[Vol. II] Mathematical Analysis for Engineers

Preface
11
Chapter 13
Number Series 13
34
Definition. Sum of a Series 13
13.2
Operations
on Series
15
13.3
Tests for Convergence of Series 18
13.4
Alternating Series. Leibniz Test 30
13.5
Series of Positive and Negative Terms.
Absolute and
Conditional Convergence 32
Exercises 35
Answers 37
Chapter
14
Functional Series 38
14.1
Convergence Domain and Convergence Hnterval
38
14.2
Uniform Convergence 49
14.3
Weierstrass Test 43
14.4
Properties of Uniformly Convergent Functional
Series 45
Exercises 50
Answers 50
Chapter
15
Power Series 51
15.1
Abel’s Theorem.
Interval
and
Radius
of Convergence
for Power Series 41
15.2
Properties of Power
Series 56
15.3
‘Taylor's Series 59 :
Exercises 70
Answers 7}
Chapter
16
Fourier Series 73
16.1
Trigonometric Series 73
16.2
Fourier Series for a Function with Period 2 76
16.3
Sufficient Conditions for the Fourier Expansion of a
Function 78
16.4
Fourier Expansions
of Odd
and
Even
Functions
82
16.5
Expansion of a Function Defined on the Given Interval
into a Series of Sines and Cosines 86
16.4
Fourier Series for a Function with Arbit
rary Period 88
16.7
Complex Representation of Fourier Series
93
16.8
Fourier Series in General Orthogonal
Systems of Func-
tions @
Exercises
104
Answers 105
hapter
17
First-Order Ordinary
Differential
Equations
106
V7.4
Basic Notions. Examples 106
2
Saiution of the Cauchy
Problem
for
First-Order
Differential Equations 109
7.3
Approximate
Methods
of Integration of the Equation
yi = fix y) 13
17.4
Some Equations Integrable by Quadratu
res il8
175
Riccati Equation 135
IhG
Differential Equations Insolvable for
the Derivative 136
1h?
Ceometrical Aspects
of
First
-Orde
r
Diffe
rential Equa-
Hens. Orthogonal
Traje
ctori
es
142
Exercises 144
Answers 145
Chapter
18
Higher-Order Differential Equations
147
18.1
Cauchy Problem [47
1B.2
Reducing the Order of Higher-Order
Equations 149
18.4
Linear Homogeneous Differential
Equations of
Order a 153
18.4
Linearly Dependent and Linearly Inde
pendent Systems
af Functions 155
18.5
Structure of General Solution of Linea
r Homogencous
Differential Equation 160
Linear
Homogeneous
Differential
Equations
with
Constant Coefficients 164
Equations Reducible to Equations
with Constant
Coefficients 172
Linear Inhomogeneous
Differential Equations
173
Integration of Linear Inhomogeneous
Equation by Var-
iation of Constants
176
18.10
inhomogeneous
Linear
Differential
Equations
with
Constant Coefficients
180
o
(8.41
Integration of Differential Equation
s
Using
Rowet
Series and Generalized Power Series 188
7
16.12
Bessel
Equation.
Bessel
Functions
190
Exercises 201
Answers
208
Chapter 19
Systems of Different al Equations 203
19.1
Essentials. Definition s 203
19.2
Methods of Integra lion of Systems of
Differential
Equations
206
19.3
Systems of Linear Differential Equations 211
19.4
Systems of Linear Differential Equations With
Con-
stant Coefficients 21 p
Exercises
224
Answers
224
Chapter 20
Stability Theory 225)
,
20.1
Preliminaries 225
20.2
Stability in the Sense of Lyapunov. Basic Concepts and
Definitions 227
20.3
Stability of Autonomous Systems. Simplest Types of
Stationary Points 23 A.
20.4
Method of Lyapuno v's Functions 244
20,5
Stability in First (Linear) Approximation 248
Exercises 253
Answers
254
.
Chapter
21
Special Topics of Di {ferential Equations 255
24.1
Asymptotic Behavio ur of Solutions of Differential
Equations as x +
255
21.2
Perturbation Method 257
21.3
Oscillations of Solutions of Differential Equations 261]
Exercises 264
Answers
264
Chapter
22
Multiple Integrals. Double Integral 265
22.1
Problem Leading to the Concept of Double Integral 265
22.2
Main Properties of Double Integral 268
22.3
Double
Integral Reduced to Iterated Integral 270
22.4
Change
of Variables in Double Integral 278
22.5
Surface Area. Surface Integral 286
22.6
Triple Integrals 292
22.7
Taking Triple Integral in Rectangular Coordinates
294
22.8
Taking Triple Integral in
Cylindr
ical
and
Spheric
al
Coordinates
296
22.9
Applications of Double and Triple integrals 302
22.10
Improper Multiple
ntegrals over Unbounded
Domains 307
Exercises 309
Answers 312
8
Contents
Chapter
23
Line Integrals 313
23.1
Line Integrals of the First Kind 313
23.2
Line Integrals of the Second Kind 318
23.3
Green’s Formula 322
23.4
Applications of Line Integrals 327
Exercises 331
Answers 333
Chapter 24
Vector Analysis 334
24.1
Sealar Field. Level Surfaces
and
Curves.
