Visual Complex Analysis

FROM THE Please do NOT buy this original 1997 edition! I have created a new, greatly improved (and cheaper!) edition, which was published on February 28th, 2023: Visual Complex 25th Anniversary Edition (with a new Foreword by Roger Penrose) ************************************************************************************************** This radical first course on complex analysis brings a beautiful and powerful subject to life by consistently using geometry (not calculation) as the means of explanation. Aimed at undergraduate students in mathematics, physics, and engineering, the book's intuitive explanations, lack of advanced prerequisites, and consciously user-friendly prose style will help students to master the subject more readily than was previously possible. The key to this is the book's use of new geometric arguments in place of the standard calculational ones. These geometric arguments are communicated with the aid of hundreds of diagrams of a standard seldom encountered in mathematical works. A new approach to a classical topic, this work will be of interest to students in mathematics, physics, and engineering, as well as to professionals in these fields. Cover......Page 1 Visual Complex Analysis - Tristan Needham (OUP, 1997)......Page 2 Preface......Page 15 Acknowledgements......Page 20 Contents......Page 6 Historical Skentch......Page 22 Bombelli's 'Wild Thought'......Page 24 Some terminology and notation......Page 27 Practice......Page 28 Equivalence of Symbolic and geometric arithmetic......Page 29 Introduction......Page 31 Moving particle argument......Page 32 Power series argument......Page 33 Introduction......Page 35 Trigonometry......Page 36 Geometry......Page 37 Calculus......Page 41 Algebra......Page 43 Vectorial operations......Page 48 Geometry through the eyes of Felix Klein......Page 51 Classifying motions......Page 55 Three reflections theorem......Page 58 Similarities and Complex arithmetic......Page 60 Spatial complex numers?......Page 64 Excercises......Page 66 Introduction......Page 76 Positive Integer Powers......Page 78 Cubics revisited *......Page 80 Cassinian Curves *......Page 81 The mystery of real power series......Page 85 The disc of convergence......Page 88 Approximating a power series with a polynomial......Page 91 Uniqueness......Page 92 Manipulating power series......Page 93 Finding the radius of convergence......Page 95 Fourier series*......Page 98 Power series approach......Page 100 The geometry of the mapping......Page 101 Another approach......Page 102 Definitions and identities......Page 105 Relation to hyperbolic functions......Page 107 The geometry of the mapping......Page 109 Example: Fractional powers......Page 111 Single-valued branches of a multifunction......Page 113 Relevance to power series......Page 116 An example with two branch points......Page 117 Inverse of the exponential function......Page 119 The logarithmic power series......Page 121 General powers......Page 122 The centroid......Page 123 Averaging over regular polygons......Page 126 Averaging over circles......Page 129 Exercises......Page 132 Connection with Einstein's theory of relativity*......Page 143 Preliminary definitions and facts......Page 145 Preservation of circles......Page 147 Construction using orthogonal circles......Page 150 Preservation of angles......Page 151 Inversion in a sphere......Page 154 A problem on touching circles......Page 157 Quadrilaterals with orthogonal diagonals......Page 158 Ptolemy's theorem......Page 159 The point at infinity......Page 160 Stereografic projection......Page 161 Transferring complex functions to the sphere......Page 164 Behaviour of functions at infinity......Page 165 Stereographic formulae......Page 167 Preservation of circles, angles and symmetry......Page 169 Non-uniqueness of the coefficients......Page 170 The group property......Page 171 Fixed points......Page 172 Fixed points at infinity......Page 173 The cross-ratio......Page 175 Evidence of a link with linear algebra......Page 177 The explanation: Homogeneous coordinates......Page 178 Eigenvectors and eigenvalues......Page 180 Rotations of the sphere......Page 182 The main idea......Page 183 Elliptic, hiperbolic, and loxodromic types......Page 185 Local geometric inerpretation of the multipler......Page 187 Parabolic transformations......Page 189 Computing the multipler......Page 190 Eingenvalue interpretation of the multipler......Page 191 Elliptic case......Page 193 Hyperbolic case......Page 194 Parabolic case......Page 195 Summary......Page 196 Counting derrees of freedom......Page 197 Finding the formula via the symmetry principie......Page 198 Interpreting the formula geometrically......Page 199 Introduction to Riemann's Mapping Theorem......Page 201 Exercises......Page 202 A puzzling phenomenon......Page 210 Introduction......Page 212 The jacobian matrix......Page 213 The amplitwist concept......Page 214 The real derivative re-examined......Page 215 The complex derivative......Page 216 Analytic functions......Page 218 A brief summary......Page 219 Some simple examples......Page 220 Introduction......Page 221 Conformality throughout a region......Page 222 Conformality and the Riemann sphere......Page 224 Degrees of crushing......Page 225 Breakdown of conformality......Page 226 Branch points......Page 227 Introduction......Page 228 The geometry of linear transformations......Page 229 The Cauchy-Riemann equations......Page 230 Exercises......Page 232 The cartesian form......Page 237 The polar form......Page 238 An intimation of rigidity......Page 240 Visual differentiation of log(z)......Page 243 Composition......Page 244 Inverse functions......Page 245 Addition and multiplication......Page 246 Polynomials......Page 247 Power series......Page 248 Rational functions......Page 249 Visual differentiation of the power function......Page 250 Visual differentiation of exp(z)......Page 252 Geometric solution of E'=E......Page 253 Introduction......Page 255 Analytic transformation of curvature......Page 256 Complex curvature......Page 259 Two kinds of elliptical orbit......Page 262 Changing the first into the second......Page 264 The geometry of force......Page 265 An explanation......Page 266 The Kasner-Arnold's theorem......Page 267 Introduction......Page 268 