Engineering Mathematics - III (M302 and M402) Second Edition WBUT – 2015

778 Pages·2015·119.847 MB·English

by  Sourav Kar

Engineering
Mathematics - III
(M302 and M402)
Second Edition
WBUT – 2015
About the Authors
Sourav Kar is presently Assistant Professor of Mathematics
at the Siliguri Institute of Technology, Siliguri, West Bengal.
He completed his postgraduation and Ph.D. in Mathematics
from the University of North Bengal, West Bengal and also
obtained a B.Ed degree from SRK BEd College, Darjeeling.
He has been involved in teaching and research for more
than eleven years and has published twelve research papers
in various national and international journals of repute. He
has also presented several research papers in national and
international conferences.
Subrata Karmakar is presently Assistant Professor of
Mathematics at the Siliguri Institute of Technology, Siliguri,
West Bengal. He is a postgraduate in Mathematics from Utkal
University, Bhubaneswar, and holds a B.Ed degree from
Regional Institute of Education (NCERT), Bhubaneswar,
Orissa. He has qualified different national-level examinations
like GATE and SLET. He has been involved in teaching and
research for more than eleven years in applied mathematics.
During his service at Siliguri Institute of Technology, as an
active academician, he has participated in different workshops/
conferences/seminars conducted by DST, AICTE and UGC. He is pursuing research
from the University of North Bengal.
Dr. Kar and Mr. Karmakar have jointly published two other books Engineering
Mathematics-I and Engineering Mathematics-II for WBUT with McGraw Hill
Education.
Engineering
Mathematics - III
(M302 and M402)
Second Edition
WBUT – 2015
Sourav Kar
Assistant Professor
Department of Mathematics
Siliguri Institute of Technology
Siliguri, West Bengal, India
Subrata Karmakar
Assistant Professor
Department of Mathematics
Siliguri Institute of Technology
Siliguri, West Bengal, India
McGraw Hill Education (India) Private Limited
NEW DELHI
McGraw Hill Education Offices
New Delhi New York St Louis San Francisco Auckland Bogotá Caracas
Kuala Lumpur Lisbon London Madrid Mexico City Milan Montreal
San Juan Santiago Singapore Sydney Tokyo Toronto
McGraw Hill Education (India) Private Limited
Published by McGraw Hill Education (India) Private Limited
P-24, Green Park Extension, New Delhi 110 016
Engineering Mathematics - III (WBUT—2015), 2e
Copyright © 2012, 2015 by McGraw Hill Education (India) Private Limited.
No part of this publication may be reproduced or distributed in any form or by any means,
electronic, mechanical, photocopying, recording, or otherwise or stored in a database or
retrieval system without the prior written permission of the publishers. The program l istings
(if any) may be entered, stored and executed in a computer system, but they may not be
reproduced for publication.
This edition can be exported from India only by the publishers,
McGraw Hill Education (India) Private Limited.
ISBN (13): 978-93-392-2207-9
ISBN (10): 93-392-2207-5
Managing Director: Kaushik Bellani
Head—Products (Higher Education and Professional): Vibha Mahajan
Assistant Sponsoring Editor: Koyel Ghosh
Editorial Executive: Piyali Chatterjee
Manager—Production Systems: Satinder S Baveja
Manager—Editorial Services: Hema Razdan
Desk Editor: Jagriti Kundu
Senior Graphic Designer—Cover: Meenu Raghav
Senior Publishing Manager (SEM & Tech. Ed.): Shalini Jha
Assistant Product Manager: Tina Jajoriya
General Manager—Production: Rajender P Ghansela
Manager—Production: Reji Kumar
Information contained in this work has been obtained by McGraw Hill Education (India),
from sources believed to be reliable. However, neither McGraw Hill Education (India) nor
its authors guarantee the accuracy or completeness of any information published herein,
and neither McGraw Hill Education (India) nor its authors shall be responsible for any
errors, omissions, or damages arising out of use of this information. This work is published
with the understanding that McGraw Hill Education (India) and its authors are supplying
information but are not attempting to render engineering or other professional services.
