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7/27/2019 Advance Em for the effect of postive coeff of answer

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Advanced ModernEngineeringMathematicsThird Edition

Glyn JamesandDavid BurleyDick ClementsPhil DykeJohn SearlNigel SteeleJerry Wright

Coventry UniversityUniversity of SheffieldUniversity of BristolUniversity of PlymouthUniversity of EdinburghCoventry UniversityAT&T

PEARSONPrenticeHall

Harlow, Efigland London N ew York Boston San Francisco Toronto Sydney Singapore Ho ng KongTokyo Seoul Taipei New Delhi Cape Town M adrid Mexico City Amsterdam M unich Paris M ilan


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ContentsPrefaceAbout the Authors XIXx x i

Chapter 1 Functions of a Complex Variable1,1 Introduction1.2 Complex functions and mappings

1.2.1 Linear mappings1.2.2 Exercises (1 -8 )1.2.3 Inversion1.2.4 Bilinear mappings1.2.5 Exercises (9 -1 9)1.2.6 The mapping w = zz1.2.7 Exercises (20-23)1.3 Complex differentiation

1.3.1 Cauchy-Riemann equations1.3.2 Conjugate and harmonic functions1.3.3 Exercises (24-32)1.3.4 Mappings revisited1.3.5 Exercises (3 3-3 7)1.4 Complex series1.4.1 Power series1.4.2 Exercises (3 8-3 9)1.4.3 Taylor series1.4.4 Exercises (4 0 -4 3 ) '.1.4.5 Laurent series1.4.6 Exercises (44-46)1.5 Singu larities, zeros and residues

1.5.1 Singu larities and zeros1.5.2 Exercises (4 7-4 9)

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V I C O N T E N T S

1.5.3 Residues1.5.4 Exercises (50-52)1.6 Contour integration

1.6.1 Contour integrals1.6.2 Cauchy's theorem1.6.3 Exercises (53-59)1.6.4 The residue theorem1.6.5 Evaluation of definite real integrals1.6.6 Exercises (60-65)

1.7 Engineering application: analysing AC circuits

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79

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86

1.8 Engineering application: use of harmonic functions1.8.1 A heat transfer problem1.8.2 Current in a field-effect transistor1.8.3 Exercises (66-72)

1.9 Review exercises (1-24)

Chapter 2 Laplace Transforms2.1 Introduction2.2 The Laplace transform

2.2.1 Definition and notation2.2.2 Transforms of simple functions2.2.3 Existence of the Laplace transform2.2.4 Properties of the Laplace transform2.2.5 Table of Laplace transforms2.2.6 Exercises (1-3)2.2.7 The inverse transform2.2.8 Evaluation of inverse transforms2.2.9 Inversion using the first shift theorem2.2.10 Exercise (4)

2.3 Solution of differential equations2.3.1 Transforms of derivatives2.3.2 Transforms of integrals2.3.3 Ordinary differential equations2.3.4 Simultaneous differential equations2.3.5 Exercises (5-6)

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CONTENTS Vii

2.4 Engineering app lications: electrical circuits andmechanical vibrations2.4.1 Electrical circuits2.4.2 Mechanical vibrations2.4.3 Exercises (7-12)

2.5 Step and impulse functions2.5.1 TheHeaviside step fun ction2.5.2 Laplace transform of unit step function2.5.3 The second shift theorem2.5.4 Inversion using the second shift theorem2.5.5 Differential equations2.5.6 Periodic functions2.5.7 Exercises (1 3-24 )2.5.8 The impulse function2.5.9 The sifting property2.5.10 Laplace transforms of impulse functions2.5.11 Relationship between Heaviside step and impulse functions2.5.12 Exercises (25-30)2.5.13 Bending of beams2.5.14 Exercises (31-33)

2.6 Transfer functions2.6.1 Definitions2.6.2 Stability2.6.3 Impulse response2.6.4 Initial- and final-value theorems2.6.5 Exercises (34-47)2.6.6 Convolution2.6.7 System response to an arbitrary input2.6.8 Exercises (48-52)

