Engineering Mathematics: Volume - 2

601 Pages·2012·46.527 MB·English

by  E. Rukmangadachari

Engineering
Mathematics-II
E. Rukmangadachari
Professor of Mathematics,
Department of Humanities and Sciences,
Malla Reddy Engineering College,
Secunderabad
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To
My Beloved Parents
Enikapati Krishnamachari,
Rajamma, Ademma
About the Author
E. Rukmangadachari is former head of Computer Science and Engineering
as well as Humanities and Sciences at Malla Reddy Engineering College,
Secunderabad. Earlier, he was a reader in mathematics (PG course) at
Government College, Rajahmundry. He is an M.A. from Osmania University,
Hyderabad, and an M.Phil. and Ph.D. degree holder from Sri. Venkateswara
University, Tirupathi.
A recent recipient of the Andhra Pradesh State Meritorius Teacher’s award
in 1981, Professor Rukmangadachari has published over 40 research papers
in national and international journals. With a rich repertoire of over 45 years’
experience in teaching mathematics to undergraduate, postgraduate and
engineering students, he is currently the vice president of the Andhra Pradesh
Society of Mathematical Sciences, Hyderabad. An ace planner with fi ne managerial skills, he was the
organizing secretary for the conduct of the 17th Congress of the Andhra Pradesh Society for Mathematical
Sciences, Hyderabad.
Contents
About the Author iv 3 Solution of Algebraic
Preface ix
and Transcendental Equations
3.1 Introduction to Numerical
1 E igenvalues
Methods 3-1
and Eigenvectors
3.2 Errors and their Computation 3-1
1.1 Introduction 1-1 3.3 Formulas for Errors 3-2
1.2 Linear Transformation 1-1 3.4 Mathematical Pre-requisites 3-3
1.3 Characteristic Value Problem 1-1 3.5 Solution of Algebraic
Exercise 1.1 1-6 and Transcendental Equations 3-4
1.4 Properties of Eigenvalues 3.6 Direct Methods of Solution 3-4
and Eigenvectors 1-7 3.7 Numerical Methods of Solution
1.5 Cayley–Hamilton Theorem 1-9 of Equations of the Form f (x) = 0 3-5
Exercise 1.2 1-12 Exercise 3.1 3-20
1.6 Reduction of a Square Matrix
to Diagonal Form 1-14
4 Interpolation
1.7 Powers of a Square Matrix A—
Finding of Modal Matrix P 4.1 Introduction 4-1
and Inverse Matrix A−1 1-18 4.2 Interpolation with Equal Intervals 4-2
Exercise 1.3 1-24 4.3 Symbolic Relations
and Separation of Symbols 4-4
Exercise 4.1 4-8
2 Quadratic Forms
4.4 Interpolation 4-8
2.1 Introduction 2-1 4.5 I nterpolation Formulas
2.2 Quadratic Forms 2-1 for Equal Intervals 4-9
2.3 Canonical Form (or) Exercise 4.2 4-13
Sum of the Squares Form 2-3 4.6 Interpolation with Unequal
2.4 Nature of Real Quadratic Forms 2-3 Intervals 4-14
2.5 Reduction of a Quadratic Form 4.7 Properties Satisfi ed by Δ′ 4-15
to Canonical Form 2-5 4.8 Divided Difference
2.6 Sylvestor’s Law of Inertia 2-6 Interpolation Formula 4-16
2.7 Methods of Reduction 4.9 Inverse Interpolation Using
of a Quadratic Form Lagrange’s Interpolation
to a Canonical Form 2-6 Formula 4-17
2 .8 Singular Value Decomposition 4 .10 C entral Difference
of a Matrix 2-9 Formulas 4-21
Exercise 2.1 2-16 Exercise 4.3 4-26
vi Contents
5 Curve Fitting 8.5 Dirichlet’s Conditions for Fourier
Series Expansion of a Function 8-4
5.1 Introduction 5-1
8.6 Fourier Series Expansions:
5.2 Curve Fitting by the Method Even/Odd Functions 8-5
of Least Squares 5-2
8.7 Simply-Defined and Multiply-
5.3 C urvilinear (or Nonlinear) (Piecewise) Defined Functions 8-7
Regression 5-8
Exercise 8.1 8-18
5.4 C urve Fitting by a Sum
8.8 Change of Interval: Fourier Series
