Verwandte Artikel zu Handbook of Complex Variables
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Inhaltsangabe
1 The Complex Plane.- 1.1 Complex Arithmetic.- 1.1.1 The Real Numbers.- 1.1.2 The Complex Numbers.- 1.1.3 Complex Conjugate.- 1.1.4 Modulus of a Complex Number.- 1.1.5 The Topology of the Complex Plane.- 1.1.6 The Complex Numbers as a Field.- 1.1.7 The Fundamental Theorem of Algebra.- 1.2 The Exponential and Applications.- 1.2.1 The Exponential Function.- 1.2.2 The Exponential Using Power Series.- 1.2.3 Laws of Exponentiation.- 1.2.4 Polar Form of a Complex Number.- 1.2.5 Roots of Complex Numbers.- 1.2.6 The Argument of a Complex Number.- 1.2.7 Fundamental Inequalities.- 1.3 Holomorphic Functions.- 1.3.1 Continuously Differentiable and Ck Functions.- 1.3.2 The Cauchy-Riemann Equations.- 1.3.3 Derivatives.- 1.3.4 Definition of Holomorphic Function.- 1.3.5 The Complex Derivative.- 1.3.6 Alternative Terminology for Holomorphic Functions.- 1.4 The Relationship of Holomorphic and Harmonic Functions.- 1.4.1 Harmonic Functions.- 1.4.2 Holomorphic and Harmonic Functions.- 2 Complex Line Integrals.- 2.1 Real and Complex Line Integrals.- 2.1.1 Curves.- 2.1.2 Closed Curves.- 2.1.3 Differentiable and Ck Curves.- 2.1.4 Integrals on Curves.- 2.1.5 The Fundamental Theorem of Calculus along Curves.- 2.1.6 The Complex Line Integral.- 2.1.7 Properties of Integrals.- 2.2 Complex Differentiability and Conformality.- 2.2.1 Limits.- 2.2.2 Continuity.- 2.2.3 The Complex Derivative.- 2.2.4 Holomorphicity and the Complex Derivative..- 2.2.5 Conformality.- 2.3 The Cauchy Integral Theorem and Formula.- 2.3.1 The Cauchy Integral Formula.- 2.3.2 The Cauchy Integral Theorem, Basic Form.- 2.3.3 More General Forms of the Cauchy Theorems.- 2.3.4 Deformability of Curves.- 2.4 A Coda on the Limitations of the Cauchy Integral Formula.- 3 Applications of the Cauchy Theory.- 3.1 The Derivatives of a Holomorphic Function.- 3.1.1 A Formula for the Derivative.- 3.1.2 The Cauchy Estimates.- 3.1.3 Entire Functions and Liouville's Theorem.- 3.1.4 The Fundamental Theorem of Algebra.- 3.1.5 Sequences of Holomorphic Functions and their Derivatives.- 3.1.6 The Power Series Representation of a Holomorphic Function.- 3.1.7 Table of Elementary Power Series.- 3.2 The Zeros of a Holomorphic Function.- 3.2.1 The Zero Set of a Holomorphic Function.- 3.2.2 Discrete Sets and Zero Sets.- 3.2.3 Uniqueness of Analytic Continuation.- 4 Isolated Singularities and Laurent Series.- 4.1 The Behavior of a Holomorphic Function near an Isolated Singularity.- 4.1.1 Isolated Singularities.- 4.1.2 A Holomorphic Function on a Punctured Domain.- 4.1.3 Classification of Singularities.- 4.1.4 Removable Singularities, Poles, and Essential Singularities.- 4.1.5 The Riemann Removable Singularities Theorem.- 4.1.6 The Casorati-Weierstrass Theorem.- 4.2 Expansion around Singular Points.- 4.2.1 Laurent Series.- 4.2.2 Convergence of a Doubly Infinite Series.- 4.2.3 Annulus of Convergence.- 4.2.4 Uniqueness of the Laurent Expansion.- 4.2.5 The Cauchy Integral Formula for an Annulus..- 4.2.6 Existence of Laurent Expansions.- 4.2.7 Holomorphic Functions with Isolated Singularities.- 4.2.8 Classification of Singularities in Terms of Laurent Series.- 4.3 Examples of Laurent Expansions.- 4.3.1 Principal Part of a Function.- 4.3.2 Algorithm for Calculating the Coefficients of the Laurent Expansion.- 4.4 The Calculus of Residues.- 4.4.1 Functions with Multiple Singularities.- 4.4.2 The Residue Theorem.- 4.4.3 Residues.- 4.4.4 The Index or Winding Number of a Curve about a Point.- 4.4.5 Restatement of the Residue Theorem.- 4.4.6 Method for Calculating Residues.- 4.4.7 Summary Charts of Laurent Series and Residues.- 4.5 Applications to the Calculation of Definite Integrals and Sums.- 4.5.1 The Evaluation of Definite Integrals.- 4.5.2 A Basic Example.- 4.5.3 Complexification of the Integrand.- 4.5.4 An Example with a More Subtle Choice of Contour.- 4.5.5 Making the Spurious Part of the Integral Disappear.- 4.5.6 The Use of the Logarithm.- 4.5.7 Summing a Series Using Residues.- 4.5.8 Summary Chart of Some Integration
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- VerlagBirkhäuser
- Erscheinungsdatum2011
- ISBN 10 1461215897
- ISBN 13 9781461215899
- EinbandPaperback
- SpracheEnglisch
- Kontakt zum HerstellerNicht verfügbar
(Keine Angebote verfügbar)
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