
# Complex Analysis

### University of Crete

Teacher: Mihalis Kolountzakis

Announcements

1. 26 Sep 2022: Lectures begin.
2. 20-9-2018:     Το μάθημα θα διδαχθεί στα Αγγλικά μια και στο ακροατήριο θα υπάρχουν φοιτητές από το πρόγραμμα Erasmus. Θα γίνει με τέτοιο τρόπο ώστε με μια στοιχειώδη γνώση Αγγλικών να μην έχει κανείς πρόβλημα να το παρακολουθήσει. Το κυρίως σύγγραμα [CB] υπάρχει και στα Αγγλικά και όλες οι ασκήσεις και τα διαγωνίσματα θα είναι και στις δύο γλώσσες.

Schedule

Mon 9-11, Wed 9-11.
Room: E204

Teacher's office hours: Thu 10-11 in Γ213 or by appointment.

Course description

Goal: Introduction to complex analysis.

Contents: Complex numbers and geometry of the complex plane. Analytic functions, contour integrals and power series. Cauchy theory and applications.

Books and lecture notes

1. [CB]: R. Churchill and J. Brown, "Complex variables and applications", 2nd edition, Crete University Press (main textbook -- 10 copies have been put on reserve in the library for use in the library only). This is a translation of the (1984) 4th edition published by McGraw-Hill.
2. [P]: M. Papadimitrakis, "Complex Analysis", lecture notes. In Greek.
3. [BN]: J. Bak και D. J. Newman, "Complex Analysis", 2004, Leader Books. This is a translation of the 1997 edition by Springer.

Student evaluation

Intermediate exam 40%, final exam 60%. This remains the same for all further examination periods.

Class diary

#### Mon, 26 Sep. 2022

We went through the definition of complex numbers and their elementary algebraic and geometric properties. We did not talk yet about the polar (trigonometric) form of the complex numbers.

#### Wed, 28 Sep. 2022

We saw the concept of the argument (angle) of a complex number and how a complex number can be written in trigonometric form $z = r(\cos\theta+i\sin\theta)$. Then we defined the symbol $e^{i\theta} = \cos\theta+i\sin\theta$ and saw, using the fact that the argument of the product of two numbers is the sum of their arguments, that it obeys the usual rule $e^{i(\theta_1+\theta_2)} = e^{i\theta_1} e^{i\theta_2}$, which allows us to write the complex number $z$ as $re^{i\theta}$ and simplifies calculations tremendously. Then we solved the equation (in $z$) $$z^n = r e^{i\phi}$$ and saw that the solutions are the $n$ complex numbers $$r^{1/n} e^{i\theta}$$ where $\theta$ can take the $n$ values $\frac{\phi}{n}+\frac{k}{n}2\pi$, for $k=0, 1, \ldots, n-1$. These are always the vertices of a regular $n$-gon centered at $0$ and at distance $r^{1/n}$ from $0$.

Problems: No 1 ( / )

#### Mon, 3 Oct. 2022

Today we examined in much more detail the roots of unity, i.e., the roots of the equation $z^n-1=0$, for the different values of $n$. Apart from repating the generalities that we saw last time we also examine the question "which are the $m$-th roots of unity that are also $n$-th roots of unity?". After seing several special cases of this problem we concluded that the answer is all the $g$-th roots of unity where $g = (m, n)$ is the greatest common divisor of $m$ and $n$.

We then talked about open and closed sets in the plane, saw several examples, and proved that arbitrary unions of open sets remain open. We also defined a connected set in the plane to be one such for every two points in that set, say $z$ and $w$, we can find a polygonal line connecting them without ever leaving the set.