of Exponentials 5-10 in Interval (a , a + 2l ) 8-19
5.5 Weighted Least Squares
Exercise 8.2 8-23
Approximation 5-13
8.9 Fourier Series Expansions of Even
Exercise 5.1 5-14
and Odd Functions in (−l, l ) 8-24
Exercise 8.3 8-26
6 N umerical Differentiation
8.10 Half-Range Fourier Sine/Cosine
and Integration
Series: Odd and Even Periodic
Continuations 8-26
6.1 Introduction 6-1
Exercise 8.4 8-33
6.2 Errors in Numerical Differentiation 6-6
6.3 Maximum and Minimum 8.11 Root Mean Square (RMS)
Values of a Tabulated Function 6-7 Value of a Function 8-34
Exercise 6.1 6-8 Exercise 8.5 8-36
6.4 Numerical Integration: Introduction 6-8
Exercise 6.2 6-21
9 Fourier Integral Transforms
6.5 Cubic Splines 6-21
6 .6 Gaussian Integration 6-27 9.1 Introduction 9-1
Exercise 6.3 6-31 9.2 Integral Transforms 9-1
9.3 Fourier Integral Theorem 9-1
7 N umerical Solution of Ordinary 9.4 Fourier Integral
in Complex Form 9-2
Differential Equations
9.5 Fourier Transform of f (x) 9-3
7.1 Introduction 7-1
9.6 Finite Fourier Sine Transform
7.2 Methods of Solution 7-1
and Finite Fourier Cosine
7.3 Predictor–Corrector Methods 7-17 Transform (FFCT) 9-4
Exercise 7.1 7-21 9.7 Convolution Theorem
for Fourier Transforms 9-5
8. Fourier Series
9.8 Properties of Fourier Transform 9-6
8.1 Introduction 8-1 Exercise 9.1 9-15
8.2 Periodic Functions, Properties 8-1 9.9 Parseval’s Identity for Fourier
8.3 Classifi able Functions—Even Transforms 9-16
and Odd Functions 8-2 9 .10 Parseval’s Identities for
8.4 Fourier Series, Fourier Fourier Sine and Cosine
Coeffi cients and Euler’s Transforms 9-17
Formulae in (a , a +2p ) 8-3 Exercise 9.2 9-18
Contents vii
10 Partial Differential 11 Z-Transforms and Solution
Equations of Difference Equations
10.1 Introduction 10-1 1 1.1 Introduction 11-1
10.2 Order, Linearity 1 1.2 Z-Transform: Definition 11-1
and Homogeneity of a 11.3 Z-Transforms of Some Standard
Partial Differential Equation 10-1 Functions (Special Sequences) 11-4
10.3 Origin of Partial Differential 1 1.4 Recurrence Formula for the Sequence
Equation 10-2 of a Power of Natural Numbers 11-5
10.4 Formation of Partial Differential 11.5 Properties of Z-Transforms 11-6
Equation by Elimination of Two
Exercise 11.1 11-11
Arbitrary Constants 10-3
11.6 Inverse Z-Transform 11-11
Exercise 10.1 10-4
Exercise 11.2 11-16
10.5 Formation of Partial Differential
11.7 Application of Z-Transforms:
Equations by Elimination
Solution of a Difference
of Arbitrary Functions 10-5
Equations by Z-Transform 11-17
Exercise 10.2 10-7
11.8 Method for Solving a Linear
10.6 Classification of First-Order
Difference Equation with
Partial Differential Equations 10-7
Constant Coeffi cients 11-18
10.7 Classifi cation of Solutions
Exercise 11.3 11-21
of First-Order Partial
Differential Equation 10-8 12 Special Functions
10.8 Equations Solvable by Direct
1 2.1 Introduction 12-1
Integration 10-9
1 2.2 Gamma Function 12-1
Exercise 10.3 10-10
1 2.3 R ecurrence Relation or
10.9 Quasi-Linear Equations
Reduction Formula 12-1
of First Order 10-11
1 2.4 Various Integral Forms
10.10 Solution of Linear, Semi-Linear
of Gamma Function 12-3
and Quasi-Linear Equations 10-11
Exercise 12.1 12-6
Exercise 10.4 10-17
1 2.5 Beta Function 12-6
10.11 Nonlinear Equations
of First Order 10-18 1 2.6 Various Integral Forms
of Beta Function 12-7
Exercise 10.5 10-22
1 2.7 Relation Between Beta
1 0.12 E uler’s Method of Separation
and Gamma Functions 12-10
of Variables 10-23
1 2.8 Multiplication Formula 12-10
Exercise 10.6 10-25