2.7 Engineering applica tion: frequency response2.8 Review exercises (1 -3 0)

124124128132

134134137139142145148152154155156159164164168

168168171176178182183186190

191

198

Chapter 3 The z Transform3.1 Introduction 2063.2 The z transform

3.2.1 Definition and notation3.2.2 Sampling: a first introduction3.2.3 Exercises (1-2)

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Vlli CONTENTS

3.3 Properties of the z transform 21 23.3.1 The jinearity property 21 23.3.2 The first shift property (delaying) 2133.3.3 The second sh ift property (advancing) 21 53.3.4 Some further properties 2163.3.5 Table of z transform s 21 63.3.6 Exercises (3 -1 0) 217

3.4 The inverse z transform 21 83.4.1 Inverse techniques 21 83.4.2 Exercises (1 1-13) 225

3.5 Discrete-time systems and difference equations 2253.5.1 Difference equations 2253.5.2 The solution of difference equations 22 73.5.3 Exercises (1 4-2 0) 232

3.6 Discrete linear systems: characterization 2333.6.1 z transfer functions 2333.6.2 The impulse response 2393.6.3 Stability 24 23.6.4 Convolution 2463.6.5 Exercises (2 1-28 ) 25 0

3.7 The relationship between Laplace and z transforms 25 03.8 Engineering app lication : design of discrete-time systems 252

3.8.1 Analogue filters 2523.8.2 Designing a digital replacement filter 2533.8.3 Possible developments 25 43.9 Engineering app lication : the delta operator andthe 3) transform 255

3.9.1 Introduction 2553.9.2 The q or shift operator and the S operator 2553.9.3 Constructing a discrete-time system model 25 63.9.4 Implem enting the design 2593.9.5 The 2 transform 2603.9.6 Exercises (29 -32 ) 2613.10 Review exercises (1 -16) 261


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C O N T E N T S IX

Chapter 4 Fourier Series4.1 Introduction4.2 Fourier series expansion4.2.1 Periodic func tions4.2.2 Fourier's theorem4.2.3 The Fourier coefficients4,2.4 Functions of period 2%4.2.5 Even and odd functions4.2.6 Even and odd harmonics4.2.7 Linearity property4.2.8 Convergence of the Fourier series4.2.9 Exercises (1 -7 )

4.2.10 Functions of period T4.2.11 Exercises (8-13)4.3 Functions defined over a finite interval

4.3.1 Full-range series4.3.2 Half-range cosine and sine series4.3.3 Exercises (14-23)4.4 Differentiation and integration of Fourier series

4.4.1 Integration of a Fourier series4.4.2 Differentiation of a Fourier series4.4.3 Coefficients in term s of jumps at discontinuities4.4.4 Exercises (24-29)4.5 Engineering app lication : frequency response andoscillating systems

4.5.1 Response to periodic input4.5.2 Exercises (30-33)

4.6 Complex form of Fourier series4.6.1 Complex representation4.6.2 The multiplica tion theorem and Parseval's theorem4.6.3 Discrete frequency spectra i4.6.4 Power spectrum4.6.5 Exercises (34-39)

4.7 Orthogonal functions4.7.1 Definitions4.7.2 Generalized Fourier series

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X C O N T E N T S

4.7.3 Convergence of generalized Fourier series4.7.4 Exercises (40-46) 3323344.8 Engineering app lication: describing functions4.9 Review exercises (1 -2 0)

Chapter 5 The Fourier Transform

337338

5.1 Introduction5.2 The Fourier transform

5.2.1 The Fourier integral5.2.2 The Fourier transfo rm pair5.2.3 The continuous Fourier spectra5.2.4 Exercises (1-10)5.3 Properties of the Fourier trans form

5.3.1 The linearity property5.3.2 Time-differentiation property5.3.3 Time-shift property5.3.4 Frequency-shift property5.3.5 The symmetry property5.3.6 Exercises (11-16)5.4 The frequency response

5.4.1 Relationship between Fourier and Laplace transform s5.4.2 The frequency response5.4.3 Exercises (1 7-2 1)5.5 Transforms of the step and impulse functions

