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Engineering Mathematicsfor Semesters III and IVPrelims CB-eM-ii.indd 14/28/2016 4:46:54 PM

About the AuthorsC B Gupta is presently working as Professor in the Department of Mathematics, Birla Institute of Technologyand Science, Pilani (Rajasthan). With over 25 years of teaching and research experience, he is a recipientof numerous awards like the Shiksha Rattan Puraskar 2011, Best Citizens of India Award 2011, Glory ofIndia Award 2013, and Mother Teresa Award 2013. He was listed in Marquis’ Who’s Who in Science andTechnology in the World 2010 and 2013, and in top 100 scientists of the world in 2012. He obtained hismaster’s degree in Mathematical Statistics and PhD in Operations Research from Kurukshetra University,Kurukshetra. His fields of specialization are Applied Statistics, Optimization, and Operations Research. Anumber of students have submitted their thesis/dissertations on these topics under his supervision. He haspublished a large number of research papers on these topics in peer-reviewed national and internationaljournals of repute. He has authored/co-authored 12 books on Probability and Statistics, QuantitativeMethods, Optimization in Operations Research, Advance Discrete Mathematics, Engineering MathematicsI–III, Advanced Mathematics, and the like. He is also on the editorial board and a reviewer of many nationaland international journals. Dr. Gupta is a member of various academic and management committees of manyinstitutes/universities. He has participated in more than 30 national and international conferences in whichhe has delivered invited talks and chaired technical sessions. He has been a member of Rajasthan Board ofSchool Education, Ajmer, and also a member of various committees of RPSC Ajmer, UPSC, New Delhi, andAICTE, New Delhi.S R Singh is presently working as an Associate Professor in the Department of Mathematics at ChaudharyCharan Singh University, Meerut (Uttar Pradesh) and has an experience of 20 years in academics and research.His areas of specialization are Inventory Control, Supply-Chain Management, and Fuzzy Set Theory. He hasattended various seminars/conferences. Fifteen students have been awarded PhD under his supervision. Hehas published more than hundred research papers in reputed national and international journals. His researchpapers have been published in International Journal of System Sciences, Asia Pacific Journal of OperationalResearch, Control and Cybernetics, Opsearch, International Journal of Operational Research, Fuzzy Setsand Systems, and International Journal of Operations and Quantitative Management. He has authored/coauthored nine books.Mukesh Kumar is presently working as an Associate Professor in the Department of Mathematicsat Graphic Era University, Dehradun (Uttarakhand). He received an MPhil in Mathematics from IndianInstitute of Technology, Roorkee, and PhD in Mathematics from HNB Garhwal Central University, Srinagar,Uttarakhand. He has academic experience of more than 12 years. His fields of specialization are InventoryControl, Supply Chain Management, Operations Research and Applied Mathematics. He has publishednumerous research papers in national and international reputed journals. He is also on the editorial boardand reviewer of many national and international journals and have authored two books on mathematics.Prelims CB-eM-ii.indd 24/28/2016 4:46:54 PM

Engineering Mathematicsfor Semesters III and IVC B GuptaProfessorDepartment of MathematicsBirla Institute of Technology and Science (BITS)Pilani, RajasthanS R SinghAssociate ProfessorDepartment of MathematicsChaudhary Charan Singh UniversityMeerut, Uttar PradeshMukesh KumarAssociate ProfessorDepartment of MathematicsGraphic Era UniversityDehradun, UttarakhandMcGraw Hill Education (India) Private LimitedNew DelhiMcGraw Hill Education OfficesNew Delhi New York St Louis San Francisco Auckland Bogotá CaracasKuala Lumpur Lisbon London Madrid Mexico City Milan MontrealSan Juan Santiago Singapore Sydney Tokyo TorontoPrelims CB-eM-ii.indd 34/28/2016 4:46:55 PM

McGraw Hill Education (India) Private LimitedPublished by McGraw Hill Education (India) Private LimitedP-24, Green Park Extension, New Delhi 110 016Engineering Mathematics for Semesters III and IVCopyright 2016, 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 ofthe publishers. The program listing (if any) may be entered, stored and executed in a computer system, but they may notbe reproduced for publication.This edition can be exported from India only by the publishers,McGraw Hill Education (India) Private Limited.Print EditionISBN 13: 978-93-858-8050-6ISBN 10: 93-858-8050-0EBook EditionISBN 13: 978-93-858-8051-3ISBN 10: 93-858-8051-9Managing Director: Kaushik BellaniDirector—Products (Higher Education and Professional): Vibha MahajanManager—Product Development: Koyel GhoshSpecialist—Product Development: Sachin KumarHead—Production (Higher Education and Professional): Satinder S BavejaSenior Copy Editor: Kritika LakheraSenior Production Executive: Suhaib AliAsst. Gen. Manager—Product Management (Higher Education and Professional): Shalini JhaManager—Product Development: Ritwick DuttaGeneral Manager—Production: Rajender P GhanselaManager—Production: Reji KumarInformation 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 informationpublished 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 Text-o-Graphics, B-1/56, Aravali Apartment, Sector-34, Noida 201 301, and printed atCover Printer:Visit us at: www.mheducation.co.inPrelims CB-eM-ii.indd 44/28/2016 4:46:55 PM