If such services are required, the assistance of an appropriate professional should be sought.
Typeset at Mukesh Technologies Pvt. Ltd., Puducherry and Printed at Adarsh Printers,
B-47, Jhilmil Industrial, D.S.I.D.C. G.T. Road, Shahdara, Delhi 110 095
Cover Printer: SDR Printers Pvt. Ltd.
Code: RZZDRRCVDRBBY
Visit us at: www.mheducation.co.in
To my teacher
Dr. Sanjib Kr. Datta
and
my beloved family members
Sourav Kar
To the Holy Mother
Ma Sarada
Subrata Karmakar

Contents
Preface xv
Roadmap xxi
1. Fourier Series 1.1 – 1.60
1.1 Introduction 1.1
1.2 Periodic Functions and Their Properties 1.1
1.3 Nonperiodic Functions and Their Properties 1.2
1.4 Special Waveforms 1.8
1.5 Waveform Synthesis 1.10
1.6 Even and Odd Functions 1.11
1.7 Fourier Series Expansion 1.14
1.8 Convergence of Fourier Series Expansion of a Function
(Discussion of Dirichlet’s Conditions) 1. 17
1.9 Fourier Series Expansion of a Function in the Interval [–l, l ] 1.20
1.10 Fourier Series Expansion of a Function in the Interval [0, 2l ] 1.23
1.11 Fourier Series Expansion of a Function in
the Interval [-p,p] 1.25
1.12 Fourier Expansions of Discontinuous Functions 1.27
1.13 Fourier Expansions of Nonperiodic Functions 1.27
1.14 Differentiation and Integration 1.28
1.15 Waveform Symmetry and Fourier Coefficients 1.30
1.16 Half-Range Series 1.31
1.17 Parseval’s Identity 1.37
Worked Out Examples 1.41
Short and Long Answer-Type Questions 1.56
Multiple-Choice Questions 1.59
2. Fourier Transforms 2.1 – 2.52
2.1 Introduction 2.1
2.2 Complex Fourier Transform 2.2
2.3 Fourier Transforms of Some Standard Functions 2.4
2.4 Properties of Fourier Transform 2.7
2.5 Fourier Sine Transform 2.15
2.6 Properties of Fourier Sine Transform 2.17
2.7 Fourier Cosine Transforms 2.19
viii Contents
2.8 Properties of Fourier Cosine Transforms 2.23
2.9 Convolution 2.24
2.10 Parseval’s Identity 2.29
2.11 Finite Fourier Transforms 2.32
2.12 Evolution of Laplace Transform from Fourier Transforms 2.34
Worked Out Examples 2.35
Short and Long Answer-Type Questions 2.50
Multiple-Choice Questions 2.52
3. Fundamentals of Complex Analysis 3.1 – 3.48
3.1 Introduction 3.1
3.2 Basic Definitions and Notations 3.3
3.3 Functions of Complex Variable 3.6
3.4 Limit and Continuity 3.7
3.5 Differentiability of a Function 3.11
3.6 Necessary Condition for Differentiability at a Point
(Cauchy–Riemann Equation) 3.12
3.7 Sufficient Condition for Differentiability at a Point 3.12
3.8 Analytic Function 3.13
3.9 Harmonic Functions 3.15
3.10 Conjugate Harmonic 3.18
3.11 Method of Constructing an Analytic Function
(Milne Thomson Method) 3.22
3.12 Polar Form of Cauchy–Riemann Equations 3.24
Worked Out Examples 3.26
Short and Long Answer-Type Questions 3.45
Multiple-Choice Questions 3.46
4. Complex Integration and Series Expansion 4.1 – 4.42
4.1 Introduction 4.1
4.2 Curves in the Complex Plane 4.1
4.3 Riemann’s Theory of Integration 4.3
4.4 Some Results on Complex Integrals 4.4
4.5 Cauchy’s Theorem 4.6
4.6 Cauchy’s Integral Formulas 4.13
