COMPLEX VARIABLES Principles and Problem Sessions A K KAPOOR University of Hyderabad, India NEW JERSEY LONDON YJ? World Scientific SINGAPORE BEIJING SHANGHAI HONG KONG TAIPEI CHENNAI
CONTENTS Preface vii Acknowledgments ix To the Reader xi Notation and Symbols xv Principles 1 1 Complex Numbers 3 11 Introduction 3 111 Complex conjugate 3 112 Polar form 3 113 Shifted polar form 5 114 Representation of angles 5 12 Examples 7 13 Linear Transformations 13 131 Complex numbers as vectors in a plane 14 132 Translation, scaling, and rotation 14 14 Reflection and Inversion 17 15 Examples 18 16 Further Topics 20 161 Point at infinity 20 162 Stereographic projection 21 163 Bilinear transformations 22 17 Notes and References 23 xvii
xviii Complex Variables: Principles and Problem Sessions 2 Elementary Functions and Differentiation 25 21 Exponential, Trigonometric, and Hyperbolic Functions 25 22 Solutions to Equations 29 23 Examples 31 24 Open Sets, Domains, and Regions 35 25 Limit, Continuity, and Differentiation 35 26 Cauchy-Riernann Equations 37 261 Cauchy-Riemann equations in polar form 38 262 Sufficiency conditions 38 27 Examples 39 28 Analytic Functions 42 281 Properties of analytic functions 42 282 Power series as an analytic function 44 29 Harmonic functions 45 210 Graphical Representation of Functions 46 211 Notes and References 49 3 Functions with Branch Point Singularity 51 31 Inverse Functions 51 311 Many-to-one functions 52 312 Inverse functions 52 313 Branch of a multivalued function 53 32 Nature of Branch Point Singularity 54 321 Multivalued function arg(z) 54 322 Single-valued branches of the square root and the cube root 56 33 Ensuring Single-Valuedness 57 331 Branch cut 60 34 Defining a Single-Valued Branch 60 341 Principal value of the logarithm and power 61 342 Examples 62 35 Multivalued Functions of z - 65 351 Sum and product of \fz + 1 and \/z 1 66 36 Discontinuity Across the Branch Cut 67 37 Examples 69 38 Inverse Trigonometric Functions 72 39 Differentiation 72 310 Riemann Surface 74 311 Summary 75 312 Notes and References 76
Contents xix 4 Integration in the Complex Plane 79 41 Improper Integrals 79 42 Definitions 82 421 Integration in the complex plane 83 43 Examples of Line Integrals in the Complex Plane 84 44 Bounds on Integrals 86 441 Jordan's lemma 88 45 Examples 89 46 Cauchy's Fundamental Theorem 92 461 Cauchy's theorem 92 462 Deformation of contours 93 463 Indefinite integral 95 47 Transforming Integrals over a Real Interval into Contour Integrals 97 471 Adding line segments or circular arcs to close the contour 97 472 Translation and rotation of the contour 98 48 Integration of Multivalued Functions 100 481 Line integrals 100 482 Indefinite integrals 101 483 Case study of the indefinite integral / y 104 484 Integration around a branch cut 107 49 Summary 110 5 Cauchy's Integral Formula Ill 51 Cauchy's Integral Formula Ill 52 Existence of Higher Order Derivatives 114 53 Taylor Series 115 54 Real Variable vs Complex Variable 117 55 Examples 118 56 Laurent Expansion 120 57 Examples 124 58 Taylor and Laurent Series for Multivalued Functions 128 59 More Results Flowing from the Integral Formula 129 510 Analytic Continuation 132 5101 Analytic continuation 132 5102 Uniqueness of analytic continuation 135 5103 Analytic function as a single entity 136 5104 Schwarz reflection principle 137 5105 An application 137 511 Summary 138