ContentsPreface xvii1. Fourier Transforms 1.11.21.31.41.51.61.71.81.91.101.11Introduction Fourier Integral Theorem Fourier Integral in Complex Form Fourier Transform Fourier Cosine and Sine Transforms Properties of Fourier Transform Parseval’s Theorem Fourier Transform of Some Basic Functions Discrete Fourier Transform Exercise 1.1 Answers Finite Fourier Transform Inverse Finite Fourier Transform Exercise 1.2 Answers Exercise 1.3 Answers Summary Objective Type Questions Answers 2. Z-Transforms 2.12.22.32.4Prelims CB-eM-ii.indd 5Introduction Basic Concept of Sequence Z-Transform Exercise 2.1 Answers The Inverse Z-Transform Exercise 2.2 Answers 12.42.92.92.102.132.134/28/2016 4:46:55 PM

vi2.52.62.72.82.9ContentsProperties of the Z-Transform Convolution of Sequences Table of Z-transforms Some Useful Inverse Z-transform Solution of Difference Equations using Z-Transforms Exercise 2.3 Answers Exercise (Mixed Problems) Answers Summary Objective Type Questions Answers 3. Complex Variables and Calculus .28Prelims CB-eM-ii.indd 6Complex Number Equality of Complex Numbers Fundamental Operations with Complex Numbers Division of Complex Numbers Modulus of a Complex Number Geometrical Representation of Complex Numbers Polar Form of a Complex Numbers Conjugate Complex Number De Moivre’s Theorem Roots of a Complex Number Euler’s Formula Exponential (or Eulerian) Form of a Complex Number Circular Functions Hyperbolic Functions Real and Imaginary Parts of Circular Function Logarithm of a Complex Number Exercise 3.1 Answers Summation of Trigonometric Series – (C iS) Method Introduction to Theory of Complex Variables Basic Concepts of the Complex Variable Cauchy–Reimann equations Harmonic and Conjugate Harmonic Functions Method of Constructing Conjugate Function Method of Constructing an Analytic Function or a Regular Function Determination of Velocity Potential and Stream Function Exercise 3.2 Answers Introduction to Complex Integration Line Integral in Complex Plane Complex Function Integrals Properties of Complex Integrals 3.403.413.423.423.433.434/28/2016 4:46:55 PM

Contents 413.423.433.44Exercise 3.3 Answer Cauchy Fundamental Theorem Cauchy’s Theorem Cauchy’s Integral Formula Cauchy Integral Formula for the Derivative of an Analytic Function Cauchy Integral Formula for Higher Order Derivatives Poisson’s Integral Formula Morera’s Theorem (Converse of Cauchy’s Theorem) Fundamental Theorem of Integral Calculus Cauchy’s Inequality Theorem Liouville’s Theorem Expansion of Analytic Functions as Power Series Exercise 3.4 Zeros of an Analytic Function Singularities Exercise 3.5 Answer The Calculus of Residues Cauchy’s Residue Theorem Exercise 3.6 Answers Evaluation of Real Definite Integrals 63.573.733.733.743.763.783.793.803.873.883.88 3.45 Improper Real Integrals of the Form3.463.473.483.493.50Úf ( z ) dz Improper Integrals with Poles on the Real Axis Exercise 3.7 Conformal Mapping Bilinear (or Mobius or Fractional) Transformation Cross Ratio Applications of Complex Variables Exercise 3.8 Answer Summary Objective-Type Questions Answers 4. Empirical Laws and Curve Fitting 4.14.24.34.44.5Prelims CB-eM-ii.indd 73.94- Introduction Scatter Diagram Curve Fitting Graphical Method Exercise 4.1 Answer Least Square Method 304.1–4.324.14.14.14.24.64.64.74/28/2016 4:46:55 PM

