Complex Analysis
Fall 2022-23
Department of Mathematics and Applied Mathematics
University of Crete
Teacher: Mihalis Kolountzakis
▶ Announcements
No class tomorrow, Monday December 5, as I am still sick. See you Wednesday.
No class tomorrow, Wednesday November 30, because of a slight emergency. Also no office hours on Thursday. See you Monday.
You can find the results of the midterm exam here.
You can have a look at the intermediate exam of this course (of 2019-20) here:
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Please come prepared to ask questions on Monday morning. Go over all homework problems as well as over all problems of the old exam.
Intermediate exam: Monday 21/11/2022, 17:30-19:30 in auditorium A201. The morning lecture on that Monday will be held normally.
Those of you who have a significant reason to be exempted from this exam must contact me by email until Sunday, Nov. 13, 2022. No exceptions beyond that date. Students granted exemption, who will only take the final exam, can be seen here.
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
▶ Student evaluation
Intermediate exam 40%, final exam 60%. This remains the same for all further examination periods.
▶ Class diary
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.
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
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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.
We solved several of the problems of the 1rst problem set. Then we talked about limits of complex sequences and continuity of complex functions at a point. We said what it means for $z_n \to \infty$ and described the stereographic projection. We saw several simple examples of functions $f:\CC\to\CC$ and saw how they transform some simple shapes. Next we saw some examples of parametrizations of curves in the complex plane. Finally what defined the derivative of a function $f:\CC\to\CC$ at a point $z_0$ where it is defined. We saw that the functions $z, z^2$ are differentiable everywhere but that the function $\overline{z}$ is not.
Problems: No 2
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We proved some theorems, mostly identical with the real case or, at least, with the corresponding result about real functions on the plane, regarding convergence, convergence to infinity, continuity and when it is preserved. The last thing we proved is that if $\Omega\subseteq\CC$ is a bounded, closed set and $z_n \in \Omega$ then the sequence $z_n$ has a convergent subsequence which converges to a point $z\in\Omega$. By now we have covered everything in [CB] up to the definition of a complex derivative.
We used the last result of last time to prove that for every continuous function $f:\Omega\to\CC$, where $\Omega \subseteq \CC$ is bounded and closed, there is $z_0 \in \Omega$ such that $\Abs{f(z_0)} = \sup_{z \in \Omega} \Abs{f(z)}$. Then we returned to the definition of a complex derivative and we indicated that the usual rules of differentiation hold, as we knew them for real functions of a real variable. We talked about polynomials, how we divide them, and why if a polynomial $P(z)$ vanishes at $\rho \in \CC$ it can be written in the form $$ P(z) = (z-\rho) Q(z) $$ where $Q(z)$ is another polynomial of degree $\deg Q = \deg P -1$. This implies that a polynomial of degree $n$ cannot vanish on $n+1$ different complex numbers unless it is constantly 0. It also implies that if two polynomials $P, Q$ of degree $\le n$ agree on $n+1$ points then they are the same polynomial.
Writing $f(z) = u(z)+iv(z)$, with real functions $u, v$, we proved that if $f'(z)$ exists at $z = x+iy$ then the Cauchy-Riemann equations hold: $$ u_x = v_y,\ \ u_y = - v_x. $$
Problems: No 3
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We solved several problems from Problem Set No 2. Then we proved that if a function $f(z) = u(z)+iv(z)$ satisfies the Cauchy-Riemann equations at a point $z=x+iy$ and if the partial derivaties $u_x, u_y, v_x, v_y$ exist and are continuous in a neighborhood of the point $x+iy$ then $f'(z)$ exists. Along the way we rememberd the notations $O(\cdot)$ and $o(\cdot)$ and what they mean.
First we saw how the Cauchy-Riemann equations transform in polar coordinates. Then we saw how to apply the C-R equations to show that several functions (such as $e^z$, $1/z$) are differentiable. We defined what it means for a function to be analytic at a point or to be anayltic on a set $S \subseteq \CC$. We talked about the function $\sqrt{z}$ and saw that, in order to have it continuous on its domain of definition, we are forced to exclude the negative semi-axis from its domain (this is the slit complex plane). We then talked about harmonic functions and their conjugates.
Problems: No 4
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We talked about the function $\exp(z) = e^z = e^{\Re{z}} e^{i \Im{z}}$. We saw some of its mapping and periodicity properties. We also saw how it grows when $z$ moves towards $\infty$ in the complex plane. Then we defined the trigonometric functions $\cos{z}$ and $\sin{z}$ using the exponential function and studied them similarly. We also talked briefly about the hyperbolic functions $\cosh{z}$ and $\sinh{z}$ which are also defined via the exponential function. We defined the logarithm function ${\rm Log}\, z = \ln\Abs{z}+i\Arg{z}$ and saw that it cannot be defined as a continuous functions on $\CC\setminus\Set{0}$ and we can have continuity and analyticity if we restrict this function to be defined on the slit plane (remove the negative half-axis plus the point 0).
