Visual complex analysis / Tristan Needham.
By: Needham, Tristan.
Material type: BookPublisher: Oxford : New York : Clarendon Press, Oxford University Press, 1997Description: xxiii, 592 p. : ill ; 24 cm.ISBN: 0198534477.Subject(s): Functions of complex variables | Mathematical analysisDDC classification: 515.9Item type | Current library | Call number | Copy number | Status | Date due | Barcode | Item holds |
---|---|---|---|---|---|---|---|
General Lending | MTU Bishopstown Library Lending | 515.9 (Browse shelf(Opens below)) | 1 | Available | 00069948 |
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Enhanced descriptions from Syndetics:
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. Although designed for use by undergraduates in mathematics and science, the novelty of the approach will also interest professional mathematicians. PDF extracts of the book can be read at http://www.usfca.edu/vcal/.
Includes bibliographical references (pages 573-578) and index.
Geometry and complex arithmetic -- Complex functions as transformations -- Mobius transformations and inversion -- Differentiation: the amplitwist concept -- Further geometry of differentiation -- Non- euclidean geometry -- Winding numbers and topology -- Complex integration: Cauchy's theorem -- Cauchy's formula and its application -- Vector fields: physics and topology -- Vector fields and complex integration -- Flows and harmonic functions.
Table of contents provided by Syndetics
- 1 Geometry and Complex Arithmetic (p. 1)
- I Introduction (p. 1)
- II Euler's Formula (p. 10)
- III Some Applications (p. 14)
- IV Transformations and Euclidean Geometry* (p. 30)
- V Exercises (p. 45)
- 2 Complex Functions as Transformations (p. 55)
- I Introduction (p. 55)
- II Polynomials (p. 57)
- III Power Series (p. 64)
- IV The Exponential Function (p. 79)
- V Cosine and Sine (p. 84)
- VI Multifunctions (p. 90)
- VII The Logarithm Function (p. 98)
- VIII Averaging over Circles* (p. 102)
- IX Exercises (p. 111)
- 3 Mobius Transformations and Inversion (p. 122)
- I Introduction (p. 122)
- II Inversion (p. 124)
- III Three Illustrative Applications of Inversion (p. 136)
- IV The Riemann Sphere (p. 139)
- V Mobius Transformations: Basic Results (p. 148)
- VI Mobius Transformations as Matrices* (p. 156)
- VII Visualization and Classification* (p. 162)
- VIII Decomposition into 2 or 4 Reflections* (p. 172)
- IX Automorphisms of the Unit Disc* (p. 176)
- X Exercises (p. 181)
- 4 Differentiation: The Amplitwist Concept (p. 189)
- I Introduction (p. 189)
- II A Puzzling Phenomenon (p. 189)
- III Local Description of Mappings in the Plane (p. 191)
- IV The Complex Derivative as Amplitwist (p. 194)
- V Some Simple Examples (p. 199)
- VI Conformal = Analytic (p. 200)
- VII Critical Points (p. 204)
- VIII The Cauchy-Riemann Equations (p. 207)
- IX Exercises (p. 211)
- 5 Further Geometry of Differentiation (p. 216)
- I Cauchy-Riemann Revealed (p. 216)
- II An Intimation of Rigidity (p. 219)
- III Visual Differentiation of log(z) (p. 222)
- IV Rules of Differentiation (p. 223)
- V Polynomials, Power Series, and Rational Functions (p. 226)
- VI Visual Differentiation of the Power Function (p. 229)
- VII Visual Differentiation of exp(z) (p. 231)
- VIII Geometric Solution of E' = E (p. 232)
- IX An Application of Higher Derivatives: Curvature* (p. 234)
- X Celestial Mechanics* (p. 241)
- XI Analytic Continuation* (p. 247)
- XII Exercises (p. 258)
- 6 Non-Euclidean Geometry* (p. 267)
- I Introduction
- II Spherical Geometry (p. 278)
- III Hyperbolic Geometry (p. 293)
- IV Exercises (p. 328)
- 7 Winding Numbers and Topology (p. 338)
- I Winding Number (p. 338)
- II Hopf's Degree Theorem (p. 341)
- III Polynomials and the Argument Principle (p. 344)
- IV A Topological Argument Principle* (p. 346)
- V Rouche's Theorem (p. 353)
- VI Maxima and Minima (p. 355)
- VII The Schwarz-Pick Lemma* (p. 357)
- VIII The Generalized Argument Principle (p. 363)
- IX Exercises (p. 369)
- 8 Complex Integration: Cauchy's Theorem (p. 377)
- I Introduction (p. 377)
- II The Real Integral (p. 378)
- III The Complex Integral (p. 383)
- IV Complex Inversion (p. 388)
- V Conjugation (p. 392)
- VI Power Functions (p. 395)
- VII The Exponential Mapping (p. 401)
- VIII The Fundamental Theorem (p. 402)
- IX Parametric Evaluation (p. 409)
- X Cauchy's Theorem (p. 410)
- XI The General Cauchy Theorem (p. 414)
- XII The General Formula of Contour Integration (p. 418)
- XIII Exercises (p. 420)
- 9 Cauchy's Formula and Its Applications (p. 427)
- I Cauchy's Formula (p. 427)
- II Infinite Differentiability and Taylor Series (p. 431)
- III Calculus of Residues (p. 434)
- IV Annular Laurent Series (p. 442)
- V Exercises (p. 446)
- 10 Vector Fields: Physics and Topology (p. 450)
- I Vector Fields (p. 450)
- II Winding Numbers and Vector Fields* (p. 456)
- III Flows on Closed Surfaces* (p. 462)
- IV Exercises (p. 468)
- 11 Vector Fields and Complex Integration (p. 472)
- I Flux and Work (p. 472)
- II Complex Integration in Terms of Vector Fields (p. 481)
- III The Complex Potential (p. 494)
- IV Exercises (p. 505)
- 12 Flows and Harmonic Functions (p. 508)
- I Harmonic Duals (p. 508)
- II Conformal Invariance (p. 513)
- III A Powerful Computational Tool (p. 517)
- IV The Complex Curvature Revisited* (p. 520)
- V Flow Around an Obstacle (p. 527)
- VI The Physics of Riemann's Mapping Theorem (p. 540)
- VII Dirichlet's Problem (p. 554)
- VIII Exercises (p. 570)
- References (p. 573)
- Index (p. 579)