## Exercises Document

The problems assigned as homeworks are found in the PDF document below. It will be updated throughout the semester with new exercises.

The problems assigned as homeworks are found in the PDF document below. It will be updated throughout the semester with new exercises.

You will find in the tables below the lecture notes for each section of each chapter. The lecture notes are based on the book **Complex Analysis** by Donald Marshall.

In the first chapter, we will introduce the basic notations we will use throughout the semester. The new material is on the Riemann Sphere and the Extended Complex Plane (section 1.3).

Section No. | Document (PDF) | Annotated Notes (PDF) |
---|---|---|

1.1 | ||

1.2 | ||

1.3 |

In the second chapter, we will start our journey in the realm of analytic functions! We will consider first the polynomial functions and then we will study power series, the natural generalization of polynomials.

Section No. | Document (PDF) | Annotated Notes (PDF) |
---|---|---|

2.1 | ||

2.2 | ||

2.3 | ||

2.4 | ||

2.5 | ||

2.6 |

In the third chapter, we will study the behavior of functions as they approach the boundary. In particular, we will prove the maximum principle for analytic functions, Liouville's Theorem, and Schwarz's Lemma.

Section No. | Document (PDF) | Annotated Notes (PDF) |
---|---|---|

3.1 | ||

3.2 | ||

3.3 |

In the fourth chapter, we will prove Runge's Theorem. This Theorem says that if a function is analytic on a compact set, then it can be uniformly approximated by a rational function.

Section No. | Document (PDF) | Annotated Notes (PDF) |
---|---|---|

4.1 | ||

4.2 | ||

4.3 | ||

4.4 |

In the fifth chapter, we will explore in more depth the winding number. We will also introduce Laurent series and different types of singularities. The important results we will cover are Riemann's Removable Singularity Theorem, PainlevĂ©'s Removability Theorem, the Argument Principle and RouchĂ©'s Theorem.

Section No. | Document (PDF) | Annotated Notes (PDF) |
---|---|---|

5.1 | ||

5.2 | ||

5.3 | ||

5.4 | ||

5.5 |