Special Functionsand

Complex Variables

Shahnaz Bathul

Second Edition

(Engineering Mathematics III)

Shahnaz BathulProfessor and Head

Department of MathematicsJNTUH College of Engineering

Kukatpally, Hyderabad

Special Functions andComplex Variables(Engineering Mathematics III)

Second Edition

New Delhi-1100012010

SPECIAL FUNCTIONS AND COMPLEX VARIABLES (Engineering Mathematics III)Second EditionShahnaz Bathul

© 2010 by PHI Learning Private Limited, New Delhi. All rights reserved. No part of this bookmay be reproduced in any form, by mimeograph or any other means, without permission inwriting from the publisher.

ISBN-978-81-203-4193-7

The export rights of this book are vested solely with the publisher.

Third Printing (Second Edition) ººººº ººººº ººººº September, 2010

Published by Asoke K. Ghosh, PHI Learning Private Limited, M-97, Connaught Circus,New Delhi-110001 and Printed by Baba Barkha Nath Printers, Bahadurgarh, Haryana-124507.

Tomy beloved parents

Late Syed Abdul Hannan Saheband

Late Smt. Saidunnisabi Saheba

v

Preface .............................................................................................................. ix

1 SPECIAL FUNCTIONS–I ............................................................. 1–52

1.1 The Gamma Function ..................................................................... 1WORKED-OUT PROBLEMS.................................................................... 3

1.2 The Beta Function .......................................................................... 81.2.1 Relation between b and G Functions ................................. 91.2.2 Applications of b and G Functions .................................... 9WORKED-OUT PROBLEMS.................................................................. 10

1.3 Bessel’s Function .......................................................................... 261.3.1 Recurrence Relations ...................................................... 281.3.2 Generating Function for Jn(x) ......................................... 30WORKED-OUT PROBLEMS.................................................................. 31

Multiple Choice Questions ..................................................................... 42True/False ............................................................................................... 48Exercises ................................................................................................. 49

2 SPECIAL FUNCTIONS–II .......................................................... 53–93

2.1 Legendre’s Equation and Legendre’s Function ............................ 532.1.1 Legendre’s Function........................................................ 552.1.2 Rodrigue’s Formula ........................................................ 56WORKED-OUT PROBLEMS.................................................................. 592.1.3 Generating Function ....................................................... 602.1.4 Recurrence Relations ...................................................... 612.1.5 Orthogonality of Pn(x) .................................................... 63WORKED-OUT PROBLEMS.................................................................. 65

CONTENTS

vi Contents

2.2 Chebyshev Polynomials ............................................................... 712.2.1 Series Solution ................................................................ 732.2.2 Generating Function ....................................................... 762.2.3 Orthogonality of Tn(x) ..................................................... 762.2.4 Recurrence Relations ...................................................... 78WORKED-OUT PROBLEMS.................................................................. 81

Multiple Choice Questions ..................................................................... 86True/False ............................................................................................... 91Exercises ................................................................................................. 91

3 COMPLEX DIFFERENTIATION ANDELEMENTARY FUNCTIONS ................................................... 94–203

3.1 Limits ............................................................................................ 943.2 Continuity ..................................................................................... 983.3 Differentiability ............................................................................ 993.4 Cauchy–Riemann Equations ...................................................... 101

3.4.1 Polar Form of Cauchy–Riemann Equations .................. 1033.5 Analytic Functions...................................................................... 104

3.5.1 Harmonic Functions ...................................................... 1043.6 Application of Analytic Functions to Flow Problems................. 113

MISCELLANEOUS WORKED-OUT PROBLEMS ....................................... 1143.7 Elementary Functions ................................................................. 157

WORKED-OUT PROBLEMS................................................................ 1583.8 Exponential Function.................................................................. 1603.9 Trigonometric Functions ............................................................ 1603.10 Hyperbolic Functions ................................................................. 163

WORKED-OUT PROBLEMS................................................................ 1643.11 Logarithmic Function of a Complex Variable ............................ 166

MISCELLANEOUS WORKED-OUT PROBLEMS ....................................... 167Multiple Choice Questions ................................................................... 182True/False ............................................................................................. 195Exercises ............................................................................................... 196

4 COMPLEX INTEGRATION .................................................... 204–252

4.1 Line Integral ............................................................................... 204WORKED-OUT PROBLEMS................................................................ 205

4.2 Cauchy’s Theorem for Multiple Connected Region ................... 2204.2.1 Cauchy’s Integral Formula ............................................ 221WORKED-OUT PROBLEMS................................................................ 222

Multiple Choice Questions ................................................................... 241True/False ............................................................................................. 247Exercises ............................................................................................... 248

Contents vii

5 COMPLEX POWER SERIES ................................................ 253–289

5.1 Taylor’s Theorem ........................................................................ 2535.2 Laurent Series ............................................................................. 254

WORKED-OUT PROBLEMS................................................................ 256Multiple Choice Questions ................................................................... 276True/False ............................................................................................. 284Exercises ............................................................................................... 285

6 CONTOUR INTEGRATION ................................................... 290–367

6.1 Singular Points ............................................................................ 2906.2 Calculus of Residues .................................................................. 291

6.2.1 Cauchy’s Residue Theorem .......................................... 292WORKED-OUT PROBLEMS................................................................ 293

6.3 Evaluation of Real Integrals in Unit Circle ................................ 315WORKED-OUT PROBLEMS................................................................ 315

6.4 Contour Integration When the Poles Lie on Imaginary Axis ...... 3306.4.1 Property of Definite Integral ......................................... 330WORKED-OUT PROBLEMS................................................................ 331

