Table of Contents:
| Preface | i | |||
| 1 | Elementary Solution Methods for Ordinary Differential Equations | 1 | ||
| 1.1 | Euler's Multiplier | 5 | ||
| 1.2 | Autonomous Systems | 8 | ||
| 1.2.1 | Class of Examples | 8 | ||
| 1.2.2 | State Space of a System of Two Homogeneous Real 1st Order Differential Equations | 12 | ||
| 1.3 | On the Stability of Equilibrium Positions | 16 | ||
| 2 | Boundary Value Problems of Ordinary Differential Equations | 21 | ||
| 2.1 | The General Linear Boundary Value Problem | 22 | ||
| 2.1.1 | Uniqueness Criterion | 23 | ||
| 2.1.2 | Green's Function for the General Semi-homogeneous (Linear) Boundary Value Problem | 26 | ||
| 2.2 | Vibrations of a String | 29 | ||
| 2.2.1 | d'Alembert's Solution Ansatz | 29 | ||
| 2.2.2 | Fourier's Solution Method | 31 | ||
| 3 | Inversive Geometry and Number Sphere | 33 | ||
| 3.1 | The Cross-ratio | 36 | ||
| 3.1.1 | The Rational Functions as Mapping of the Number Sphere \(\mathbb{P}\) in Itself | 38 | ||
| 3.2 | A Class \(\mathcal{H}\) of Rational Functions | 39 | ||
| 3.2.1 | Elementary Properties of \(\mathcal{H}\) | 39 | ||
| 3.2.2 | Partial Fraction Decomposition of the Function \(h(z) \in \mathcal{H}\) | 40 | ||
| 3.2.3 | Division Algorithm for Functions in \(\mathcal{H}\) | 41 | ||
| 3.2.4 | Application: The Stability Criterion | 42 | ||
| 4 | Holomorphic or Analytic Functions | 45 | ||
| 4.1 | Geometric Interpretation of Holomorphy | 46 | ||
| 4.2 | Holomorphy and Cauchy-Riemann Differential Equations | 47 | ||
| 4.2.1 | Harmonic Functions | 48 | ||
| 4.2.2 | Transplantation of Harmonic Functions | 50 | ||
| 4.3 | Power Series, the Basic Example of Holomorphic Functions | 52 | ||
| 4.3.1 | Theorem on the Holomorphy of Power Series | 53 | ||
| 5 | Integration of Complex-valued Functions | 55 | ||
| 5.1 | Integral and Primitive Function | 56 | ||
| 5.2 | Cauchy's Integral Formula | 58 | ||
| 5.2.1 | Theorem on Taylor Expansion of Holomorphic Functions | 58 | ||
| 5.2.2 | Liouville's Theorem | 60 | ||
| 5.3 | Identity Theorem for Analytic Functions | 61 | ||
| 5.3.1 | Theorem about the Zeros of Analytic Functions | 61 | ||
| 5.3.2 | Cauchy's Integral Formula for \(z = z_{0}\) | 62 | ||
| 5.3.3 | Principle of Maximum and Minimum for Analytic Functions | 63 | ||
| 5.4 | The Poisson Integral Formula | 65 | ||
| 5.4.1 | Solution of Dirichlet's Boundary Value Problem for a Circular Disk | 66 | ||
| 6 | Extending the Theory of Analytic Functions | 69 | ||
| 6.1 | The Holomorphic Logarithm | 69 | ||
| 6.2 | The Winding Number | 72 | ||
| 6.2.1 | General Version of Cauchy's Integral Formula | 73 | ||
| 6.3 | General Version of Cauchy's Integral Theorem | 75 | ||
| 6.3.1 | Existence of Global Primitive Functions | 76 | ||
| 6.3.2 | Isolated Singularities | 77 | ||
| 6.3.3 | A Formula for Residues | 79 | ||
| 6.3.4 | Logarithmic Derivatives | 80 | ||
| 6.4 | Residue Theorem | 81 | ||
| 6.4.1 | Classification of Isolated Singularities | 82 | ||
| 6.4.2 | The Zeros- and Poles-counting Integral | 84 | ||
| 6.4.3 | Ronché's Theorem | 84 | ||
| 7 | Applications of the Residue Theorem | 87 | ||
| 7.1 | Applications I | 87 | ||
| 7.2 | Applications II | 89 | ||
| 7.3 | Applications III | 94 | ||