Measure Theory and Complex Analysis

**Instructor:** Bjoern Muetzel

**E-Mail:** bjorn.mutzel-at-dartmouth.edu

**Office:** 318 Kemeny Hall

**Office Hours:** Wednesday 3:30 -5

**Lectures:** MWF 2:10-3:15 in 004 Kemeny Hall

**x-hour:**Thursday 1:20-2:10 in 004 Kemeny Hall

During the x-hour I will be usually available in my office, but x-hours will be occasionally used for class work.

This course is an introduction to graduate level analysis. Divided roughly in half, the first part of the course covers abstract measure theory.

This part starts with the introduction of measurable sets and measures on spaces. This will allow us to define the Lebesque integral which has a number of advantages over the Riemann integral. We will then take a look at \(L^1 \) spaces which are very useful in functional analysis and then end this part with the proof of the Radon-Nikodym theorem.

The second half of the course covers complex analysis. It starts with the definition of holomorphic functions and then covers the most important theorems in complex analysis, Cauchy's theorem, the Residue theorem and Laurent series expansions.

A prior knowledge of basic set theory, topology and real analysis is required for this course. Additionally you should also be familiar with complex numbers.

*Real and Complex Analysis* (Third edition) by W. Rudin

*Real Analysis* (Fourth edition) by H.L. Royden and P.M. Fitzpatrick

*Real Analysis, Modern techniques and their applications* (Second edition) by G.B. Folland

Midterm Exam |
Monday, October 15 from 4:30-6:30 pm | 105 Kemeny Hall |

Final Exam |
Friday, November 16 from 3-5 pm | 004 Kemeny Hall |

If you have a conflict with the midterm exam because of a religious observance, scheduled extracurricular activity such as a game or performance [not practice], scheduled laboratory for another course, or similar commitment, please see your instructor as soon as possible.

Written homework | 40% |

Midterm exam | 30% |

Final exam | 30% |

Academic integrity is at the core of our mission as mathematicians and educators, and we take it very seriously. We also believe in working and learning together.

Collaboration on homework is permitted and encouraged, but obviously it is a violation of the honor code for someone to provide the answers for you.

On written homework, you are encouraged to work together, and you may get help from others, but you must write up the answers yourself.

On exams, you may not give or receive help from anyone. Exams in this course are closed book, and no notes, calculators or other electronic devices are permitted.

Students with disabilities who will be taking this course and may need disability-related classroom accommodations are encouraged to make an appointment to see their instructor as soon as possible. Also, they should stop by the Academic Skills Center in Collis Center to register for support services.