Complex Analysis
本課程是由 國立陽明交通大學應用數學系 提供。
本課程主要目的是探討解析函數之微分及積分性質，作為後續研究之基礎。
課程用書：Complex Variables and Applications,James Ward Brown & Ruel V. Churchill,6th edition, McGraw-Hill, 1996.
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| Instructor(s) | Department of Applied Mathematics Prof. Pei Yuan Wu |
|---|---|
| Course Credits | 3 Credits |
| Academic Year | 96 Academic Year |
| Level | Third year university students |
| Prior Knowledge | Advanced Calculus |
| Related Resources | Course Video Course Syllabus Course Calendar |
| Week | Course Content | Course Video |
|---|---|---|
| 第一章 Complex Numbers Section1 Sums and Products Section2 Basic Algebraic Properties Further Properties Section3 Moduli Section4 Complex Conjugates | Watch Online | |
| Section5 Polor Form Section6 Exponential Form Section7 Powers and Roots | Watch Online | |
| 第一章 Complex Numbers Section8 Regions in the Complex Plane 第二章 Analytic Functions Section9 Functions of a Complex Variable Section10 Mappings Section11 Mappings by the Exponential Function | Watch Online | |
| Section12 Limits Section13 Theorems in Limits Section14 Derivatives Section15 Differentiation Formulas | Watch Online | |
| Section16 Limits Involving the Points at Infinity Section17 Continuity Section18 Differentiatility | Watch Online | |
| Section18 Differentiatility Section19 Differentiatility Formulas Section20 Cauchy-Riemann Equations Section21 Reflectiotn Principle Section22 C-R equation in Polar Form Section23 Analytic Function | Watch Online | |
| Section23 Analytic Functions Section24 Harmonic Functions | Watch Online | |
| Section25 Harmonic Functions-1 Section26 Uniquely Determined Analytic Functions Section27 Refelction Principle 第三章 Elementary Functions Section28 The Exponential Func | Watch Online | |
| Section29 The Logarithmic Function Section31 Branches and Derivatives of Logarithms Section32 Complex Exponents Functions Section33 Trigonometric Functions | Watch Online | |
| Section34 Hyperbolic Functions Section35 Inverse Trigonometric and Hyperbolic Functions Section36 Derviatives of Functions Section37 Integrals Section38 Contours Section39 Contours Integrals | Watch Online | |
| Section40 Examples Section41 Antiderivatives Section42 Fundamental Theory of Integral Section43 Examples | Watch Online | |
| Section44 Cauchy-Goursat Theorem (VersionI) | Watch Online | |
| Section45 Cauchy-Goursat Theorem (VersionII) Section46 Cauchy-Goursat Theorem (VersionIII) Section47 Cauchy Integral Formula Section48 Derivatives of Analytic Functions Section49 Liouvilles Theorem | Watch Online | |
| Section50 Maximum Modulus Principle | Watch Online | |
| Section50 Maximum Modulus Principle 第五章 Series Section51 Convergence of Sequences Section52 Convergence of Series Section53 Taylor Series Section54 Examples | Watch Online | |
| Section55 Laurent Series Section56 Examples | Watch Online | |
| Section56 Examples Section57 Absolate amd Uniform Convergence of Power Series | Watch Online | |
| Section58 Continuity of Sums of Power Series Section59 Integration and Differentiation of Power Series | Watch Online | |
| Section59 Integration and Differentiation of Power Series Section60 Uniqueness of Series Representations Section61 Multiplication and Division of Power Series 第六章 Residues and Poles Section62 Residues Section63 Cauchy | Watch Online | |
| Section61 Multiplication and Division of Power Series 第六章 Residues and Poles Section62 Residues Section63 Cauchys Residue Theorem | Watch Online | |
| Section63 Cauchys Residue Theorem Section64 Using a single Residue Section65 The Three Types of Isolated Singular Points | Watch Online | |
| Section66 Residues at Poles Section67 Examples Section68 Zeros of Analytic Functions | Watch Online | |
| Section69 Zeros and Poles Section70 Behavior of f Near Isolated Singular Points | Watch Online | |
| Section70 Behavior of f Near Isolated Singular Points 第七章 Residues and Poles Section71 Evaluation of Improper Integrals Section72 Example | Watch Online | |
| Section73 Improper Integrals from Fourier Analysis Section74 Jordans Lemma Section75 Indented PAths Section76 An Indentation Around a Branch Point | Watch Online | |
| Section77 Integration Along a Branch Cut Section78 Definite Integrals Involving Sines and Cosines | Watch Online | |
| Section79 Argument Principle Section80 Rouches Theorem Section81 Inverse Laplace Transforms Section82 Examples | Watch Online | |
| 第八章 Mapping by Elementary Functions Section86 Linear Fractional Transformations 第九章Conformal Mapping Section94 Preservation of Angles | Watch Online |
課程目標
本課程主要目的是探討解析函數之微分及積分性質，作為後續研究之基礎 。
課程章節
| 章節 | 主題內容 |
| 第一章 | Complex numbers |
| 第二章 | Analytic functions |
| 第三章 | Elementary functions |
| 第四章 | Integrals |
| 第五章 | Series |
| 第六章 | Residues and poles |
| 第七章 | Applications of residues |
| 第八章 | Mapping by elementary functions |
課程書目
Complex Variables and Applications,James Ward Brown & Ruel V. Churchill,6th edition, McGraw-Hill, 1996.
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