xx Complex Variables: Principles and Problem Sessions 6 Residue Theorem 141 61 Classification of Singular Points 141 611 Behavior near an isolated singular point 142 62 Finding the Order of Poles and Residues 143 63 Residue at an Isolated Singular Point 145 631 Residue at a pole 146 64 Computing the Residues 148 65 Cauchy's Residue Theorem 150 651 An integral for indented contours 151 66 Residue at Infinity 152 67 Illustrative Examples 154 68 Residue Calculus and Multivalued Functions 156 69 Zeros and Poles of a Meromorphic Function 159 610 Notes and References 160 7 Contour Integration 163 71 Rational and Trigonometric Functions 164 72 Integration Around a Branch Cut 168 73 Using Indented Contours for Improper Integrals 172 731 An example 172 74 Indented Contours for Singular Integrals 174 741 Definition using indented contours 174 742 Using the ie prescription 176 743 Cauchy principal value 177 75 Miscellaneous Contour Integrals 178 76 Series Summation and Expansion 184 761 Summation of series 184 762 Mittag-Leffler expansion 186 77 A Summary 187 8 Asymptotic Expansion 191 81 Properties of Asymptotic Expansions 192 82 Integration by Parts 193 83 Laplace's Method 195 831 Dominant term of asymptotic expansion 195 832 Full asymptotic expansion 197 84 Method of Stationary Phase 201 85 Method of Steepest Descent 203 851 Central idea 203 852 Local properties of steepest paths 204 853 Change of variable 206
Contents xxi 86 Saddle Point Method 206 87 Examples 206 88 Topics for Further Study 213 9 Conformal Mappings 215 91 Conformal Mappings 215 92 Bilinear Transformations 219 93 Examples 222 94 Mapping by Elementary Functions 228 941 Mapping w zn = 228 942 Exponential map 229 943 Map io Log = z 230 944 Map io sin2; = 230 95 Joukowski Map 231 96 Examples 235 97 Schwarz-Christoffel Transformation 236 98 Examples 239 99 Notes and References 243 10 Physical Applications of Conformal Mappings 245 101 Model Problems 245 102 Physical Applications 248 103 Steady State Temperature Distribution 249 104 Electrostatic Potential 252 105 Flow of Fluids 255 1051 Stream function and stream lines 255 106 Solutions Described by Simple Complex Potentials 258 107 Using Conformal Mappings 260 108 Method of Images 265 109 Using the Schwarz-Christoffel Transformation 267 1010 Notes and References 269 Problem Sessions 271 1 Complex Numbers 273 11 Exercise: Polar Form of Complex Numbers 273 12 Exercise: Curves in the Complex Plane 275 13 Exercise: Complex Numbers and Geometry 278 14 Tutorial: Geometric Representation 280 15 Quiz: Transformations 282 16 Exercise: Linear Transformations 284 17 Exercise: Reflections 286
xxii Complex Variables: Principles and Problem Sessions 18 Mined: Complex numbers 287 19 Mixed Bag: Complex Numbers and Transformations 290 2 Elementary Functions and Differentiation 295 21 Exercise: De Moivre's Theorem 295 22 Exercise: Real and Imaginary Parts 296 23 Questions: Hyperbolic and Trigonometric Functions 298 24 Exercise: Solutions to Equations 299 25 Mined: Solutions to Equations 301 26 Tutorial: Roots of a Complex Number 302 27 Quiz: Roots of Unity 304 28 Exercise: Continuity and Differentiation 305 29 Questions: Cauchy-Riemann Equations 307 210 Quiz: Cauchy-Riemann Equations 308 211 Tutorial: Analytic Functions 309 212 Exercise: Harmonic Functions 311 213 Mixed Bag: Differentiation and Analyticity 312 3 Functions with Branch Point Singularity 317 31 Questions: Branch Point 317 32 Tutorial: Square Root Branch Cut 320 33 Exercise: Branch Cut for JO y Z-\-0 322 34 Quiz: Discontinuity and Branch Cut 325 35 Exercise: Logarithmic Function 327 36 Exercise: Discontinuity Across the Branch Cut 329 37 Mined: Branch Point Singularity 331 38 Mixed Bag: Multivalued Functions 333 4 Integration in the Complex Plane 337 41 Questions: Range of Parameters in Improper Integrals 337 42 Tutorial: Computing Line Integrals in the Complex plane 338 43 Exercise: Evaluation of Line Integrals 339 44 Questions: Deformation of Contours 341 45 Exercise: Deformation of Contours 343 46 Exercise: Cauchy's Theorem 344 47 Tutorial: Shift of a Real Integration Variable by a Complex Number 346 48 Tutorial: Scaling of a Real Integration Variable by a Complex Number 348 49 Exercise: Shift and Scaling by a Complex Number 350
