.3 Solutions

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