1 2.9 L egendre’s Duplication Formula 12-11
1 0.13 C lassifi cation of Second-Order
Partial Differential Equations 10-25 Exercise 12.2 12-16
Exercise 10.7 10-33 12.10 L egendre Functions 12-16
Exercise 10.8 10-42 Exercise 12.3 12-30
1 0.14 T wo-dimensional Wave 1 2.11 Bessel Functions 12-31
Equation 10-46 Exercise 12.4 12-42
Exercise 10.9 10-49 Exercise 12.5 12-50
viii Contents
13 Functions of a Complex 1 6.8 Laurent’s Series 16-9
Variable 1 6.9 Higher Derivatives of Analytic
Functions 16-18
1 3.1 Introduction 13-1
Exercise 16.1 16-19
1 3.2 Complex Numbers–Complex
Plane 13-1 17 Calculus of Residues
Exercise 13.1 13-3
1 7.1 Evaluation of Real Integrals 17-1
Exercise 13.2 13-19
Exercise 17.1 17-7
1 3.3 L aplace’s Equation: Harmonic
Exercise 17.2 17-14
and Conjugate Harmonic
Functions 13-19 Exercise 17.3 17-35
Exercise 13.3 13-26
18 Argument Principle
14 Elementary Functions and Rouche’s Theorem
1 4.1 Introduction 14-1 18.1 Introduction 18-1
1 4.2 Elementary Functions 1 8.2 Meromorphic Function 18-1
of a Complex Variable 14-1
1 8.3 Argument Principle
Exercise 14.1 14-16 (Repeated Single Pole/Zero) 18-1
1 8.4 G eneralised Argument Theorem 18-2
15 Complex Integration
1 8.5 Rouche’s Theorem 18-3
1 5.1 Introduction 15-1 1 8.6 Liouville Theorem 18-4
1 5.2 Basic Concepts 15-1 1 8.7 Fundamental Theorem of Algebra 18-4
1 5.3 Complex Line Integral 15-2 1 8.8 Maximum Modulus Theorem
for Analytic Functions 18-4
1 5.4 Cauchy–Goursat Theorem 15-13
Exercise 18.1 18-9
1 5.5 C auchy’s Theorem for Multiply-
Connected Domain 15-14
1 5.6 Cauchy’s Integral Formula (C.I.F.) 19 Conformal Mapping
or Cauchy’s Formula Theorem 15-22
1 9.1 Introduction 19-1
1 5.7 Morera’s Theorem (Converse
1 9.2 C onformal Mapping:
of Cauchy’s Theorem) 15-23
Conditions for Conformality 19-2
1 5.8 Cauchy’s Inequality 15-23
1 9.3 Conformal Mapping by
Exercise 15.1 15-31 Elementary Functions 19-3
1 9.4 Some Special Transformations 19-6
16 Complex Power Series
1 9.5 Bilinear or Mobius or Linear
Fractional Transformations 19-17
1 6.1 Introduction 16-1
1 9.6 Fixed Points of the Transformation
1 6.2 Sequences and Series 16-1
az+b
1 6.3 Power Series 16-2 w= 19-18
cz+d
1 6.4 Series of Complex Functions 16-2
Exercise 19.1 19-26
1 6.5 U niform Convergence of a Series
of Functions 16-2
Question Bank Q-1
1 6.6 Weierstrass’s M-Test 16-3
1 6.7 Taylor’s Theorem (Taylor Series) 16-3 Index I-1
Preface
I am pleased to present this edition of Engineering Mathematics-II to the fi rst year B.Tech. students.
The topics have been dealt with in a coherent manner, supported by illustrations for better comprehension.
Each chapter is replete with examples and exercises, along with solutions and hints, wherever necessary.
The book also has a question bank, with numerous Multiple Choice Questions, Fill in the Blanks, and
Match the Following and True or False statements, thus providing the student with an abundant repository of
exam specifi c problems.
Suggestions for the improvement of the book are welcome and will be gratefully acknowledged.
Acknowledgements
I express my deep sense of gratitude to Sri. Ch. Malla Reddy, Chairman, and Sri. Ch. Mahender Reddy,
Secretary, Malla Reddy Group of Institutions (MGRI), whose patronage has given me the opportunity to
write a book.
I am thankful to Prof. Madan Mohan, Director (Academics) and Col. G. Ram Reddy, Director
(Administration), MRGI; and Dr R. K. Murthy, Principal, Malla Reddy Engineering College, Secunderabad,
for their kindness, guidance and encouragement.
E. Rukmangadachari