Rigidity......Page 270 Uniqueness......Page 271 Preservation of indentities......Page 272 Analytic continuation via reflections......Page 273 Exercises......Page 279 The parallel axiom......Page 288 Some facts from non-euclidean geometry......Page 290 Geometry on a curved surface......Page 292 Gaussian curvature......Page 294 Surfaces of constant curvature......Page 296 The connection with Moebius transformations......Page 298 The angular excess of a spherical triangle......Page 299 Motions of the sphere......Page 300 A conformal map of the sphere......Page 304 Spatial rotations as Moebius transformations......Page 307 Spatial Rotations and quaternions......Page 311 The tractix and the pseudosphere......Page 314 The constant curvature of the pseudosphere......Page 316 A conformal map of the pseudosphere......Page 317 Beltrami's hiperbolic plane......Page 319 Hiperbolic lines and reflections......Page 322 The Bolyai-Lobachevsky formula......Page 326 The three types of direct motion......Page 327 Decomposition into two reflections......Page 332 The angular excess of a hiperbolic triangle......Page 334 The Poincare disc......Page 337 Motions of the Poincare disc......Page 340 The hemisphere model and hyperbolic space......Page 343 Exercises......Page 349 Definition......Page 359 What does 'inside' mean?......Page 360 Finding winding numbers quickly......Page 361 The result......Page 362 Loops as mappings of the circle*......Page 363 The explanation*......Page 364 Polynomials and the argument principie......Page 365 Counting preimages algebraically......Page 367 Counting preimages geometrically......Page 368 Topological characteristics of analyticity......Page 370 A topological argument principie......Page 371 Two examples......Page 373 The result......Page 374 Brouwer's fixed point theorem*......Page 375 Maximum-modulus theorem......Page 376 Schwarz's lemma......Page 378 Liouville's theorem......Page 380 Pick's result......Page 381 Rational functions......Page 384 Poles and essential singularities......Page 386 The explanation*......Page 388 Exercises......Page 390 Introduction......Page 398 The Riemann sum......Page 399 The trapezoidal rule......Page 400 Geometric estimation of errors......Page 401 Complex Riemann sums......Page 404 A useful inequality......Page 407 Rules of integration......Page 408 A circular arc......Page 409 General loops......Page 411 Winding number......Page 412 Introduction......Page 413 Area interpretation......Page 414 Integration along a circular arc......Page 416 General contours and the deformation theorem......Page 418 A further extension of the theorem......Page 420 Residues......Page 421 The exponential mapping......Page 422 Introduction......Page 423 An example......Page 424 The fundamental theorem......Page 425 The integral as antiderivate......Page 427 Logaritm as integral......Page 429 Parametric evaluation......Page 430 Some preliminaries......Page 431 The explanation......Page 433 The result......Page 435 The explanation......Page 436 A simpler explanation......Page 438 The general formula of contour integration......Page 439 Exercises......Page 441 First explanation......Page 448 General Cauchy formula......Page 450 Infinity differentiability......Page 452 Taylor series......Page 453 Laurent series centred at a pole......Page 455 A formula for calculating residues......Page 456 Application to real integrals......Page 457 Calculating residues using taylor series......Page 459 Application to summation of series......Page 460 Laurent's theorem......Page 463 Exercises......Page 467 Complex functions as vector fields......Page 471 Physical vector fields......Page 472 Flows and force fields......Page 474 Sources and sinks......Page 475 The index of a singular point......Page 477 The index according to Poincare......Page 480 The index theorem......Page 481 Formulation of the Poincare-Hopf theorem......Page 483 Defining the index on a surface......Page 485 An explanation fo the Poincare-Hopf theorem......Page 486 Exercises......Page 489 Flux......Page 493 Work......Page 495 Local flux and local work......Page 497 Divergence and crul in geometric form*......Page 499 Divergence-free and crul-free vector fields......Page 500 The Polya vector field......Page 502 Cauchy's theorem......Page 504 Example: Area as flux......Page 505 Example: Winding number as flux......Page 506 Local behaviour of vector fields*......Page 507 Cauchy's formula......Page 509 Positive powers......Page 510 Negative powers and multipoles......Page 511 Multipoles at infinity......Page 513 Laurent's series as a multipole expansion......Page 514 The stream function......Page 515 The gradient field......Page 518 The potential function......Page 519 The complex potential function......Page 521 Examples......Page 524 Exercises......Page 526 Dual flows......Page 529 Harmonic duals......Page 532 Conformal invariance of harmonicity......Page 534 Conformal invariance of the Laplacian......Page 536 The meaning fo the Laplacian......Page 537 A powerful computational tool......Page 538 The curvature of harmonic equipotentials......Page 541 Further complex curvature calculations......Page 544 Further geometry of the complex curvature......Page 546 Introduction......Page 548 An example......Page 549 The metoth of images......Page 553 Mapping one flow onto another......Page 559 Introduction......Page 561 Exterior mappings and flows round obstacles......Page 562 Interior mappings and dipoles......Page 565 Interior mappings, vortices, and sources......Page 567 An example: automorphisms of the disc......Page 570 Green's function......Page 571 Introduction......Page 575 Schwarz's interpretation......Page 577 Dirichlet's problem for the disc......Page 579 The interpretations of Neumann and Boecher......Page 581 Green general formula......Page 586 Exercises......Page 591 References......Page 594 Index......Page 600 This radical approach to complex analysis replaces the standard calculational arguments with new geometric ones. With several hundred diagrams, and far fewer prerequisites than usual, this is the first visually intuitive introduction to complex analysis. As a new approach to a classical topic, this work will be of interest to professionals in mathematics, physics, and engineering, as well as to students in these fields.