Contents xxiii 410 Exercise: Rotation of the Contour 352 411 Mixed Bag: Integration in the Complex Plane 353 5 Cauchy's Integral Formula 355 51 Exercise: Cauchy's-Integral Formula 355 52 Quiz: Circle of Convergence of Taylor Expansion 356 53 Exercise: Using MacLaurin's Theorem 358 54 Exercise: Taylor Series Representation 359 55 Tutorial: Series Expansion from the Binomial Theorem 56 Exercise: Laurent Expansion using the Binomial Theorem 361 363 57 Quiz: Subsets for Convergence of Laurent Expansions 366 58 Exercise: Laurent Expansion Near a Singular Point 367 59 Quiz: Regions of Convergence; Taylor and Laurent Series 510 Questions: Region of Convergence for Laurent Expansion 369 370 6 Residue Theorem 373 61 Questions: Classifying Singular Points 373 62 Tutorial: Isolated Singular Points 375 63 Questions: Selecting Functions with Singularities Specified 377 64 Tutorial: Residues at Simple Poles 379 65 Tutorial: Residues at Multiple Poles 381 66 Exercise: Computation of Integrals 382 67 Questions: Residue Theorem 384 68 Tutorial: Integrals of Trigonometric Functions 385 69 Exercise: Integrals of the Type f** /(cos d, sin 8)dQ 386 610 Exercise: Integrals Using the Residue at Infinity 387 611 Quiz: Finding Residues 388 612 Mined: How to Compute Residues 389 613 Mixed Bag: Residues and Integration in the Complex Plane 391 7 Contour Integration 395 71 Tutorial: fq(x)dx 395 72 Tutorial: Improper Integrals of Rational Functions 397 73 Exercise: Integrals of Type JQ(x)dx 400 74 Exercise: Integrals of with Rational Functions 401 \COS ie/ 75 Tutorial: Integration Around the Branch Cut 403 76 Exercise: Integrals of the Type J xaq{x)dx 406 77 Exercise: Integrals of the Type JlogxQ(x)dx 408 78 Tutorial: Hyperbolic Functions 409 79 Exercise: Integrals Involving Hyperbolic Functions 411 710 Tutorial: Principal Value Integrals 413
xxiv Complex Variables: Principles and Problem Sessions 711 Exercise: Integrals Requiring the Use of Indented Contours 416 712 Exercise: Series Summation and Expansion 418 713 Exercise: What You See Is Not What You Get 419 714 Exercise: Integrals from Statistical Mechanics 422 715 Exercise: Alternate Routes to Improper Integrals 424 716 Open-Ended: Killing Two Birds with One Stone 425 717 Open-Ended: Food for Your Thought 426 718 Mixed Bag: Improper Integrals 428 8 Asymptotic Expansions 431 81 Exercise: Integration by Parts 431 82 Exercise: Dominant Term 433 83 Exercise: Laplace's Method 434 84 Exercise: Steepest Paths 436 85 Tutorial: Method of Steepest Descent 439 86 Tutorial: Saddle Point Method 441 87 Exercise: Steepest Descent and Saddle Point Method 443 9 Conformal Mapping 447 91 Tutorial: Inversion Map 447 92 Exercise: Map 448 93 Exercise: Bilinear Transformation I 450 94 Questions: Bilinear Transformation II 451 95 Exercise: Symmetry Principle 453 96 Exercise: Elementary Functions 454 97 Exercise: Finding Maps 456 98 Quiz: Matching Aerofoils 458 99 Tutorial: Schwarz-Christoffel Transformation 460 910 Quiz: Schwarz-Christoffel Mapping 462 911 Exercise: Schwarz-Christoffel Mapping 463 912 Mixed Bag: Conformal Mappings 465 10 Physical Applications of Conformal Mappings 469 101 Tutorial: Temperature Distribution 469 102 Exercise: Steady State Temperature 471 103 Exercise: Electrostatics 474 104 Quiz: Nine Problems and 476 105 Exercise: Flow of Fluids 477 106 Exercise: Method of Images 479 107 Open-Ended: Using Ideas From Gauss' Law 481 108 Tutorial: Boundary Value Problems 482
Contents xxv 109 Exercise: Using Schwarz-Christoffel Mapping 484 1010 Exercise: Utilizing Conservation of Flux of Fluids 487 1011 Mixed Bag: Boundary Value Problems 489 Bibliography 491 Index 493