LEC # | TOPICS | READINGS | SUPPLEMENTARY NOTES |
---|---|---|---|

L1 | The algebra of complex numbers: the geometry of the complex plane, the spherical representation | Ahlfors, pp. 1-11 and 19-20 | (PDF) |

L2 | Exponential function and logarithm for a complex argument: the complex exponential and trigonometric functions, dealing with the complex logarithm | (PDF) | |

L3 | Analytic functions; rational functions: the role of the Cauchy-Riemann equations | Ahlfors, pp. 21-32 | (PDF) |

L4 | Power series: complex power series, uniform convergence | Ahlfors, pp. 33-42 | (PDF) |

L5 | Exponentials and trigonometric functions | Ahlfors, pp. 42-47 | (PDF) |

L6 | Conformal maps; linear transformations: analytic functions and elementary geometric properties, conformality and scalar invariance | Ahlfors, pp. 69-80 | (PDF) |

L7 | Linear transformations (cont.): cross ratio, symmetry, role of circles | Ahlfors, pp. 80-89 | (PDF) |

L8 | Line integrals: path independence and its equivalence to the existence of a primitive | Ahlfors, pp. 101-108 | (PDF) |

L9 | Cauchy-Goursat theorem | Ahlfors, pp. 109-115 | (PDF) |

L10 | The special cauchy formula and applications: removable singularities, the complex taylor's theorem with remainder | Ahlfors, pp. 118-126 | (PDF) |

L11 | Isolated singularities | Ahlfors, pp. 126-130 | (PDF) |

L12 | The local mapping; Schwarz's lemma and non-Euclidean interpretation: topological features, the maximum modulus theorem | Ahlfors, pp. 130-136 | (PDF) |

L13 | The general Cauchy theorem | (PDF) | |

L14 | The residue theorem and applications: calculation of residues, argument principle and RouchÃ©'s theorem | (PDF) | |

L15 | Contour integration and applications: evaluation of definite integrals, careful handling of the logarithm | Ahlfors, pp. 154-161 | (PDF) |

L16 | Harmonic functions: harmonic functions and holomorphic functions, Poisson's formula, Schwarz's theorem | (PDF) | |

L17 | Mittag-Leffer's theorem: Laurent series, partial fractions expansions | Ahlfors, pp. 187-190 | (PDF) |

L18 | Infinite products: Weierstrass' canonical products, the gamma function | Ahlfors, pp. 191-200 | (PDF) |

L19 | Normal families: equiboundedness for holomorphic functions, Arzela's theorem | (PDF) | |

L20 | The Riemann mapping theorem | Ahlfors, pp. 229-231 | (PDF) |

L21-L22 | The prime number theorem: the history of the theorem and the proof, the details of the proof | (PDF) | |

L23 | The extension of the zeta function to C, the functional equation |
Ahlfors, pp. 214-217 For the original proof, see p. 146 of Weber, Heinrich, ed. |
(PDF) |