Complex Analysis, Fall 2019-20

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 ( / )

Thu, 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 ( / )

Tue, 22 Oct. 2019:

[CB, Ch. 4, § 32-33] We stated and proved the triangle inequality for contour integrals, which we write in the form $$ \Abs{\oint_\Gamma f(z)\,dz} \le \oint_\Gamma \Abs{f(z)} \cdot \Abs{dz}. $$ From this we proved the so-called M-L inequality $$ \Abs{\oint_\Gamma f(z)\,dz} \le M \cdot L, $$ where $M = \sup_{z \in \Gamma}\Abs{f(z)}$ and $L$ is the length of the curve $\Gamma$. Then we did several examples of contour integrals. We also proved that if $f(z) = F^\prime(z)$ on the curve $\Gamma$ (which goes from point $a$ to point $b$) then $$ \oint_\Gamma f(z) dz = F(b)-F(a). $$ If $\Gamma$ is a closed curve then this integral is 0. This also means that the integral depends only on the endpoints of $\Gamma$ in this case, not on the actual shape of $\Gamma$ itself. /p>

Problems: No 8 ( / )

Thu, 24 Oct. 2019:

[CB, Ch. 4, § 34-35] Today we finished the proof of the Theorem in [CB, Ch. 4, § 34]. More specifically, we proved that if a function $f$, defined and continuous everywhere in a domain $D \subseteq \CC$ has the property that its integrals do not depend on the path but only on the endpoints then there exists an analytic function $F:D\to\CC$ for which $F^\prime = f$ everywhere in $D$.

Then we revisited Green's formula from Calculus: $$ \iint_\Omega Q_x - P_y \,dx dy = \int_{\partial\Omega} Pdx+Qdy, $$ where $\Omega \subseteq \RR^d$ is a region in the plane and $\partial\Omega$ is its boundary curve, positively oriented, and $P, Q$ are two real functions defined in an open neighborhood of $\Omega$ and having continuous first derivatives. Then we used Green's formula to prove Cauchy's formula theorem for analytic functions: if a function $f$ is analytic in the interior of a simple closed curve $\Gamma$ then $$ \oint_\Gamma f(z)\,dz = 0. $$ To be precise, our proof uses some extra assumptions: that $f$ has continuous derivatives and that it is defined and analytic in an open neighborhood of $\Omega$. The proof without these assumptions can be found in [CB, Ch. 4, § 36-37] but we are not going to go through it. It is a very nice proof and I encourage you, or those of you who like mathematics very much, to read it and understand it. We are going to use Cauchy's theorem as stated here from now on.

We also saw an example of how using Green's formula can help us find a formula for the area of a polygon in the plane of which we know the coordinates of its vertices. You can read about it in my problem blog here. You can also find a continuation of this technique (nothing to do with complex variables) here. (Sorry, Greek only.)

Problems: No 9 ( / )

Tue, 29 Oct. 2019:

[CB, Ch. 4, § 36-39] Today we applied Cauchy's theorem to the evaluation of contour integrals. We did several examples, some from Problem Set 9 and some from the midterm exam of last year. We saw how to change the contour without changing the integral in several cases, depending on where our function is analytic.

Problems: Try to do the midterm exam from last year. With what you know you can do all problems except 5 and 6. This midterm will contain more stuff from what we did before coming to Cauchy's theorem. ( / )

Thu, 31 Oct. 2019:

[CB, Ch. 4, § 40-41] We proved Cauchy's formulas for the derivatives of an analytic function: $$ f^{(n)}(z) = \frac{n!}{2\pi i}\oint_C \frac{f(w) dw}{(w-z)^{n+1}}, $$ where $C$ is a simple closed curve with $z$ in its interior and with $f$ being analytic on $C$ and inside $C$. This implies that every analytic function has derivatives of all orders. We also proved Morera's theorem: if $f$ is such that its contour integrals on closed curves are all 0 then $f$ is analytic.

Please come to my office hours on Monday if you have questions regarding some problems.

Tue, 5 Nov. 2019:

[CB, Ch. 4, § 42] We proved the Maximum Modulus Priciple for analytic functions: if $f$ is analytic at $z$ then $\Abs{f}$ cannot attain a local maximum at $z$. In other words, if $\Abs{f(z)}$ is $\ge \Abs{f(w)}$ for all $w$ in an open neighborhood of $z$ then $f$ is constant in that neighborhood.

Problems: No 10 ( / )

Thu, 7 Nov. 2019:

[CB, Ch. 4, § 43] We proved Liouville's theorem (and some variations of it) and we used it to prove the Fundamental Theorem of Algebra, which says that every polynomial with complex coefficients has a root in $\CC$. This implies that every polynomial of degree $n$ can be written in the form $$ p(z) = a(z-\rho_1)\cdots(z-\rho_n), $$ where $a$ is the top degree coefficient of the polynomial and the $\rho_j$ are its $n$ roots (not necessarily distinct). If the polynomial has real coefficients then its non-real roots come in complex conjugate pairs. This implies that every polynomial with real coefficients can be written as a product of real linear or quadratic factors.

