Advanced Engineering Mathematics, Student Solutions Manual and Study Guide, Volume 2: Chapters 13 - 25

Erwin O. Kreyszig was a German Canadian applied mathematician and the Professor of Mathematics at Carleton University in Ottawa, Ontario, Canada. He was a pioneer in the field of applied mathematics: non-wave replicating linear systems.

PART D: COMPLEX ANALYSIS…257

Chapter 13. Numbers and Functions. Complex Differentiation…257

13.1 Complex Numbers and Their Geometric Representation…258

13.2 Polar Form of Complex Numbers. Powers and Roots…261

13.3 Derivative. Analytic Function…267

13.4 Cauchy–Riemann Equations. Laplace’s Equation…269

13.5 Exponential Function…274

13.6 Trigonometric and Hyperbolic Functions. Euler’s Formula…277

13.7 Logarithm. General Power. Principal Value…279

Chapter 14: Complex Integration…283

14.1 Line Integral in the Complex Plane…283

14.2 Cauchy’s Integral Theorem…288

14.3 Cauchy’s Integral Formula…291

14.4 Derivatives of Analytic Functions…295

Chapter 15: Power Series, Taylor Series…298

15.1 Sequences, Series, Convergence Tests…298

15.2 Power Series…303

15.3 Functions Given by Power Series…306

15.4 Taylor and Maclaurin Series…309

15.5 Uniform Convergence. Optional…312

Chapter 16: Laurent Series. Residue Integration…316

16.1 Laurent Series…316

16.2 Singularities and Zeros. Infinity…320

16.3 Residue Integration Method…322

16.4 Residue Integration of Real Integrals…326

Chapter 17: Conformal Mapping…332

17.1 Geometry of Analytic Functions: Conformal Mapping…333

17.2 Linear Fractional Transformations. (Möbius Transformations)…339

17.3 Special Linear Fractional Transformations…343

17.4 Conformal Mapping by Other Functions…347

17.5 Riemann Surfaces. Optional…352

Chapter 18: Complex Analysis and Potential Theory…353

18.1 Electrostatic Fields…354

18.2 Use of Conformal Mapping. Modeling…358

18.3 Heat Problems…359

18.4 Fluid Flow…361

18.5 Poisson’s Integral Formula for Potentials…364

18.6 General Properties of Harmonic Functions. Uniqueness Theorem for the Dirchlet Problem…367

PART E: NUMERIC ANALYSIS…373

Chapter 19: Numerics in General…373

19.1 Introduction…374

19.2 Solution of Equations by Iteration…379

19.3 Interpolation…384

19.4 Spline Interpolation…389

19.5 Numeric Integration and Differentiation…393

Chapter 20: Numeric Linear Algebra…400

20.1 Linear Systems: Gauss Elimination…400

20.2 Linear Systems: LU-Factorization, Matrix Inversion…404

20.3 Linear Systems: Solution by Iteration…410

20.4 Linear Systems: Ill-Conditioning, Norms…415

20.5 Least Squares Method…419

20.6 Matrix Eigenvalue Problems: Introduction…424

20.7 Inclusion of Matrix Eigenvalues…424

20.8 Power Method for Eigenvalues…429

20.9 Tridiagonalization and QR-Factorization…434

Chapter 21: Numerics for ODEs and PDEs…442

21.1 Methods for First-Order ODEs…442

21.2 Multistep Methods…445

21.3 Methods for Systems and Higher Order ODEs…446

21.4 Methods for Elliptic PDEs…452

21.5 Neumann and Mixed Problems. Irregular Boundary…454

21.6 Methods for Parabolic PDEs…459

21.7 Method for Hyperbolic PDEs…462

PART F: OPTIMIZATION, GRAPHS…465

Chapter 22: Unconstrained Optimization. Linear Programming…465

22.1 Basic Concepts. Unconstrained Optimization: Method of Steepest Descent…465

22.2 Linear Programming…471

22.3 Simplex Method…474

22.4 Simplex Method. Difficulties…479

Chapter 23: Graphs. Combinatorial Optimization…482

23.1 Graphs and Digraphs…482

23.2 Shortest Path Problems. Complexity…484

23.3 Bellman’s Principle. Dijkstra’s Algorithm…487

23.4 Shortest Spanning Trees: Greedy Algorithm…490

23.5 Shortest Spanning Trees: Prim’s Algorithm…493

23.6 Flows in Networks

23.7 Maximum Flow: Ford–Fulkerson Algorithm…497

23.8 Bipartite Graphs. Assignment Problems…499

PART G: PROBABILITY, STATISTICS…502

Chapter 24: Data Analysis, Probability Theory…502

24.1 Data Representation. Average. Spread…502

24.2 Experiments, Outcomes, Events…507

24.3 Probability…509

24.4 Permutations and Combinations…512

24.5 Random Variables. Probability Distributions…516

24.6 Mean and Variance of a Distribution…520

24.7 Binomial, Poisson, and Hypergeometric Distributions…523

24.8 Normal Distribution…526

24.9 Distribution of Several Random Variables…530

Chapter 25: Mathematical Statistics…533

25.1 Introduction. Random Sampling…533

25.2 Point Estimation of Parameters…533

25.3 Confidence Intervals…536

25.4 Testing of Hypotheses. Decisions…540

25.5 Quality Control…543

25.6 Acceptance Sampling…544

25.7 Goodness of Fit. Chi-Square Test…547

25.8 Nonparametric Tests…549

25.9 Regression. Fitting Straight Lines. Correlation…551

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