5.5.1 Energy and power5.5.2 Convolution5.5.3 Exercises (22-27)5.6 The Fourier transform in discrete tim e ;

5.6.1 Introduction5.6.2 A Fourier transform for sequences5.6.3 The discrete Fourier transform5 6.4 Estimation of the continuous Fourier transform5 6.5 The fast Fourier transform5.6.6 Exercises (28-31)

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CONTENTS XI

5.7 Engineering applica tion: the design of analogue filters 4065.8 Engineering applica tion: modulation, dem odulation andfrequency-domain filtering 409

5.8.1 Introduction5.8.2 Modulation and transmission5.8.3 Identification and isolation of the info rmation-carrying signal5.8.4 Demodulation stage5.8.5 Final signal recovery5.8.6 Further developments5.9 Review exercises (1 -2 5)

409411412413414415415

Chapter 6 Matrix Analys6.1 Introduction6.2 Review of matrix algebra

6.2.1 Definitions6.2.2 Basic operations on matrices6.2.3 Determinants6.2.4 Adjoint and inverse matrices6.2.5 Linear equations6.2.6 Rank of a matrix

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6.3 Vector spaces6.3.1 Linear independence6.3.2 Transformations between bases6.3.3 Exercises (1-4)

427428429431

6.4 The eigenvalue problem6.4.1 The characteristic equation t6.4.2 Eigenvalues and eigenvectors6.4.3 Exercises (5 -6 )6.4.4 Repeated eigenvalues6.4.5 Exercises (7 -9 )6.4.6 Some useful properties of eigenvalues6.4.7 Symmetric matrices6.4.8 Exercises (10-13)

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X i i C O N T E N T S

6.5 Numerical methods 44 66.5.1 The power method 44 66.5.2 Gerschgorin circles ,4 5 26.5.3 Exercises (1 4-1 9) 454

6.6 Reduction to canonical form 4556.6.1 Reduction to diagonal form 4556.6.2 The Jordan canonical form 45 86.6.3 Exercises (2 0-2 7) 4626.6.4 Quadratic forms 4636.6.5 Exercises (2 8-3 3) 469

6.7 Functions of a matrix 4706.7.1 Exercises (3 4 -4 1 ) 481

6.8 State-space representation 4826.8.1 Sing le-input-s ingle-output (SISO) systems 48 26.8.2 Multi-inpu t-mu lti-outpu t (MIMO) systems' 4876.8.3 Exercises (4 2-4 7) 48 9

6.9 Solution of the state equation 49 06.9.1 Direct form of the solution 49 06.9.2 The transition matrix 4926.9.3 Evaluating the transition matrix 49 26.9.4 Exercises (4 8- 53 ) 4956.9.5 Laplace transform solution 4956.9.6 Exercises (5 4 -5 7 ) 5006.9.7 Spectral representation of response 5016.9.8 Canonical representation 5046.9.9 Exercises (5 8 -6 6 ) 509

6.10 Discrete-time systems 5106.10.1 State-space model 5106.10.2 Solution of the discrete-time state equation 5136.10.3 Exercises (67-7 0) 517

6.11 Engineering app lication: capacitor microphone 5186.12 Engineering application: pole placement 521

6.12.1 Poles and eigenvalues 5216.12.2 The pole placement or eigenvalue location technique 5226.12.3 Exercises (71-76 ) 523

6.13 Review exercises (1 -22) 525


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CONTENTS Xii l

Chapter 7 Vector Calculus7.1 Introduction

7.1.1 Basic concepts7.1.2 Exercises (1-10)7.1.3 Transformations7.1.4 Exercises (11-17)7.1.5 The total d ifferential7.1.6 Exercises (18-20)7.2 Derivatives of a scalar poin t function

7.2.1 The gradient of a scalar point func tion7.2.2 Exercises (21-30)7.3 Derivatives of a vector point function

7.3.1 Divergence of a vector field7.3.2 Exercises (31-37)7.3.3 Curl of a vector field7.3.4 Exercises (38-45)7.3.5 Further properties of the vector operator V7.3.6 Exercises (46-55)7.4 Topics in integration

