▶ Announcements
20-9-2018: Το μάθημα θα διδαχθεί στα Αγγλικά μια και στο ακροατήριο θα υπάρχουν 1-3 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 (Bak & Newman) υπάρχει και στα Αγγλικά και όλες οι ασκήσεις και τα διαγωνίσματα θα είναι και στις δύο γλώσσες.
▶ Time Schedule
Room Α 208. Monday 11-1, Friday 9-11.
Teacher office hours: Monday 9-11, Γ213.
▶ Course description
Goals: Introduction to the basic techniques of Complex Analysis, primarily from a computational standpoint.
Content: Topology of the complex plane. Analytic functions, contour integrals and power series. Cauchy theory and applications.
▶ Books and lecture notes
▶ Student evaluation
▶ Class diary
We covered, more or less, § 1.1 and 1.2 of [BN]. We talked about the properties of complex numbers, how we add and multiply them, their conjugate, their polar form and how the polar form interacts with multiplication. The property \[ \cis(\theta_1) \cdot \cis(\theta_2) = \cis(\theta_1+\theta_2) \] is particularly important (here \( \cis\theta \) is shorthand for \( \cos\theta+i\sin\theta \)). We showed how to compute the roots of the general quadratic equation \[ a z^2 + b z + c = 0 \] and also how to find the square roots of a given complex number and also how to find the three cubic roots of 1 (that is, all solutions to the equation \(z^3=1\)).
Problems:
To prepare for Friday please do problems 1-3 from [BN, Chapter 1].
(
1-3
)
We spoke about how to interpret geometrically (as parts of the complex plane) several algebraic descriptions of sets of complex numbers (such \(\Set{z:\ \Abs{z-1-i}\lt 2}\) or \(\Set{z: \Re{z}\gt 0}\)).
We saw why the fact that a number cannot have more than \(n\) roots of order \(n\) is a consquence of polynomial division: If \(A(z), B(z)\) are two polynomials with complex coefficients then there are unique polynomials \(Q(z)\) (the quotient) and \(R(z)\) (the remainder) such that \[ A(z) = Q(z) B(z) + R(z),\ \ \text{ and } \deg R(z) \lt \deg B(z). \] Applying this to the polynomial of degree one \[ B(z) = z - z_0 \] we obtain that if \(A(z_0)=0\) (we say that \(z_0\) is a root of \(A(z)\)) then \[ A(z) = (z-z_0) Q(z),\ \ \ \text{ for some polynomial } Q(z), \] and it is easy to see that \(\deg Q(z) = \deg A(z)-1\). Repeating this proves that any polynomial of degree \(n\) can have \(\le n\) roots (we will see later that it always has \(n\) complex roots) which, in turn, implies that a complex number can only have \(\le n\) roots of order \(n\). Of course, we saw, using the polar form of complex numbers, that there are exactly \(n\) such roots and we described them completely.
The \(n\)-th roots of the non-zero complex number \(z = r\cdot \cis\theta = r(\cos\theta + i \sin\theta)\) are the \(n\) numbers \[ r^{1/n} \cis\left(\frac{\theta}{n}+k\frac{2\pi}{n}\right),\ \ \ \text{ where } k=0, 1, 2, \ldots, n-1. \] These numbers form a regular polygon of \(n\) sides inscribed in the circle \(\Set{\zeta:\ \Abs{\zeta}=r^{1/n}}\).
Next, we started talking about the topology of \(\CC\) as a metric space. We reminded the definition of a Cauchy sequence in a metric space and proved that \(\CC\) is a complete metric space (here we assumed the fact that \(\RR\) is a complete metric space). We spoke about open and closed sets in \(\CC\) and saw several examples.
Problems:
Please do problems 4, 5, 10, 11, 12, 13 from [BN, Chapter 1].
(
4-5
10-13
)
We continued our discussion of convergence and continuity for complex numbers and functions of complex numbers. We also discussed the concept of connectivity and polygonal connectivity [BN, § 1.3]. These two concepts are identical for open sets \(D \subseteq \CC\) and such sets we call regions from now on (i.e. a region is an open connected set).
Next we talked about the convergence of series of complex numbers \[ \sum_{n=0}^\infty z_n \] and our emphasis was in power series, centered at 0 \[ \sum_n a_n z^n \] or centered at \(a \in \CC\) \[ \sum_n a_n (z-a)^n. \] This discussion is essentially a repetition of what we have learned about power series in our (real) analysis courses, the only difference being that instead of an interval of convergence centered at \(a\) we now have a disk of convergence again centered at \(a\). You can read the relevant material in [BN, § 2.2]. The main theorem about power series is Theorem 2.8 in there. Please read also carefully the example applications that follow that theorem. About absolute convergence you can read in [BN, § 1.3].
Problems:
Please do problems 7, 8, 9, 12, 13 from [BN, Chapter 2].
(
7-9
12-13
)
Today we defined what it means for a function \(f:\CC\to\CC\) to be differentiable at a point \(z \in \CC\). The definition is completely analogous to the one for real functions: \(f\) is differentiable at \(z\) if the limit \[\frac{f(z+h)-f(z)}{h}\] exists as \(h \to 0\). If the limit exists we call it the derivative of \(f\) at \(z\) and denote it by \(f'(z)\). Here \(h\) is a complex quantity so it can tend to 0 in many different ways, and that is why this notion of differentiability is, in a sense, rare. For instance the functions \(f(z) = \overline{z}\) and \(g(z) = \Abs{z}^2\) are NOT differentiable (but, for instance, all polynomials of \(z\) are differentiable).
