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
Eulers Formula | 10 |
Transformations and Euclidean Geometry | 30 |
Exercises | 45 |
Complex Functions as Transformations | 55 |
Power Series | 64 |
The Exponential Function | 79 |
Multifunctions | 90 |
The Logarithm Function | 98 |
Exercises | 328 |
Winding Numbers and Topology | 338 |
Polynomials and the Argument Principle | 344 |
Rouchés Theorem | 353 |
The Generalized Argument Principle | 363 |
Exercises | 369 |
Cauchys Theorem | 377 |
The Complex Integral | 383 |
Exercises | 111 |
Möbius Transformations and Inversion | 122 |
Construction Using Orthogonal Circles | 128 |
Three Illustrative Applications of Inversion | 136 |
Basic Results | 148 |
Möbius Transformations as Matrices | 156 |
Visualization and Classification | 162 |
Decomposition into 2 or 4 Reflections | 172 |
Exercises | 181 |
The Amplitwist Concept | 189 |
The CauchyRiemann Equations | 207 |
Further Geometry of Differentiation | 216 |
Visual Differentiation of logz | 222 |
Visual Differentiation of the Power Function | 229 |
Celestial Mechanics | 241 |
Analytic Continuation | 247 |
Exercises | 258 |
NonEuclidean Geometry | 267 |
Spherical Geometry | 278 |
Hyperbolic Geometry | 293 |
Conjugation | 392 |
The Exponential Mapping | 401 |
Parametric Evaluation | 409 |
The General Formula of Contour Integration | 418 |
Cauchys Formula and Its Applications | 427 |
Calculus of Residues | 434 |
Annular Laurent Series | 442 |
Physics and Topology | 450 |
Winding Numbers and Vector Fields | 456 |
Flows on Closed Surfaces | 462 |
Exercises | 468 |
Exercises | 505 |
Conformal Invariance | 513 |
The Complex Curvature Revisited | 520 |
Flow Around an Obstacle | 527 |
The Physics of Riemanns Mapping Theorem | 540 |
Dirichlets Problem | 554 |
Exercises | 570 |
| 579 | |
Other editions - View all
Common terms and phrases
algebraic amplitwist analytic function analytic mapping angle arbitrary Argument Principle branch point C₁ Cauchy's centred Chapter complex function complex numbers complex plane complex potential conformal mapping consider constant contour convergence critical point curvature curve deduce defined derivative dipole direct motion disc of convergence distance equal equation Euclidean Euclidean geometry example exercise fact Figure fixed points flow flux formula geometric h-lines harmonic hyperbolic geometry hyperbolic plane illustrated infinitesimal infinity inside integral interior intersection line-segment Möbius transformation multiplication obtain origin orthogonal pole Pólya vector field polynomial power series preimages pseudosphere radius real axis real number reflection region result Riemann Riemann sphere rotation round simple loop singularity sphere square stereographic projection streamlines surface symmetric tangent Theorem triangle unit circle unit disc upper half-plane vanish verify vertical winding number yields