PART D: COMPLEX ANALYSISâ¦257
Chapter 13. Numbers and Functions. Complex Differentiationâ¦257
13.1 Complex Numbers and Their Geometric Representationâ¦258
13.2 Polar Form of Complex Numbers. Powers and Rootsâ¦261
13.3 Derivative. Analytic Functionâ¦267
13.4 Cauchyâ"Riemann Equations. Laplaceâs Equationâ¦269
13.5 Exponential Functionâ¦274
13.6 Trigonometric and Hyperbolic Functions. Eulerâs Formulaâ¦277
13.7 Logarithm. General Power. Principal Valueâ¦279
Chapter 14: Complex Integrationâ¦283
14.1 Line Integral in the Complex Planeâ¦283
14.2 Cauchyâs Integral Theoremâ¦288
14.3 Cauchyâs Integral Formulaâ¦291
14.4 Derivatives of Analytic Functionsâ¦295
Chapter 15: Power Series, Taylor Seriesâ¦298
15.1 Sequences, Series, Convergence Testsâ¦298
15.2 Power Seriesâ¦303
15.3 Functions Given by Power Seriesâ¦306
15.4 Taylor and Maclaurin Seriesâ¦309
15.5 Uniform Convergence. Optionalâ¦312
Chapter 16: Laurent Series. Residue Integrationâ¦316
16.1 Laurent Seriesâ¦316
16.2 Singularities and Zeros. Infinityâ¦320
16.3 Residue Integration Methodâ¦322
16.4 Residue Integration of Real Integralsâ¦326
Chapter 17: Conformal Mappingâ¦332
17.1 Geometry of Analytic Functions: Conformal Mappingâ¦333
17.2 Linear Fractional Transformations. (M¶bius Transformations)â¦339
17.3 Special Linear Fractional Transformationsâ¦343
17.4 Conformal Mapping by Other Functionsâ¦347
17.5 Riemann Surfaces. Optionalâ¦352
Chapter 18: Complex Analysis and Potential Theoryâ¦353
18.1 Electrostatic Fieldsâ¦354
18.2 Use of Conformal Mapping. Modelingâ¦358
18.3 Heat Problemsâ¦359
18.4 Fluid Flowâ¦361
18.5 Poissonâs Integral Formula for Potentialsâ¦364
18.6 General Properties of Harmonic Functions. Uniqueness Theorem for the Dirchlet Problemâ¦367
PART E: NUMERIC ANALYSISâ¦373
Chapter 19: Numerics in Generalâ¦373
19.1 Introductionâ¦374
19.2 Solution of Equations by Iterationâ¦379
19.3 Interpolationâ¦384
19.4 Spline Interpolationâ¦389
19.5 Numeric Integration and Differentiationâ¦393
Chapter 20: Numeric Linear Algebraâ¦400
20.1 Linear Systems: Gauss Eliminationâ¦400
20.2 Linear Systems: LU-Factorization, Matrix Inversionâ¦404
20.3 Linear Systems: Solution by Iterationâ¦410
20.4 Linear Systems: Ill-Conditioning, Normsâ¦415
20.5 Least Squares Methodâ¦419
20.6 Matrix Eigenvalue Problems: Introductionâ¦424
20.7 Inclusion of Matrix Eigenvaluesâ¦424
20.8 Power Method for Eigenvaluesâ¦429
20.9 Tridiagonalization and QR-Factorizationâ¦434
Chapter 21: Numerics for ODEs and PDEsâ¦442
21.1 Methods for First-Order ODEsâ¦442
21.2 Multistep Methodsâ¦445
21.3 Methods for Systems and Higher Order ODEsâ¦446
21.4 Methods for Elliptic PDEsâ¦452
21.5 Neumann and Mixed Problems. Irregular Boundaryâ¦454
21.6 Methods for Parabolic PDEsâ¦459
21.7 Method for Hyperbolic PDEsâ¦462
PART F: OPTIMIZATION, GRAPHSâ¦465
Chapter 22: Unconstrained Optimization. Linear Programmingâ¦465
22.1 Basic Concepts. Unconstrained Optimization: Method of Steepest Descentâ¦465
22.2 Linear Programmingâ¦471
22.3 Simplex Methodâ¦474
22.4 Simplex Method. Difficultiesâ¦479
Chapter 23: Graphs. Combinatorial Optimizationâ¦482
23.1 Graphs and Digraphsâ¦482
23.2 Shortest Path Problems. Complexityâ¦484
23.3 Bellmanâs Principle. Dijkstraâs Algorithmâ¦487
23.4 Shortest Spanning Trees: Greedy Algorithmâ¦490
23.5 Shortest Spanning Trees: Primâs Algorithmâ¦493
23.6 Flows in Networks
23.7 Maximum Flow: Fordâ"Fulkerson Algorithmâ¦497
23.8 Bipartite Graphs. Assignment Problemsâ¦499
PART G: PROBABILITY, STATISTICSâ¦502
Chapter 24: Data Analysis, Probability Theoryâ¦502
24.1 Data Representation. Average. Spreadâ¦502
24.2 Experiments, Outcomes, Eventsâ¦507
24.3 Probabilityâ¦509
24.4 Permutations and Combinationsâ¦512
24.5 Random Variables. Probability Distributionsâ¦516
24.6 Mean and Variance of a Distributionâ¦520
24.7 Binomial, Poisson, and Hypergeometric Distributionsâ¦523
24.8 Normal Distributionâ¦526
24.9 Distribution of Several Random Variablesâ¦530
Chapter 25: Mathematical Statisticsâ¦533
25.1 Introduction. Random Samplingâ¦533
25.2 Point Estimation of Parametersâ¦533
25.3 Confidence Intervalsâ¦536
25.4 Testing of Hypotheses. Decisionsâ¦540
25.5 Quality Controlâ¦543
25.6 Acceptance Samplingâ¦544
25.7 Goodness of Fit. Chi-Square Testâ¦547
25.8 Nonparametric Testsâ¦549
25.9 Regression. Fitting Straight Lines. Correlationâ¦551