We started talking about the complex integral along a curve $\Gamma$ in the complex plane. We defined the integral via a parametrization of $\Gamma$ and saw, by an example, that the parametrization used on the curve does not affect the outcome. We will prove this later.
We first saw that, in the definition of the contour integral, the parametrization of the curve does not affect the outcome. Then we talked about the triangle inequality $$ \Abs{\oint_\Gamma f(z) dz} \le \oint_\Gamma \Abs{f(z)} \Abs{dz} $$ where $\Abs{dz} = \Abs{z'(t)}dt$, and saw the so-called M-L bound $$ \Abs{\oint_\Gamma f(z) dz} \le M L $$ where $L = \oint_\Gamma \Abs{dz}$ is the length of the curve and $M$ is an upper bound for $\Abs{f(z)}$ on $\Gamma$.
We computed several examples of contour integrals using the definition.
Then we saw that if $f(z) = F'(z)$ on $\Gamma$ we have $$ \oint_\Gamma f(z) dz = F(b)-F(a), $$ where $a$ and $b$ are the two endpoints of the curve $\Gamma$ (from $a$ to $b$). Using this we can understand many of the zeros that we got in our example contour integrals.
Problems: No 5
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We started by computing a few more examples of contour integrals (e.g. the integral of $1/z^2$ over the upper semicircle of radius 3 centered at the origin). We then saw a few cases where we applied the so-called $M\cdot L$ bound to obtain upper bounds for some integrals (not precise values).
We then proved that for a continuous function $f$ on a domain $D$ the following are equivalent:
Then we remembered the real contour integrals of Calculus, i.e. the expressions of the form $\oint_{C} P(x, y)dx + Q(x,y)dy$ as well as Green's theorem which relates such contour integrals over simple closed curves the the double integral of the interior of the curve. We used Green's theorem, coupled with the Cauchy-Riemann equations to show that every (complex) contour integral $\oint_C f(z)\,dz=0$ over any closed curve $C$ for any function $f$ that is analytic in the interior of $C$ (and on $C$) and for which we assume that it has a continuous derivative $f\prime$ there. This last requirement is not really necessary and we will see a proof of this next.
We first proved Cauchy's theorem on a rectangle with no continuity assumption for the derivative of $f$ (we followed the proof in Newman's book). Then we extended this to every simple closed curve $C$ such that $f$ is analytic on $C$ and in the interior of $C$ (the conclusion is that $\oint_C f(z)\,dz = 0$). We saw what are simply connected domains and proved that any function that is analytic on a simply connected domain $D$ has an antiderivative on $D$ (and, therefore, its integrals on any closed curve in $D$ is 0). Finally we saw that if $C_1$ is a simple closed curve which contains in its interior some simple closed curves $C_2, \ldots, C_n$ (such that all curves $C_j$ are exterior to any curve $C_i$, $i=2,\ldots,n$) then $$ \oint_{C_1} f(z)\,dz = \oint_{C_2} f(z)\,dz + \cdots + \oint_{C_n} f(z)\,dz. $$
Problems: No 6
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Today we proved Cauchy's integral formula for the function $f$ and its derivatives of all orders. Along the way we proved that if a function $f$ is analytic at a point $z_0$ then it has derivatives of all orders at $z_0$. The formula is $$ f^{(n)}(z_0) = \frac{n!}{2\pi i} \oint_C \frac{f(z)\,dz}{(z-z_0)^{n+1}}, $$ where $C$ is a simple closed curve with $z_0$ in its interior and $f$ is a function that is analytic inside and on $C$. The left-hand side $f^{(n)}(z_0)$ is the $n$-th derivative of $f$ at $z_0$ (by definition the $0$-th derivative is the function $f$ itself). We then used these formulas to compute several concrete integrals.
We solved several problems today, from problems sets 5 and 6 and also some problems involving using Caychy's integral formulas.
Problems: No 7
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We proved Morera's theorem, which says that if a continuous function $f$ has 0 integral over any closed curve then it is analytic. Then we proved the Maximum Modulus Principle, which says that if a function is analytic at $z_0$ and $\Abs{f}$ has a local maximum at $z_0$ then $f$ is constant in a neighborhood of $z_0$. If a function is analytic and not constant on a connected open $D \subseteq \CC$ then we also proved that $\Abs{f}$ cannot have any local maxima in $D$.
We used the maximum modulus principle to prove a similar statement for the minimum modulus (provided the function does not vanish) and also for the real and imaginary parts of an analytic function. Then we proved the so-called Cauchy estimates which bound the function and its derivatives at a point using the maximum of the function on a circle centered at the point and used them to prove Liouville's theorem (a bounded entire function is necessarily a constant). Then we used Liouville's theorem to prove the Fundamental Theorem of Algebra (every non-constant polynomial has a root in $\CC$).
