# Student Solutions Manual Advanced Engineering Mathematics, Volume 2

## Preface

This is the student Solutions Manual to accompany Advanced Engineering Mathematics, Volume 2, Tenth Edition. This market-leading text is known for its comprehensive coverage, careful and correct mathematics, outstanding exercises, and self contained subject matter parts for maximum flexibility. The new edition continues with the tradition of providing instructors and students with a comprehensive and up-to-date resource for teaching and learning engineering mathematics, that is, applied mathematics for engineers and physicists, mathematicians and computer scientists, as well as members of other disciplines.

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## About the Author

## Table of Contents

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. (Mobius 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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