Introduction to Methods of Applied Mathematics, 2004

I. Algebra

1. Sets and Functions

1.1 Sets

1.2 Single Valued Functions

1.3 Inverses and Multi-Valued Functions

1.4 Transforming Equations

1.5 Exercises

1.6 Hints

1.7 Solutions

2. Vectors

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

2.3 Exercises

2.4 Hints

2.5 Solutions

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

3.9 Hints

3.10 Solutions

3.11 Quiz

3.12 Quiz Solutions

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

5.1 Vector Functions

5.2 Gradient, Divergence and Curl

5.3 Exercises

5.4 Hints

5.5 Solutions

5.6 Quiz

5.7 Quiz Solutions

III. Functions of a Complex Variable

6. Complex Numbers

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

6.8 Hints

6.9 Solutions

7. Functions of a Complex Variable

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

7.11 Hints

7.12 Solutions

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

8.7 Hints

8.8 Solutions

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

9.5 Hints

9.6 Solutions

10. Contour Integration and the Cauchy-Goursat Theorem

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

10.9 Exercises

10.10 Hints

10.11 Solutions

11. Cauchy’s Integral Formula

11.1 Cauchy’s Integral Formula

11.2 The Argument Theorem

11.3 Rouche’s Theorem

11.4 Exercises

11.5 Hints

11.6 Solutions

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

12.8 Hints

12.9 Solutions

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

13.10 Infinite Sums

13.11 Exercises

13.12 Hints

13.13 Solutions

IV. Ordinary Differential Equations

14. First Order Differential Equations

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

14.10 Hints

14.11 Solutions

14.12 Quiz

14.13 Quiz Solutions

15. First Order Linear Systems of Differential Equations

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

15.5 Exercises

15.6 Hints

15.7 Solutions

16. Theory of Linear Ordinary Differential Equations

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

16.9 Hints

16.10 Solutions

16.11 Quiz

16.12 Quiz Solutions

17. Techniques for Linear Differential Equations

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

17.8 Hints

17.9 Solutions

18. Techniques for Nonlinear Differential Equations

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

18.9 Hints

18.10 Solutions

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

19.5 Exercises

19.6 Hints

19.7 Solutions

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

20.5 Exercises

20.6 Hints

20.7 Solutions

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

21.10 Exercises

21.11 Hints

21.12 Solutions

21.13 Quiz

21.14 Quiz Solutions

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

22.7 Exercises

22.8 Hints

22.9 Solutions

23. Series Solutions of Differential Equations

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

23.5 Exercises

23.6 Hints

23.7 Solutions

23.8 Quiz

23.9 Quiz Solutions

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

25.12 Exercises

25.13 Hints

25.14 Solutions

26. Self Adjoint Linear Operators

26.1 Adjoint Operators

26.2 Self-Adjoint Operators

26.3 Exercises

26.4 Hints

26.5 Solutions

27. Self-Adjoint Boundary Value Problems

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

27.6 Exercises

27.7 Hints

27.8 Solutions

28. Fourier Series

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

28.11 Exercises

28.12 Hints

28.13 Solutions

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

29.4 Exercises

29.5 Hints

29.6 Solutions

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

31.2.1 $\hat f(s)$ with Poles

31.2.2 $\hat f(s)$ with Branch Points

31.2.3 Asymptotic Behavior of $\hat f(s)$

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

32.2.1 A Word of Caution

32.3 Evaluating Fourier Integrals

32.3.1 Integrals that Converge

32.3.2 Cauchy Principal Value and Integrals that are Not Absolutely Convergent

32.3.3 Analytic Continuation

32.4 Properties of the Fourier Transform

32.4.1 Closure Relation

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 $xf(x)$

32.5 Solving Differential Equations with the Fourier Transform

32.6 The Fourier Cosine and Sine Transform

32.6.1 The Fourier Cosine Transform

32.6.2 The Fourier Sine Transform

32.7 Properties of the Fourier Cosine and Sine Transform

32.7.1 Transforms of Derivatives

32.7.2 Convolution Theorems

32.7.3 Cosine and Sine Transform in Terms of the Fourier Transform

32.8 Solving Differential Equations with the Fourier Cosine and Sine Transforms

32.9 Exercises

32.10 Hints

32.11 Solutions

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

33.6 Exercises

33.7 Hints

33.8 Solutions

34. Bessel Functions

34.1 Bessel’s Equation

34.2 Frobenius 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

34.8 Exercises

34.9 Hints

34.10 Solutions

V. Partial Differential Equations

35. Transforming Equations

35.1 Exercises

35.2 Hints

35.3 Solutions

36. Classification of Partial Differential Equations

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

36.3 Exercises

36.4 Hints

36.5 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

37.8 Exercises

37.9 Hints

37.10 Solutions

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

40.1 Introduction

40.2 Fundamental Solution

40.2.1 Two Dimensional Space

40.3 Exercises

40.4 Hints

40.5 Solutions

41. Waves

41.1 Exercises

41.2 Hints

41.3 Solutions

42. Similarity Methods

42.1 Exercises

42.2 Hints

42.3 Solutions

43. Method of Characteristics

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

43.10 Solutions

44. Transform Methods

44.1 Fourier Transform for Partial Differential Equations

44.2 The Fourier Sine Transform

44.3 Fourier Transform

44.4 Exercises

44.5 Hints

44.6 Solutions

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