7.4.1 Line integrals7.4.2 Exercises (5 6 -6 4 )7.4.3 Double integrals7.4.4 Exercises (6 5-7 6)7.4.5 Green's theorem in a plane7.4.6 Exercises (7 7- 82)7.4.7 Surface integrals7.4.8 Exercises (8 3-91 )7.4.9 Volume integrals7.4.10 Exercises (92-102)7.4.11 Gauss's divergence theorem7.4.12 Stokes' theorem7.4.13 Exercises (103-112)

7.5 Engineering app lication: streamlines in fluid dynamics7.6 Engineering app lication : heat transfer7.7 Review exercises (1 -2 0)

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XJV CONTENTS

Chapter 8 Numerical Solution of Ordinary Differen tial Equations 6 0 18.18.28.3

8.4

8.58.68.7

IntroductionEngineering application: m otion in a viscous fluidNumerical solution of first-order ordinary differential equations8.3.1 A simple solution me thod: Euler's method8.3.2 Analysing Euier's method8.3.3 Using num erical methods to solve engineering problems8.3.4 Exercises (1 -7 )8.3.5 More accurate solution methods: mu ltistep methods8.3.6 Local and global truncation errors8.3.7 More accurate solution methods: predictor-co rrectormethods8.3.8 More accurate solution methods: Runge-Kutta methods8.3.9 Exercises (8 -1 7)8.3.10 Stiff equations8.3.11 Computer software libraries and the 'state of the a rt'Num erical solution of second- and higher-orderdifferential equations8.4.1 Num erical solution of coupled first-order equations8.4.2 State-space representation of higher-order systems8.4.3 Exercises (18-23)8.4.4 Boundary-value problems8.4.5 The method of shooting8.4.6 Function approximation methodsEngineering application: oscillations of a pendulumEngineering application: heating of an electrical fuseReview exercises (1-12)

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Chapter 9 Partial ^ jl fe ri n ti a l Equations9.1 Introduction 6629.2 General discussion

9.2.1 Wave equation9.2.2 Heat-conduction or diffusion equation663663666


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CONTENTS XV

9.2.3 Laplace equation9.2.4 Other and related equations9.2.5 Arbitrary functions9.2.6 Exercises (1 -1 4)9.3 Solution of the wave equation

9.3.1 D'Alembert solution and characteristics9.3.2 Separated solutions9.3.3 Laplace transform solution9.3.4 Exercises (1 5-2 7)9.3.5 Numerical solution9.3.6 Exercises (2 8 -3 1)9.4 Solution of the heat-conduction/diffusion equation

| 9.4.1 Separation methodI 9.4.2 Laplace trans form methodI 9.4.3 Exercises (3 2-4 0)1 9.4.4 Num erical solutiont 9.4.5 Exercises (4 1-4 3 )j | 9.5 So lution of the Laplace equationI I 9.5.1 Separated solutionsi t 9.5.2 Exercises (4 4 -5 4 )mm 9.5.3 Numerical solutionK 9.5.4 Exercises (5 5-5 9) ... E L 9.6 Finite elementsB j l 9.6.1 Exercises (60-62)m 9.7 General considerations P | 9.7.1 Formal classification^Ef~ 9.7.2 Boundary conditionsH | 9.7.3 Exercises (63 -69 )^ H l r 9.8 Engineering application: wave propagation under a^H f | | ' moving load^ H K | 9-9 Engineering application: blood-flow model^ ^ B j ; ' 9-10 Review exercises (1-21)

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X V i C O N T E N T S

Chapter 10 Op timization10.1 Introduction10.2 Linear programming

10.2.1 Introduction10.2.2 Simplex algorithm: an example10.2.3 Simplex algo rithm : general theory10.2.4 Exercises (1-11)10.2.5 Two-phase method10.2.6 Exercises (12-20)

764

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10.3 Lagrange multipliers10.3.1 Equality constraints10.3.2 Inequality constraints10.3.3 Exercises (21-28)

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10.4 Hill climbing10.4.1 Single-variable search10.4.2 Exercises (29-33)10.4.3 Simple multivariable searches10.4.4 Exercises (34-38)10.4.5 Advanced multivariable searches10.4.6 Exercises (39-42)

10.5 Engineering application: chemical processing plant10.6 Engineering app lication: heating fin10.7 Review exercises (1-26)