Directiona!
Derivative 334
24.2
Gradient of a Scalar Field 339
24.3
Vector Field. Vector Lines and Their Differential Equa-
tions 344
24.4
Vector Flux Through a Surface and Its Properties 349
24,5
Flux of a Vector Through an Open Surface 354
24.6
Flux of a Vector Through a Closed Surface. Ostrograd-
sky-Gauss Formula 363
24.7
Divergence of a Vector Field 371
24.8
Circulation of a Vector Field. Curl of a Vector. Stokes
Theorem 378
24.9
Independence of the Line Integral of Integration
Path 386
24.10
Potential
Field 391
24.11
Hamiltonian 398
24.12
Differential Operations
of the Second
Order.
Laplace
Operator 402
2413
Curvilinear Coordinates 406
24.14
Basic Vector Operations in Curvilinear Coordinates 408
Exercises: 416
Answers 419
Chapter
25
Integrals Depending on Parameter 420
25.4
Proper Integrals Depending on Parameter 420
25.2.
Improper Integrals Depending on Parameter 425
25.3
Euler Integrals. Gamma
Function. Beta Function 431
Exercises 436
Answers 438
Chapter
26
Functions
of a Complex
Variable 441
-
26.1
Essentials. Derivative. Cauchy-Riemann Equations 44]
26.2
Elementary Functions of a Complex Variable 453
26.3
Integration
with
Respect
to
a
Complex
. Argument.
Cauchy Theorem. Cauchy Integral Formula 461
26.4
Complex Power Series. Taylor Series 476
Contents
26.5
Laurent Series. Isolated Singularities and Their Classifi-
cation 491
26.6
Residues. Basic Theorem on Residues. Application of
Residues to Integrals 503
Exercises
519
Answers 522
Chapter 27
Integral Transforms.
Fourier Transforms 424
;
27.1
Fourier
Integral
524
27.2
Fourier Transform,
Fourier Sine and Cosine
Transforms 528
27.3
Properties of the Fourier Transform
535
27.4
Applications 539
27,5
Multiple Fourier Transforms
543
Exercises 544
:
Answers 545
Chapter 28
Laplace Transform 546 :
28.1
Basic Definitions 546
28.2
Properties of Laplace Transform 551
28.3
Inverse Transform 560
28.4
Applications of Laplace Transform (Operational
Cal-
culus) 565
Exercises 372
Answers 573
Chapter
29
Partial Differential Equations 575
29.4
Essentials. Examples 575
29.2
Linear Partial Differential
Equations.
Properties
of
Their Solutions 577
29.3
Classification
of
Second~ Order
Linear
Differential
Equations in Two Independent
Variables 579
Exercises 583
Answers 584
Chapter 30
Hyperbolic Equations 585
30.1
Essentials 585
30.2
Solution of the Cauchy Problem (fnitial Value Problem)
for an Infinite String 587
30.3
Examination of the D’Alembert Formula 591
30.4
Well-Posedness of a Problem. Hadamard’s Example of
Hi-Posed Problem 594
30.5
Free Vibrations of a String Fixed at Both Ends. Fourier
Method 598 |
30.6
Forced Vibrations of a String Fixed at Both Ends 606
30.7
Forced
Vibrations-of a String
with
Unfixed
Ends 61]
30.8
General Scheme of the Fourier Method 613
30.9
‘Uniqueness of Solution of a Mixed Problem 621
JG.10
Vibrations of a Round Membrane 623
FOUL
Application of Laplace Transforms to Solution, of
Mixed Problems 627
Exercises 60
Answers 632
Chapler
3]
Parabolic Equations 633
3h
Heat Equation 633
SEZ
Cauchy’ Problera for Heat Equation 634
313
Heat
Propagation
in a Finite Rod
640
Fd
Fourier Method
Por Heat Equation 643
Exercises 649
Answers 649
Chapler 32
Elliptic Equations 656
32.1
Definitions. Formulation of Boundary Problems 650
32.2
Fundamental Solution of Laplace Equation 652
32.3
Green's Formulas 653
324
Basic Integral Creen’s Formula 654
42.5
Properties of Harmonic Functions 657 :
52.6
Solution of the Mirichlet Problem for a Circle Using the
Fourier Method 661
32.7
Poisson: Integral 664
Exercises 666
Answers 666
Appendix TI Conformal Mappings 667
index 6o4