4.7 Series Expansion of an Analytic Function 4.19
Worked Out Examples 4.22
Short and Long Answer-Type Questions 4.39
Multiple-Choice Questions 4.41
Contents ix
5. Zeros, Singularities and Residues 5.1 – 5.34
5.1 Introduction 5.1
5.2 Zeros of an Analytic Function 5.1
5.3 Singularities 5.2
5.4 Classification of Isolated Singular Points 5.3
5.5 Limit Point of Zeros and Poles 5.7
5.6 Zeros and Singularities of a Function at Infinity 5.9
5.7 Entire and Meromorphic Functions 5.9
5.8 Residues 5.10
5.9 Cauchy’s Residue Theorem 5.12
5.10 Principle of Argument 5.13
Worked Out Examples 5.13
Short and Long Answer-Type Questions 5.31
Multiple-Choice Questions 5.33
6. Conformal Mapping 6.1 – 6.24
6.1 Introduction 6.1
6.2 Conformal Mapping 6.1
6.3 Bilinear Transformation 6.4
6.4 Equation of Straight Line and Circle in Complex Plane 6.10
6.5 Cross-Ratio of Four Points 6.13
Worked Out Examples 6.15
Short and Long Answer-Type Questions 6.22
Multiple-Choice Questions 6.23
7. Fundamentals of Probability Theory 7.1 –7.62
7.1 Introduction 7.1
7.2 Preliminary Notions of Sets 7.4
7.3 Event Space 7.8
7.4 Classical Definition of Probability 7.8
7.5 Statistical Regularity and Frequency Definition of Probability 7.9
7.6 Axiomatic Definition of Probability 7.11
7.7 Some Simple Deductions Considering Axiomatic Definition 7.12
7.8 Elements of Combinatorial Analysis 7.16
7.9 Conditional Probability 7.23
7.10 Baye’s Theorem 7.25
7.11 Stochastic Independence 7.28
7.12 Repeated Independent Trials 7.33
7.13 Bernoulli Trials 7.33
x Contents
7.14 Drawing with Replacement and without Replacement 7.35
Worked Out Examples 7.38
Short and Long Answer-Type Questions 7.58
Multiple-Choice Questions 7.62
8. Probability Distributions 8.1 – 8.45
8.1 Introduction 8.1
8.2 Random Variables 8.1
8.3 Distribution Function and its Properties 8.4
8.4 Discrete Distribution 8.7
8.5 Some Important Discrete Distributions 8.15
8.6 Continuous Distribution 8.15
8.7 Important Continuous Distributions 8.21
8.8 Mixed Distribution 8.23
8.9 Transformation of Random Variable 8.24
Worked Out Examples 8.26
Short and Long Answer-Type Questions 8.40
Multiple-Choice Questions 8.43
9. Mathematical Expectations and Moments 9.1 – 9.40
9.1 Introduction 9.1
9.2 Expectation or Mean of a Random Variable 9.1
9.3 Properties of Expectation or Mean of Random Variables 9.3
9.4 Variance and Standard Deviation (SD) of Random Variables 9.5
9.5 Properties of Variance and Standard Deviation (SD)
of Random Variables 9.6
9.6 Moments of a Random Variable 9.12
9.7 Central Moments 9.13
9.8 General Relation Between Raw Moments and Central Moments
of a Distribution 9.13
9.9 Skewness and Kurtosis of a Distribution 9.15
9.10 Moment Generating Function 9.19
9.11 Properties of Moment Generating Function 9.19
9.12 Characteristic Functions 9.24
9.13 Properties of Characteristic Functions 9.24
9.14 Median and Mode of a Distribution 9.24
Worked Out Examples 9.29
Short and Long Answer-Type Questions 9.36
Multiple-Choice Questions 9.39