
# Complex Analysis

### University of Crete

Teacher: Mihalis Kolountzakis

Announcements

1. The final exam for the course will be on 14 Jan 2020, from 13:00 to 16:00.
2. Intermediate exam: Monday Nov. 4, 2019, 5-7pm, in Auditorium A201.
Those of you who have a significant reason to be exempted from this exam must contact me by email until Sunday, Oct. 23, 2019. No exceptions beyond that date.
Students granted exception can be seen here.
3. 20-9-2018:     Το μάθημα θα διδαχθεί στα Αγγλικά μια και στο ακροατήριο θα υπάρχουν 2 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

Room A208. Tue and Thu, 9-11.

Teacher's office hours: Mon 9-11..

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). 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

#### Tue, 24 Sep. 2019:

[CB, Ch. 1, §1-4]: We saw what are complex numbers and how we to algebraic operations with them. We also saw what conjugate numbers are, the concept of the modulus of a complex number and several relations between them. We saw how to solve quadratic equations in complex numbers.

Problems: [CB, Ch. 1, p. 13-14]: 1, 2, 4, 10, 13.
( 13-14 )

#### Thu, 26 Sep. 2019:

We went over several examples and problems covering the material we did last time. Towards the end of the lecture we spoke about the polar form of a complex number [CB, Ch. 1, § 5] and we proved the basic relation: $$\arg(z w) = \arg z + \arg w.$$

Problems: No 1 ( / )

#### Tue, 1 Oct. 2019:

[CB, Ch. 1, §5-7]: We defined what is the meaning of $e^z$, when $z \in \CC$: $$e^z = e^{\Re{z}}(\cos{\Im{z}} + i \sin{\Im{z}}).$$ We saw that with this definition the important property of the exponential function $$e^{z+w} = e^z e^w$$ is preserved. Then we used this form of representing a complex number $$z = r e^{i\theta},\ \ \ \text{ where r = \Abs{z}, \theta = \arg{z}},$$ to solve several problems. The last problem we solved was how to find the $n$-th roots (there are $n$ of them) of a given complex number $z_0$.

Then we saw that a function of the form $$z(t) = R e^{it},\ \ \ 0\le t \lt 2\pi,$$ parametrizes a circle of center 0 and radius $R\gt 0$. We also saw how to get parametrizations of different circles.

Problems: No 2 ( / )

#### Thu, 3 Oct. 2019:

We did several examples involving the polar form and roots of numbers. Then we talked about limits and continuity of sequences and functions, and we saw that nothing really changes from the case of real valued functions of a real variable. Changing the values of the function to be complex, instead of real, creates no problems at all. Limits and continuity can also be defined exactly as in the real case, for functions of a complex variable. What is different is the notion of a derivative when the variable, as well as the value, is complex.

If $f:\CC\to\CC$ is a complex valued function of a complex variable then we say that $f$ is differentiable at $z_0 \in \CC$ if the limit $$\lim_{z \to z_0} \frac{f(z)-f(z_0)}{z-z_0}$$ exists. If it does we denote it by $f'(z_0)$ and call it the derivative of $f$ at $z_0$. Then we tried some simple functions: $f(z) = z, g(z) = z^2$ and $h(z) = \overline{z}$. We proved that the limit exists in the first two cases, and that $f'(z) = 1, g'(z) = 2z$, and that the limit does not exist in the case of $h(z)$.

Problems: No 3 ( / )

#### Tue, 8 Oct. 2019:

Today we did several problems from the first 2 problem sets. Then we talked about mapping properties of simple functions of the form $w = f(z)$.

Problems: No 4 ( / )

#### Thu, 10 Oct. 2019:

We started by discussing the function $f(z) = z^2$ and its mapping properties (how it transforms various regions in the complex plane). We also investigated on which domains this function is one-to-one and, therefore, on which domains we can define its inverse $z \to \sqrt{z}$. We did not give a complete answer to this last question but we saw that while it is possible to define the square root on the upper unit disk (via the formula $r e^{i\theta} \to \sqrt{r} e^{i \theta/2}$, where $0 \le \theta \le \pi$) it appears to be (and it is) impossible to define the square root on the entire unit disk as a continuous function.

Next we proved that if a complex function $f$ is defined in a neighborhood of a point $z \in \CC$ and is differentiable at $z$ then, writing $$f(z) = f(x+iy) = u(x, y) + i v(x, y),\ \ \ \text{ where u, v \in \RR},$$ we have the Cauchy-Riemann (C-R) partial differential equations: $$u_x = v_y,\ \ u_y = -v_x,$$ at the point $z = z+iy$.

We then proved a converse to this: if the C-R equations hold and the partial derivatives of $u, v$ are assumed to not just exist but also to be continuous at $(x, y)$ then the function $f$ is differentiable at $z = x+iy$.

Problems: No 5 ( / )

#### Tue, 15 Oct. 2019:

[CB, Ch. 3, § 22-26] First we used the Cauchy-Riemann equations to prove that the function $e^z$ is analytic everywhere. We defined the trigonometric functions $\cos$ and $\sin z$ via the exponential functions by the formulas $$\cos z = \frac{e^{iz}+e^{-iz}}{2},\ \ \ \sin z = \frac{e^{iz}-e^{-iz}}{2i}.$$ Then we used these formulas to prove many properties of these trigonometric functions defined now on the whole complex plane. We also defined the hyperbolic cosine and sine $\cosh z$ and $\sinh z$. Last we started talking about the inverse function of $e^z$, the logarithm, and saw that it is a multi-valued function, which is impossible to define on the whole punctured plane (i.e. $\CC\setminus\Set{0}$) as a continuous function.

Problems: No 6 ( / )

#### Tue, 17 Oct. 2019:

[CB, Ch. 3, § 26-28 and Ch. 4, § 30-32] We finished our discussion about the logarithm as a multi-valued function. Then we started talking about curve parametrizations and the so-called contour integrals, that is integrals of the form $$\oint_\Gamma f(z)\,dz,$$ where $\Gamma$ is a curve in the complex plane and $f$ is a function on $\Gamma$. We saw that this integral does not depend on the parametrization of the curve and computed some examples.

Problems: No 7 ( / )