دانلود کتاب Twenty-One Lectures on Complex Analysis: A First Course

فهرست مطالب :

Front Matter ....Pages i-xii
Complex Numbers. The Fundamental Theorem of Algebra (Alexander Isaev)....Pages 1-8
ℝ- and ℂ-Differentiability (Alexander Isaev)....Pages 9-16
The Stereographic Projection. Conformal Maps. The Open Mapping Theorem (Alexander Isaev)....Pages 17-23
Conformal Maps (Continued). Möbius Transformations (Alexander Isaev)....Pages 25-32
Möbius Transformations (Continued). Generalised Circles. Symmetry (Alexander Isaev)....Pages 33-39
Domains Bounded by Pairs of Generalised Circles. Integration (Alexander Isaev)....Pages 41-48
Primitives Along Paths. Holomorphic Primitives. The Existence of a Holomorphic Primitive of a Function Holomorphic on a Disk. Goursat’s Lemma (Alexander Isaev)....Pages 49-55
Proof of Lemma 7.2. Constructible Primitives of Holomorphic Functions along Paths. Integration of Holomorphic Functions over Arbitrary Paths. Homotopy. Simply-Connected Domains. The Riemann Mapping Theorem (Alexander Isaev)....Pages 57-66
Cauchy’s Independence of Homotopy Theorem. Integration over Piecewise C1-paths. Jordan Domains and Integration over their Boundaries (Alexander Isaev)....Pages 67-76
Cauchy’s Integral Theorem. Proof of Theorem 3.1. Cauchy’s Integral Formula (Alexander Isaev)....Pages 77-85
Morera’s Theorem. Sequences and Series of Functions. Uniform Convergence Inside a Domain. Power Series. Abel’s Theorem. Disk of Convergence. Radius of Convergence (Alexander Isaev)....Pages 87-95
Proof of Theorem 11.9. Power Series (Continued). Taylor Series. Local Power Series Expansion of a Holomorphic Function. Cauchy’s Inequalities. The Uniqueness Theorem (Alexander Isaev)....Pages 97-105
Liouville’s Theorem. Laurent Series. Annulus of Convergence. Laurent Series Expansion of a Function Holomorphic on an Annulus. Cauchy’s Inequalities. Isolated Singularities of Holomorphic Functions (Alexander Isaev)....Pages 107-116
Isolated Singularities of Holomorphic Functions (Continued). Characterisation of an Isolated Singularity via the Laurent Series Expansion. Orders of Poles and Zeroes. Casorati-Weierstrass’ Theorem. Isolated Singularities of Holomorphic Functions at ∞ and their Characterisation via Laurent Series Expansions (Alexander Isaev)....Pages 117-125
Isolated Singularities of Holomorphic Functions at ∞ (Continued). Orders of Poles at ∞. Casorati-Weierstrass’ Theorem for an Isolated Singularity at ∞. Residues. Cauchy’s Residue Theorem. Computing Residues (Alexander Isaev)....Pages 127-135
Computing Residues (Continued). Computing Integrals over the Real Line Using Contour Integration. The Argument Principle (Alexander Isaev)....Pages 137-145
Index of a Path. The Argument Principle (Continued). Rouché’s Theorem. Theorem 1.1 Revisited. Proof of Theorem 3.2. The Maximum Modulus Principle. Proof of Theorem 3.3 (Alexander Isaev)....Pages 147-155
Schwarz’s Lemma. Conformal Maps of the Unit Disk and the Upper Half-Plane. (Pre)-Compact Subsets of a Metric Space. Continuous Linear Functionals on H(D). Arzelà-Ascoli’s Theorem. Montel’s Theorem. Hurwitz’s Theorem (Alexander Isaev)....Pages 157-165
Analytic Continuation (Alexander Isaev)....Pages 167-173
Analytic Continuation (Continued). The Monodromy Theorem (Alexander Isaev)....Pages 175-182
Proof of Theorem 8.3. Conformal Transformations of the Canonical Simply-Connected Domains (Alexander Isaev)....Pages 183-190
Back Matter ....Pages 191-194