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813

Chapter 11 App lied Probab ility and Statistics11.1 Introduction11.2 Review of basic probability theory

11.2.1 The rules of probability11.2.2 Random variables11.2.3 The Bernoulli, binom ial and Poisson distributions11.2.4 The normal distribution11.2.5 Sample measures

82282:823]823825827


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CONTENTS XVii

11.3 Estimating parameters11.3.1 Interval estimates and hypothesis tests11.3.2 Distribution of the sample average11.3.3 Confidence interval for the mean11.3.4 Testing simple hypotheses11.3.5 Other confidence intervals and tests concerning means11.3.6 Interval and test for proportion11.3.7 Exercises (1-13)

11.4 Joint distributions and correlation11.4.1 Joint and marginal distributions11.4.2 Independence11.4.3 Covariance and correlation11.4.4 Sample correlation11.4.5 Interval and test for co rrelation11.4.6 Rank correlation11.4.7 Exercises (14-24)

11.5 Regression11.5.1 The method of least squares11.5.2 Normal residuals11.5.3 Regression and correlation11.5.4 Nonlinear regression11.5.5 Exercises (25-33)

11.6 Goodness-of-fit tests11.6.1 Chi-square distribution and test11.6.2 Contingency tables11.6.3 Exercises (34-42)

11.7 Moment generating functions11.7.1 Definition and simple applications11.7.2 The Poisson approximation to the binomial11.7.3 Proof of the central lim it theorem11.7.4 Exercises (43-47)

11.8 Engineering application: analysis of engine performance data11.8.1 Introduction11.8.2 Difference in mean runn ing times and temperatures11.8.3 Dependence of running time on temperature11.8.4 Test for normality11.8.5 Conclusions

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CONTENTS XVli

11.3 Estimating parameters11.3.1 Interval estimates and hypothesis tests11.3.2 Distribution of the sample average11.3.3 Confidence interval for the mean11.3.4 Testing simple hypotheses11.3.5 Other confidence intervals and tests concerning means11.3.6 Interval and test for proportion11.3.7 Exercises (1-13)

11.4 Joint distributions and correlation11.4.1 Joint and marginal distributions11.4.2 Independence11.4.3 Covariance and correlation11.4.4 Sample correlation11.4.5 Interval and test for correlation11.4.6 Rank correlation11.4.7 Exercises (14-24)

11.5 Regression11.5.1 The method of least squares11.5.2 Normal residuals11.5.3 Regression and correlation11.5.4 Nonlinear regression11.5.5 Exercises (25-33)

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841842844845849851852853

854855857859859861

11 .6 Goodness-of-fit tes ts11.6.1 Chi-square distribution a'nd test11.6.2 Contingency tables11.6.3 Exercises (34-42)

11.7 Moment generating functions11.7.1 Definition and simple applications11.7.2 The Poisson approximation to the binomial11.7.3 Proof of the central limit theorem11.7.4 Exercises (43-47)

11.8 Engineering app lication: analysis of engine performance data11.8.1 Introduction11.8.2 Difference in mean running times and temperatures11.8.3 Dependence of running time on temperature11.8.4 Test for normality11.8.5 Conclusions

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XViii CONTENTS

11.9 Engineering app lication: statistical quality control 88011.9.1 Introduction 88011.9.2 Shewhart attribute control charts 88011.9.3 Shewhart variable control charts 88311.9.4 Cusum control charts 884 ;11.9.5 Moving-average control charts 88711.9.6 Range charts 88911.9.7 Exercises (4 8- 59 ) 889

11.10 Poisson processes and the theory of queues 89011.10.1 Typical queueing problems 89011.10.2 Poisson processes 89111.10.3 Single service channel queue 8 9 4 ;11.10.4 Queues with multiple service channels 898ll . lO iS Queueing system simulation 89911.10.6 Exercises (60-67) 901

11.11 Bayes'theo rem and its applications 90211.11.1 Derivation and simple examples 9 0 2 ,11.11.2 App lications in probabilistic infe ren ce/ 904311.11.3 Exercises (68-78) 907!

11.12 Review exercises (1-10) 908 |Answers to Exercises 9 1 1 |Index 937|

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advance em for the effect of postive coeff of answer

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