Problems: No 8
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In our morning session we solved the problems from the homeworks that had not gone over yet, as well as the problems from the midterm exam
of 2019-20. In the evening we held our midterm exam: /
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After a brief introduction to series of complex numbers we proved Taylor's theorem, which states that if $f$ is analytic in the open disk $$ D = \Set{z:\ \Abs{z-z_0} \lt R} $$ then for all $z \in D$ we have the Taylor series expansion $$ f(z) = \sum_{n=0}^\infty \frac{f^{(n)}(z_0)}{n!} (z-z_0)^n. $$ We also showed that if $f$ is analytic in the annulus (ring) $$ A = \Set{z:\ R_1 \lt \Abs{z-z_0} \lt R_2} $$ (where $0\le R_1 \lt R_2 \le +\infty$) then for each $z \in A$ we have the Laurent expansion $$ f(z) = \sum_{n=-\infty}^\infty a_n (z-z_0)^n, $$ for some complex numbers $a_n$, $n \in \ZZ$.
We then saw several examples of how we compute such Taylor or Laurent expansions for specific functions $f(z)$.
Problems: No 9
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We went through some of the midterm exam problems. Then we described some more examples of Laurent series of functions, how they are derived and where they hold. Then we proved that if a power series $\sum_{n=0}^\infty a_n z^n$ converges for a point $z_1$ then it converges absolutely for all $z$ with $\Abs{z} < \Abs{z_1}$. If $0<\rho < \Abs{z_1}$ the convergence is even uniform for $\Abs{z} \le \rho$. We showed how this theorem transforms for Laurent series.
Problems: No 10
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No class today as the instructor was sick.
No class today as the instructor was sick.
We proved the formula $1/R = \limsup_{n\to\infty} \Abs{a_n}^{1/n}$ for the radius of convergence of the power series $\sum_{n=0}^\infty a_n (z-z_0)^n$. We saw that the uniform convergence in each proper subdisk of the disk of convergence implies that the resulting function is continuous. We proved that a power series $\sum_{n=0}^\infty a_n (z-z_0)^n$ represents an analytic function in the circle of its convergence. A similar result holds for a Laurent series $\sum_{n=-\infty}^\infty a_n (z-z_0)^n$ in its annulus of convergence. Then we saw that a power series can be differentiated term by term (in its region of convergence). This allows for easier computations in several cases. We also proved the uniqueness of expansion in power series: if two power series $\sum_{n=0}^\infty a_n (z-z_0)^n$ and $\sum_{n=0}^\infty b_n (z-z_0)^n$ both converge for $\Abs{z-z_0}< R$ to the same function then $a_n = b_n$ for all $n \ge 0$.
Problems: No 11
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We showed the uniqueness of Laurent expansion. Then we saw how to handle the product of two Taylor (or Laurent) series, and especially how to compute a few of the coefficients of the product. We also saw how to invert a power series, at least how to find some of its low coefficients. Then we talked about isolated singularities and saw several examples of functions and their isolated and non-isolated singularities. We defined the residue of a function at one of its isolated singularities and saw that to compute $\oint_C f(z)\,dz$ all we have to do, if there are finitely many singularities inside $C$, is to find the residue at each singularity.
We defined the three kinds of isolated singularities: removable, poles and essential singularities. We proved that at a pole the limit is $\infty$. We mentioned the theorem of Picard, which says that in every neighborhood of an isolated singularity the function takes all complex values except, perhaps, one. This theorem implies that the limit of the function at an essential singularity does not exist. We saw what it means for a function to have a root at $a$ of order $m$: it means that the function and its derivatives of order $< m$ are all 0 at $a$. Equivalently it means that its power series around $a$ starts with the term $(z-a)^m$. For a function to have a pole of order $m$ at $a$ it means that its Laurent series at $a$ starts with the term $(z-a)^{-m}$ and contains no terms of the form $(z-a)^{-k}$ with $k>m$. For a function to have a zero of order $m$ at $a$ it is equivalent that we can write (in some neightborhood of $a$) $f(z)=(z-a)^mg(z)$ where $g(z)$ is analytic at $a$ and $g(a) \neq 0$. Equivalently, for a function to have a pole at $a$ of order $m$ it is equivalent that we can write (in some neighborhood of $a$) $f(z) = \frac{g(z)}{(z-a)^m}$, with $g$ analytic at $a$ and $g(a)\neq 0$. We proved that if a function is analytic in an open, connected region $D$ and $z_n \in D$ converges to $z \in D$, and $f(z_n)=0$ for all $n$ then $f$ is identically 0 in $D$. In other words the zeros of an analytic function cannot have a point of accumulation $z$ at which the function is analytic. For entire functions this implies that in any bounded disk we can only have finitely many roots inside. Finally we proved that if $f$ is analytic at $D \subseteq \CC$ then the function $g(z) = \overline{f(\overline{z})}$ is analytic on $\overline{D}$.
Problems: No 12
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We proved that the number of roots of $f(z)$ inside a closed curve $C$ is $\frac{1}{2\pi i}\oint_C \frac{f\prime(z)}{f(z)}\,dz$. We then used this to prove the theorem of Rouche and saw how to use it in an example. We spent the second hour to go over the problems of problem set No 10.
Today we saw how to compute some kinds of improper integrals (such as $\int_{-\infty}^\infty \cdots$) using the residue calculus. We also solved some of the homework problems (up to homework No 11).
Problems: No 13
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