Visual Complex AnalysisThis 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. |
Contents
Geometry and Complex Arithmetic | 1 |
5 | 22 |
Transformations and Euclidean Geometry | 30 |
4 | 39 |
Complex Functions as Transformations | 56 |
1 | 79 |
Multifunctions | 90 |
An Example with Two Branch Points | 96 |
43 | 317 |
IV | 328 |
Winding Numbers and Topology | 338 |
IV | 346 |
3 | 349 |
VI | 355 |
The Generalized Argument Principle | 363 |
Exercises | 369 |
IX | 102 |
Möbius Transformations and Inversion | 122 |
23 | 136 |
3 | 143 |
VI | 156 |
Visualization and Classification | 162 |
123 | 169 |
5 | 175 |
X | 181 |
14 | 184 |
The Amplitwist Concept | 198 |
V | 199 |
The CauchyRiemann Equations | 207 |
III | 211 |
3 | 216 |
Rules of Differentiation | 223 |
VI | 229 |
Celestial Mechanics | 235 |
30 | 239 |
34 | 245 |
XI | 247 |
XII | 258 |
NonEuclidean Geometry | 267 |
A Conformal Map of the Sphere | 283 |
37 | 303 |
Cauchys Theorem | 377 |
III | 383 |
V | 388 |
3 | 395 |
The Exponential Mapping | 401 |
5 | 408 |
XI | 414 |
Exercises | 420 |
Cauchys Formula and Its Applications | 432 |
Annular Laurent Series | 442 |
Physics and Topology | 450 |
Winding Numbers and Vector Fields | 456 |
Flows on Closed Surfaces | 462 |
Exercises | 468 |
3 | 496 |
6 | 503 |
IV | 518 |
The Method of Images | 532 |
VI | 540 |
Dirichlets Problem | 554 |
Exercises | 570 |
| 579 | |
| 581 | |
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Common terms and phrases
algebraic amplification amplitwist analytic function analytic mapping angle arbitrary arrow branch point centred Chapter complex function complex numbers complex plane conformal mapping consider const constant convergence corresponding critical point curvature curve deduce defined derivative differential direct motion distance elliptic equal equation Euclidean geometry example exercise fact Figure fixed points flow flux formula h-lines h-reflections h-rotation h-translation hyperbolic geometry hyperbolic plane illustrated image points infinitely infinitesimal infinity inside integral intersection line-segment linear log(z loop matrix Möbius transformation multiplier non-Euclidean geometry obtain orbit origin origin-centred orthogonal Poincaré disc polynomial power series pseudosphere quaternion radius real axis reflection region result Riemann Riemann sphere rotation round segment simple singularity sphere stereographic projection streamlines surface symmetric tangent Theorem tractrix triangle unit circle unit disc upper half-plane vanish vector field verify vertical winding number