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Tables of Contents for Advanced Engineering Mathematics
Chapter/Section Title
Page #
Page Count
Acknowledgments
Introduction
1 Complex Variables
1
50
1.1 Complex Numbers
1
5
1.2 Finding Roots
6
3
1.3 The Derivative in the Complex Plane: The Cauchy-Riemann Equations
9
11
1.4 Line Integrals
20
6
1.5 Cauchy-Goursat Theorem
26
4
1.6 Cauchy's Integral Formula
30
3
1.7 Taylor and Laurent Expansions and Singularities
33
7
1.8 Theory of Residues
40
6
1.9 Evaluation of Real Definite Integrals
46
5
2 Fourier Series
51
62
2.1 Fourier Series
51
15
2.2 Properties of Fourier Series
66
9
2.3 Half-Range Expansions
75
6
2.4 Fourier Series with Phase Angles
81
3
2.5 Complex Fourier Series
84
4
2.6 The Use of Fourier Series in the Solution of Ordinary Differential Equations
88
9
2.7 Finite Fourier Series
97
16
3 The Fourier Transform
113
48
3.1 Fourier Transform
113
8
3.2 Fourier Transforms Containing the Delta Function
121
3
3.3 Properties of Fourier Transforms
124
13
3.4 Inversion of Fourier Transforms
137
13
3.5 Convolution
150
3
3.6 Solution of Ordinary Differential Equations by Fourier Transforms
153
8
4 The Laplace Transform
161
70
4.1 Definition and Elementary Properties
161
8
4.2 Heaviside Step and Dirac Delta Functions
169
6
4.3 Some Useful Theorems
175
8
4.4 The Laplace Transform of a Periodic Function
183
3
4.5 Inversion by Partial Fractions: Heaviside's Expansion Theorem
186
9
4.6 Convolution
195
4
4.7 Integral Equations
199
4
4.8 Solution of Linear Differential Equations with Constant Coefficients
203
13
4.9 Transfer Functions, Green's Function, and Indicial Admittance
216
6
4.10 Inversion by Contour Integration
222
9
5 The Z-Transform
231
38
5.1 The Relationship of the Z-Transform to the Laplace Transform
232
7
5.2 Some Useful Properties
239
8
5.3 Inverse Z-Transforms
247
10
5.4 Solution of Difference Equations
257
6
5.5 Stability of Discrete-Time Systems
263
6
6 The Sturm-Liouville Problem
269
56
6.1 Eigenvalues and Eigenfunctions
269
13
6.2 Orthogonality of Eigenfunctions
282
3
6.3 Expansion in Series of Eigenfunctions
285
4
6.4 A Singular Sturm-Liouville Problem: Legendre's Equation
289
17
6.5 Another Singular Sturm-Liouville Problem: Bessel's Equation
306
19
7 The Wave Equation
325
64
7.1 The Vibrating String
325
3
7.2 Initial Conditions: Cauchy Problem
328
1
7.3 Separation of Variables
329
21
7.4 D'Alembert's Formula
350
7
7.5 The Laplace Transform Method
357
21
7.6 Numerical Solution of the Wave Equation
378
11
8 The Heat Equation
389
76
8.1 Derivation of the Heat Equation
390
1
8.2 Initial and Boundary Conditions
391
2
8.3 Separation of Variables
393
33
8.4 The Laplace Transform Method
426
17
8.5 The Fourier Transform Method
443
5
8.6 The Superposition Integral
448
10
8.7 Numerical Solution of the Heat Equation
458
7
9 Laplace's Equation
465
42
9.1 Derivation of Laplace's Equation
466
2
9.2 Boundary Conditions
468
1
9.3 Separation of Variables
469
23
9.4 The Solution of Laplace's Equation on the Upper Half-Plane
492
3
9.5 Poisson's Equation on a Rectangle
495
4
9.6 The Laplace Transform Method
499
3
9.7 Numerical Solution of Laplace's Equation
502
5
10 Vector Analysis
507
52
10.1 Review
507
8
10.2 Divergence and Curl
515
5
10.3 Line Integrals
520
5
10.4 The Potential Function
525
2
10.5 Surface Integrals
527
9
10.6 Green's Lemma
536
5
10.7 Stokes' Theorem
541
8
10.8 Divergence Theorem
549
10
11 Linear Algebra
559
40
11.1 Fundamentals of Linear Algebra
559
8
11.2 Determinants
567
6
11.3 Cramer's Rule
573
2
11.4 Row Echelon Form and Gaussian Elimination
575
10
11.5 Eigenvalues and Eigenvectors
585
8
11.6 Systems of Linear Differential Equations
593
6
Answers to the Odd-Numbered Problems
599
28
Index
627