viiiContentsExercise 4.2 Answers 4.6 Fitting of Other Curves Exercise 4.3 Answers 4.7 Group Averages Method 4.8 Fitting of a Parabola Exercise 4.4 Answers 4.9 Moments Method Exercise 4.5 Answers Summary Objective Type Questions Answers 5. Statistical Methods ction Steps of Statistical Methods Graphical Representation of Frequency Distribution Comparison of Frequency Distribution Measures of Central Tendency Exercise 5.1 Answers Measures of Dispersion Coefficient of Variation Variance of the Combined Series Exercise 5.2 Answers Skewness Kurtosis Exercise 5.3 Answers Correlation Rank Correlation Regression Exercise 5.4 Answers Summary Objective Type Questions Answers 6. Probability and Distribution 6.1 Introduction 6.2 Terminology Exercise 6.1 Prelims CB-eM-ii.indd 5.505.525.546.1–6.866.16.16.54/28/2016 4:46:55 PM

Contents .176.186.196.206.216.22Answers Definition of Probability Addition Law of Probability or Theorem of Total Probability Conditional Probability Exercise 6.2 Answers Baye’s Theorem Exercise 6.3 Answers Random Variable Types of Random Variable Discrete Probability Distribution Continuous Probability Distribution Expectation and Variance Moment Generating Function Exercise 6.4 Answers Some Important Distributions Bernoulli Distribution Binomial Distribution Exercise 6.5 Answers Poisson Distribution Constants of Poisson Distribution Exercise 6.6 Answers Uniform Distribution Exponential Distribution Exercise 6.7 Answers Normal Distribution Exercise 6.8 Answers Normal Approximation to Binomial Distribution Chebyshev’s Inequality Exercise 6.9 Answers Summary Objective Type Questions Answers 7. Sampling and Inference and Testing of Hypothesis 7.17.27.37.4Prelims CB-eM-ii.indd 9Introduction Types of Sampling Parameter and Statistic Statistical Inference 86.796.867.1–7.357.17.17.27.24/28/2016 4:46:55 PM

167.177.187.197.207.217.227.237.247.257.26Sampling Distribution Standard Error Testing a Hypothesis Errors Null and Alternate Hypothesis Level of Significance Tests of Significance Confidence Limits Simple Sampling of Attributes Test of Significance for Large Samples Comparison of Large Samples Exercise 7.1 Answers Sampling of Variables Sampling Distribution of the Mean Central Limit Theorem Confidence Limits for Unknown Mean Test of Significance for Difference of Means Exercise 7.2 Answers Sampling of Variables – Small Samples Student’s t-Distribution Significant Test of Difference between Two Samples Exercise 7.3 Answers Chi-square (c2) test Exercise 7.4 Answers F-Distribution Fisher’s Z-distribution Exercise 7.5 Answers Summary Objective Type Questions Answers 8. Finite Differences and Interpolation 8.18.28.38.48.58.68.7Prelims CB-eM-ii.indd 10Introduction Floating Point Representations Rounding-Off and Chopping Error General Error Formula Errors in Numerical Computations Error in A Series Approximation Exercise 8.1 Answers 8.68.78.98.138.144/28/2016 4:46:55 PM

Contents 8.218.228.238.24Finite Differences Relation Between Operators Factorial Notation Reciprocal Factorial Express of Any Polynomial f (x) in Factorial Notation Exercise 8.2 Answers Interpolation Newton’s-Gregory Forward Interpolation Formula Newton’s-Gregory Backward Difference Interpolation Formula Error’s in Newton’s Interpolation Polynomial Central Difference Interpolation Formulae Guidelines for the Choice of Interpolation Exercise 8.3 Answers Interpolation For unequal Intervals Lagrange’s Interpolating Polynomials Error in Lagrange’s interpolation formula Divided Differences Inverse Interpolation Hermite Interpolation Polynomial Exercise 8.4 Answers Summary Objective Type Questions Answers 9. Numerical Solution of Equations 59.16Prelims CB-eM-ii.indd 11Introduction Some Basic Properties of an Equation Bisection Method Fixed Point Iteration Method Geometrical Interpretation of Iteration Method Iteration Method for the System of Non-Linear Equations Newton’s Method Regula Falsi Method Secant Method Convergence for Iterative Methods Exercise 9.1 Answers Introduction Linear System of Equations Gaussian Elimination Method Gauss’s–Jordan Method Crout’s Method Lu Decomposition Method 19.249.269.269.269.279.309.329.354/28/2016 4:46:55 PM

