- Lecturer
- Prof. Dr. Paul Biran
- Coordinator
- Younghan Bae

The following material will be distributed during the final exam and you are not allowed to use your printed copy of this material. Also, it is not allowed to use your own auxiliary materials.

AuxiliaryWe will discuss the following subjects: Complex functions of one variable, Cauchy-Riemann equations, Cauchy theorem and integral formula, singularities, residue theorem, index of closed curves, analytic continuation, special functions, conformal mappings, Riemann mapping theorem.

week | material covered |
---|---|

1 | Basic topology of the complex plane. Definitions from topology (open, closed, compact, path connected, continuity of a function). |

2 | Complex derivative functions and the Cauchy-Riemann equation. Comparing linear algebra over the real numbers and the complex numbers. The first order approximation of a complex function. |

3 | Consequences of the Cauchy-Riemann equation. A definition of path integrals. |

4 | Definition of a primitive function. Cauchy theorem for a holomorphic function on an unit circle. |

5 | Function of winding numbers. Global Cauchy theorem. |

6 | Cauchy estimate theorem. Maximum principle. Lemma of Schwarz. |

7 | A criterion when a continuous function has primitives. Homotopy between curves. |

8 | Convergence of a sequence of functions. |

9 | Taylor expansion. Radius of convergence. Convergence of a sequence of functions. |

10 | Laurent expansion. |

11 | Zeros of holomorphic functions. Singularities of holomorphic functions. Riemann extension theorem for removable singularity |

12 | Meromorphic functions. Casorati–Weierstrass theorem. Residue theorem. Argument principle. |

13 | Rouche's theorem. Branched covering. Open mapping theorem. Hurwitz theorem |

The new exercises will be posted here on every weekend. We expect you to look at the problems and to prepare questions for the exercise class on Tuesday.

Please hand in your solutions by the following Friday at 13:00 in your assistant's box in HG J68. Your solutions will usually be corrected and returned in the following exercise class or, if not collected, returned to the box in HG J68.

exercise sheet | due by | solutions |
---|---|---|

Exercise sheet 1 | September 20 | Solution 1 |

Exercise sheet 2 | September 27 | Solution 2 |

Exercise sheet 3 | October 4 | Solution 3 |

Exercise sheet 4 | October 11 | Solution 4 |

Exercise sheet 5 | October 18 | Solution 5 |

Exercise sheet 6 | October 25 | Solution 6 |

Exercise sheet 7 | November 1 | Solution 7 |

Exercise sheet 8 | November 8 | Solution 8 |

Exercise sheet 9 | November 15 | Solution 9 |

Exercise sheet 10 | November 22 | Solution 10 |

Exercise sheet 11 | November 29 | Solution 11 |

Exercise sheet 12 | December 06 | Solution 12 |

Exercise sheet 13 | December 13 | Solution 13 |

Exercise sheet 14 | December | Solution 14 |

time | room | assistant | language |
---|---|---|---|

Tu 13-15 | HG D 5.2 | Subhajit Jana | en |

Tu 13-15 | HG D 7.2 | Niclas Küpper | en |

Tu 13-15 | HG G 26.3 | Levie Bringmans | en |

Tu 13-15 | LEE D 101 | Rush Brown | en |

Tu 13-15 | LEE D 105 | Daniel Hainschink | en |

Tu 13-15 | LFW C 11 | Valeria Ambrosio | en |

Tu 13-15 | ML J 34.3 | Fabian Jin | en |

Tu 13-15 | NO C 6 | Filippo Berta | en |

Tu 15-17 | NO C 44 | Paula Truöl | en |

- "An introduction to complex function theory" by B. Palka (Undergraduate Texts in Mathematics. Springer-Verlag, 1991.)