We then showed for a function \[ f(z) = f(x+iy) = u(x+iy) + i v(x+iy) \] to be differentiable at \(z = x+iy\) a necessary condition is that the so-called Cauchy-Riemann equations \[ u_x = v_y,\ \ u_y - v_x \] are satisfied (here \(u = \Re{f}, v = \Im{f}\) and \(\cdot_x, \cdot_y\) denotes partial differentiation).
We then proved that if \(f(z)\) is given by a power series \[ f(z) = \sum_{n=0}^\infty a_n (z-a)^n \] in its circle of convergence \(\Set{z:\ \Abs{z-a} \lt R}\) then \(f\) is differentiable in that disk and the derivative is also given by a power series with the same radius of convergence \[ f'(z) = \sum_{n=1}^\infty n a_n(z-a)^{n-1}.\] Iterating this theorem we conclude that any power series is infinitely differentiable in its circle of convergence. We also derived the following formula that connects the coefficients of the power series with the values of its derivatives at the center: \[ a_n = \frac{f^{(n)}(a)}{n!},\ \ \ n=0, 1, 2, \ldots. \]
Read [BN, Chapter 2] and parts of [BN, § 3.1]. We did not say anything about § 2.1 but you can read it yourselves.
Problems:
Please do problems 1-3 from [BN, Chapter 3].
(
1-3
)
We first proved a uniqueness theorem [BN, Theorem 2.12] for power series: if $z_k \to a$ is a sequence of complex numbers, different from $a$, which converge to $a \in \CC$ and the power series $$ \sum_{n=0}^\infty a_n (z-a)^n $$ is equal to $0$ at each $z_k$ then $a_n = 0$ for all $n=0, 1, 2, \ldots$. A consequence of this is that if two power series are identical (as functions) on a disk around their common center then they have the same coefficients (hence the name uniqueness).
We then mentioned, without proof, the theorem that say that the Cauchy-Riemann conditions together with continuity of the partial derivatives guarantee complex differerentiability [BN, Proposition 3.2].
Afterwards we mentioned two properties of analytic functions that are unusual and do not hold for functions of a real variable, even if they have infinitely many derivatives [BN, Prop. 3.6 and 3.7]: an analytic function with constant real part or with constant modulus is necessarily constant.
Then we defined the entire functions $e^z, \cos z$ and $\sin z$, which extend, for $z \in \CC$, the well known functions of a real variable $e^x, \cos x$ and $\sin x$. We examined several properties of these funtions, and especially their range (which values the take, when $z \in \CC$) and their periodicity (a function $f:\CC\to\CC$ is called periodic with period $T \in \CC\setminus\Set{0}$ if $\forall z:\ f(z+T)=f(z)$). We saw that for each $\theta \in \RR$ we have $$ \cis{\theta} = e^{i\theta}, $$ so, from now on, we will not use the notation $\cis\theta$ anymore but use $e^{i\theta}$ instead [BN, § 3.2].
Problems:
Please do problems 18, 20 from [BN, Chapter 2] and 2, 5, 6, 7 from [BN, Chapter 3].
(
18-20
2
18-20
)
We derived (using Taylor's theorem) the power series for the function $e^x$, for real $x$, $$ e^x = \sum_{n=0}^\infty \frac{x^n}{n!}, $$ which converges for all real $x$. We then proved that the function $$ f(z) = \sum_{n=0}^\infty \frac{z^n}{n!} $$ (we can prove that this power series converges for all $z \in \CC$), which agrees with $e^x$ for $z \in \RR$, satisfies the functional equation $$ f(z+w) = f(z) f(w). $$ We had proved last time that there is precisely one function $f(z)$ with this property which extends $e^x$ to the complex plane, and we called this function $\exp(z)$ or $e^z$. Therefore we have obtained the power series representation of $e^z$: $$ e^z = \sum_{n=0}^\infty \frac{z^n}{n!} $$ valid for all $z \in \CC$. Using our definition for the functions $\cos{z}$ and $\sin{z}$ $$ \cos{z} = (e^{iz}+e^{-iz})/2,\ \ \ \sin{z} = (e^{iz}-e^{-iz})/(2i), $$ we therefore obtained the power series representation of these functions. We also observed that since $\cos{z}$ is an even function (i.e., $\cos(-z) = \cos{z}$) the implication is that all odd powers of $z$ should be missing from the power series of $\cos{z}$. Similarly all even powers are missing from the power series for $\sin{z}$. (This is a simple consequence of the uniqueness theorem for power series.)
We then moved on to [BN, Chapter 4] and defined contour (or line) integrals of the form $$ \oint_{C} f(z)\,dz, $$ where $C$ is a curve in the complex plane given by a parametrization $z(t)$, where $t$ belongs to some interval $[a, b]$. We did several examples of such integrals. Please read [BN, § 4.1] up to Prop. 4.11.
Problems:
Please do problems 14-18 from [BN, Chapter 3] and 2, 3, 6 from [BN, Chapter 4].
(
14-18
2-3
6
)