xii9.179.189.19ContentsIterative Methods Exercise 9.2 Answers Matrix Inversion Exercise 9.3 Answers Eigen Value Problems Summary Objective Type Questions Answers 10. Numerical Differentiation and Integration 689.7010.1–10.34Numerical Differentiation Numerical Differentiation Using the Following Interpolation Formulae Exercise 10.1 Answers Numerical Integration Newton–Cote’s Quadrature Formula Exercise 10.2 Answers Summary Objective Type Questions Answers .3411. Numerical Solution of Ordinary Differential Equations 11.1–11.4711.111.211.311.4Introduction Initial and Boundary Value Problems Ordinary Differential Equations of First Order and First Degree Exercise 11.1 Answers Exercise 11.2 Answers Exercise 11.3 Answers Exercise 11.4 Answers Exercise 11.5 Answers Exercise 11.6 Answers Numerical Solution of Simultaneous First Order Ordinary Differential Equations Exercise 11.7 Answers 11.5 Numerical Solution of Second Order Ordinary Differential Equations Exercise 11.8 Answers Prelims CB-eM-ii.indd 404/28/2016 4:46:55 PM

Contents Summary Objective Type Questions Answers 12. Numerical Solution of Partial Differential Equations 12.112.212.312.412.512.612.712.8Introduction Classification of Partial Differential Equation Some Standard PDE’s Finite Difference Method Parabolic Partial Differential Equations Exercises 12.1 Answers Solution of Hyperbolic Equations Exercise 12.2 Answers Numerical Solution of Elliptic Partial Differential Equations Exercise 12.3 Answers Solution of Poisson’s Equation Exercise 12.4 Answers Summary Objective Type Questions Answers 13. Linear Programming 13.113.213.313.413.513.613.713.813.913.10Prelims CB-eM-ii.indd 13Introduction General Form of Linear Programming Formulation of Model Exercise 13.1 Answers Standard Form or Equation Form of Linear Programming Problem Exercise 13.2 Answers Some Important Terms Solution of a Linear Programming Problem Exercise 13.3 Answers Algebraic Solution of a Programming Problem Exercise 13.4 Answers Simplex Method Exercise 13.5 Answers Artificial Variable Technique Exceptional Cases in LPP 2213.2213.2813.2913.2913.374/28/2016 4:46:55 PM

xiv13.1113.1213.1313.14ContentsExercise 13.6 Answers Duality in Linear Programming Exercise 13.7 Answers Exercise 13.8 Answers Dual Simplex Method Exercise 13.9 Answers Transportation Problem Exercise 13.10 Answers Assignment Problem Exercise 13.11 Answers Summary Objective Type Questions Answers 14. Method of Variational with Fixed Boundaries 14.1 Introduction 14.2 Function 14.3 Functional 14.4 Difference Between Function and Functional 14.5 Closeness of Curves 14.6 Continuity of Functional 14.7 Variation of Functional 14.8 Maxima or Minima of Functionals 14.9 Fundamental Lemma of Calculus of Variation 14.10 Extermal 14.11 Some Important Lemmas 14.12 Euler’s Equation 14.13 Alternative Forms of Euler’s Equation 14.14 Variational Problems for Functional Involving Several Dependent Variablesof the Form 14.15 Functional Dependent on Higher Order Derivatives 14.16 Functionals Dependent on the Functions of Several Independent Variables 14.17 Variational Problems in Parametric Form 14.18 Isoperimetric Problems Exercise 14.1 Answers 14.19 Sufficient Conditions for an Extremum 14.20 Jacobi Condition 14.21 Weierstrass Function 14.22 Legendre Conditions Prelims CB-eM-ii.indd 3914.4114.4414.514/28/2016 4:46:55 PM

14.23Contents xvLegendre Condition for Quadratic Function Exercise 14.2 Answer Summary Objective Type Questions Answers 14.5714.5814.6014.6014.6614.6915. Integral Equations 15.1 Introduction 15.2 Integral Equation 15.3 Differentiation of a Function Under an Integral Sign 15.4 Relation between Differential and Integral Equations Exercise 15.1 Exercise 15.2 Answers Exercise 15.3 15.5 Solution of Non-Homogenous Volterra’s Integral Equation of Second Kindby the Method of Successive Substitution 15.6 Solution of Non-Homogeneous Volterra’s Integral Equation of Second Kindby the Method of Successive Approximation 15.7 Determination of Some Resolvent Kernels Exercise 15.4 Answers 15.8 Solution of the Fredholm Integral Equation by the Method ofSuccessive Substitutions 15.9 Iterated Kernels 15.10 Solution

Methods, Optimization in Operations Research, Advance Discrete Mathematics, Engineering Mathematics I–III, Advanced Mathematics, and the like. He is also on the editorial board and a reviewer of .

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