Complex Analysis
1. Basic Information
Unit: |
| Course Code: MATH 1221 |
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Course Name: | Complex Analysis | Course Name: | 复分析 |
Credit: |
| Period: | 48 |
Teaching object: |
| Teaching Lagrange: | Chinese & English |
Previous Courses: | 数学分析I, II 线性代数I |
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2. Course introduction
This is a course on complex analysis at the advanced undergraduate level. We plan to include the following important properties of holomorphic functions: the power series expansion, Cauchy’s theorem, the conformal map, the calculus of residues, and the uniqueness of analytic continuation. If time permits, we will also talk about various topics about analytic functions. It is assumed that the students, who will take this class, have been trained in mathematical analysis and linear algebra.
3. Teaching content, method and time arrangement
(each table is meant for 1.5 week, 6 periods )
Chapter 1 | Complex Functions |
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1.1 | Complex Valued Functions |
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1.2 | Complex Differentiability |
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1.3 | The Cauchy-Rieman Equations |
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1.4 | Angles Under Holomorphic Maps |
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Chapter 2 | Power Series |
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2.1 | Formal Power Series |
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2.2 | Convergent Power Series |
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2.3 | Relations Between Formal and Convergent Power Series |
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2.4 | Differentiation of Power Series |
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Chapter 3 | Power Series Continued |
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3.1 | Analytic Functions |
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3.2 | The Inverse and Open Mapping Theorems |
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3.3 | The Local Maximum Modulus Principle |
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Chapter 4 | Cauchy’s Theorem |
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4.1 | Holomorphic Functions |
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4.2 | Integrals Over Paths |
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4.3 | Local Primitive for a Holomorphic Function |
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4.4 | Another Description of the Integral Along a Path |
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Chapter 5 | Cauchy’s Theorem Continued |
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5.1 | The Homotopy Form of Cauchy’s Theorem |
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5.2 | Existence of Global Primitives |
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5.3 | The Local Cauchy Formula |
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Chapter 6 | Winding Numbers |
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6.1 | The Winding Numbers |
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6.2 | The Global Cauchy Theorem |
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6.3 | Artin’s Proof |
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Chapter 7 | Application of Cauchy’s Integral Formula |
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7.1 | Uniform Limits of Analytic Functions |
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7.2 | Laurent Series |
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7.3 | Isolated Singularities |
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Chapter 8 | Calculus of Residues |
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8.1 | The Residue Formula |
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8.2 | Evaluations of Definite Integrals |
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8.3 | Fourier Transforms |
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8.4 | Mellin Transforms |
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Chapter 9 | Conformal Mappings |
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9.1 | Schwarz Lemma |
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9.2 | Analytic Automorphisms of the Disk |
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9.3 | The Upper Half Plane |
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Chapter 10 | Conformal Mappings Continued |
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10.1 | Other Examples |
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10.2 | Fractional Linear Transformations |
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Chapter 11 | Harmonic Functions |
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11.1 | Definitions |
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11.2 | Examples |
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11.3 | Basic Properties of Harmonic Functions |
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11.4 |
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Chapter 12 | Harmonic Functions Continued |
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12.1 | The Poisson Formula |
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12.2 | Construction of Harmonic Functions |
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12.3 | ( Differentiating under the Integral sign ) |
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Chapter 13 | Various Analytic Topics |
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13.1 | Jensen’s Formula |
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13.2 | The Picard-Borel Theorem |
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Chapter 14
| Various Analytic Topics Continued |
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14.1 | Bounds by the Real Part |
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14.2 | The Use of Three Circles |
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Final Exam | The 17th. Week |
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