6.5 Evaluation of the Integrals of the Type ( )imx

-e f x dx

•

•Ú ............. 342

WORKED-OUT PROBLEMS................................................................ 3436.6 Integration by Indentation .......................................................... 348

WORKED-OUT PROBLEMS................................................................ 349Multiple Choice Questions ................................................................... 353True/False ............................................................................................. 359Exercises ............................................................................................... 361

7 THE CONFORMAL MAPPINGS ........................................... 368–445

7.1 Introduction ................................................................................ 3687.2 Conformal Transformation ......................................................... 3687.3 Some Special Conformal Mappings (Transformations) ............. 3707.4 The Bilinear Transformation ...................................................... 393

7.4.1 Invariant Points or Fixed Points of BilinearTransformation .............................................................. 396

WORKED-OUT PROBLEMS................................................................ 399Multiple Choice Questions ................................................................... 432True/False ............................................................................................. 438Exercises ............................................................................................... 439

8 ELEMENTARY GRAPH THEORY ........................................ 446–505

8.1 Some Definitions ........................................................................ 4468.2 Planar Graphs ............................................................................. 451

viii Contents

8.3 Complete Graph ......................................................................... 4528.4 Adjacency and Incidence Matrices ............................................. 453

8.4.1 Advantages of Adjacency Matrix .................................. 4548.4.2 Incidence Matrix ........................................................... 455

8.5 Distance between Two Vertices .................................................. 4558.6 Labelled and Unlabelled Graphs ................................................ 4568.7 Euler formula .............................................................................. 458

WORKED-OUT PROBLEMS................................................................ 4598.8 Trees ........................................................................................... 465

8.8.1 Binary Trees .................................................................. 4678.8.2 Spanning Trees .............................................................. 4698.8.3 Minimal Spanning Tree ................................................. 470WORKED-OUT PROBLEMS................................................................ 471

8.9 Eulerian and Hamiltonian Circuits ............................................. 483WORKED-OUT PROBLEMS................................................................ 485

Multiple Choice Questions ................................................................... 487True/False ............................................................................................. 499Exercises ............................................................................................... 501

9 ARGUMENT PRINCIPLE AND ROUCHE’S THEOREM...... 506–521

9.1 Introduction ................................................................................ 5069.2 Morera’s Theorem: The Converse of Cauchy’s Theorem ........... 5069.3 Liouville’s Theorem ................................................................... 5079.4 Fundamental Theorem of Algebra .............................................. 5089.5 The Argument Theorem .............................................................. 5099.6 Rouche’s Theorem ...................................................................... 510WORKED-OUT PROBLEMS .......................................................................... 485Multiple Choice Questions ................................................................... 517True/False ............................................................................................. 519Exercises ............................................................................................... 520

INDEX ............................................................................................. 523–525

ix

Special functions and complex variables are two very important and usefultopics in engineering mathematics. This book, now in its Second Edition,provides a detailed discussion on the important concepts of special functions,complex variables and graph theory and analyzes their applications in a guidedmanner. The book comprises nine chapters, in which the first two chapters dealwith special functions, third to seventh and ninth with complex variables andeight with the graph theory.

Chapters 1 and 2 provide a thorough analysis of various special functionssuch as gamma and beta functions and their relations, Legendre’s equation andLegendre’s function, and Bessel’s function. The properties, recurrence relations,generating functions, and orthogonality of different functions are evaluated andare illustrated with the help of worked-out problems.

Chapter 3 discusses complex differentiation. Analytic and harmonicfunctions are defined and Cauchy–Riemann equations in Cartesian and polarcoordinates are analyzed. It also deals with elementary functions. The exponentialfunction is introduced, and with the help of this function, the trigonometric andhyperbolic and the inverse of these functions are examined. The theories ofcomplex integration are covered in Chapter 4. Cauchy’s integral theorem,Cauchy’s integral formula and the generalized integral formula are explainedin detail.

Chapter 5 provides an exclusive analysis of the complex power series. Itdiscusses Taylor’s and Laurent series in detail and explains the concept ofsingularity. The calculus of residue is the topic discussed in Chapter 6. Theresidue theorem is evaluated with regard to real integrals in unit circle and inupper half plane. Chapter 7 covers the concept of conformal mappings anddevelops the theory of bilinear transformation. Chapter 8 deals with graph

PREFACE

x Preface

theory and its application. Chapter 9 explains the argument principle andRouche’s theorem for the determination of the number of zeros of complexpolynomials. The fundamental theorem of algebra is also discussed.

Throughout the book, the concepts are discussed with the help of worked-out problems. These problems are very important and standard, and hence, willhelp students in mastering the concepts. At the end of each chapter, a set eachof multiple-choice questions and exercises is provided. The objective questionswill help students in preparing for the competitive examinations.

The topics covered in this book are taught in most of the engineeringcourses in Indian universities, and hence, the book will be an important textbookfor the engineering students. Besides, undergraduate and postgraduate studentsof mathematics will also be benefited by the book.

I am thankful to PHI Learning, the publishers, for accepting to publish thisbook. My sincere thanks are due to the staff of the editorial and productiondeapartment, for bringing out this book in a nice format and within a short spanof time.

Suggestions for the improvement of the book are most welcome.

Shahnaz Bathul

Special Functions And Complex Variables: (Engineering Mathematics Iii)

Publisher : PHI Learning ISBN : 9788120341937 Author : BATHUL,SHAHNAZ

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