Problems: No 11 ( / )

Tue, 12 Nov. 2019:

[CB, Ch. 5, § 44] Today we mostly remembered material that we have learned in our (real) Analysis courses regarding the convergence of series of numbers and of functions. We proved that absolute convergence implies convergence of a series of functions. We remembered the difference between pointwise and uniform convergence (in a domain) of a sequence or a series of functions. We proved that uniform limits of analytic functions are also analytic, using uniform convergence and Morera's theorem. We also saw that if $f_n \to f$ uniformly then $f_n'(z) \to f'(z)$.

Problems: No 12 ( / )

Thu, 14 Nov. 2019:

[CB, Ch. 5, § 45-46] We went over Problem Set 11 today. Then we proved Taylor's theorem (in § 45). Please read the examples given in § 46 before doing the homework.

Problems: No 13 ( / )

Tue, 19 Nov. 2019:

[CB, Ch. 5, § 47-48] During the first hours we went over several examples of how to compute a Taylor series of a given function (as well as to find its radius of convergence). During the next hour we proved that a function which is analytic in an annulus $$ R_1 \lt \Abs{z-z_0} \lt R_2, $$ has an expansion in a Laurent series of the form $$ \sum_{n=-\infty}^\infty a_n (z-z_0)^n. $$ Please read the examples given in § 48 before doing the homework.

Problems: No 14 ( / )

Thu, 21 Nov. 2019:

[CB, Ch. 5, § 48] Today we did several examples of finding the Laurent series for some functions in all the relevant annuli.

Problems: No 15 ( / )

Tue, 26 Nov. 2019:

[CB, Ch. 5, § 49-52] We showed that a power series converges in an open disk (its disk of convergence) and defines there an analytic function, whose Taylor series is the original power series itself. The same is true for Laurent series (power series where negative powers are allowed) but instead of in a disk a Laurent series converges to an analytic function in an annulus, where the subseries of positive powers determines the outer radius and the subsequence of negative powers determines the inner radius. Thus we have established a correspondence between analytic functions on a disk and power series with center at the center of that disk. Similarly we have a correspondence between analytic functions in an annulus and Laurent series with center at the center of that annulus. We also proved that power series (and Laurent series) can be differentiated termwise.

Problems: No 16 ( / )

Thu, 28 Nov. 2019:

[CB, Ch. 6, § 43-57] We talked about the 3 different kinds of isolated singularities of an analytic function and we saw how to distinguish them by looking at the principal part of the Laurent series of the function around the singularity (the singular part is the part of the Laurent series with negative powers). We defined the residue of a function at a singularity and we saw that knowing the residue is enough to allow us to compute the integral of the function on a curve around the singularity. In other words, whenever we are given to compute the integral of a function along a simple closed curve it is enough to find the residues of the function inside that curve. We did several examples. We also saw that a function $f(z)$ with a pole of order $m$ at a point $z_0$ can be written in the form $$ f(z) = \frac{g(z)}{(z-z_0)^m} $$ where $g$ is analytic at $z_0$ and $g(z_0) \neq 0$.

Problems: No 17 ( / )

Tue, 3 Dec. 2019:

[CB, Ch. 6, § 43-57] We stated without proof Picard's theorem that if $z_0$ is an essential singularity of $f(z)$ then $f(z)$, in any neighborhood of $z_0$, takes all the values in $\CC$, except possibly one, infinitely many times. We verified it for the function $e^{1/z}$ at the essential singularity 0 and for the value 1.

Next we saw how to find the residue of a function $f$ at $z_0$ if we know that $f$ has a pole of order $m$ at $z_0$ (§ 56). We did several examples.

Finally we went over the problems from problem set 16.

Problems: No 18 ( / )

Thu, 5 Dec. 2019:

[CB, Ch. 6, § 58, 63] First we went over problem 3 of problem set 16 again and explained why the Laurent series we found for the annulus $0 < \Abs{z} < 2\pi$ is not the same with the Laurent series for the next annulus $2\pi < \Abs{z} < 4\pi$. Problem 1 of problem set 18 is precisely about this.

We defined the logarithmic derivative of a function and saw that its integral around a simple curve counts the zeros minus the poles of the functions inside. We used this to prove the theorem of Rouché. We saw an example of using that theorem and reproved the fundamental theorem of algebra using it (problem 7 of § 63).

We started talking about how we evaluate integrals of the form $\int_{-\infty}^\infty f(x) \,dx$ using residue theory. We continue next time.

Problems: No 19 ( / )

Tue, 10 Dec. 2019:

[CB, Ch. 6, § 58, 59] We saw how to compute integrals of the form $\Ds \int_{-\infty}^\infty \frac{p(x)}{q(x)}\,dx$ ($p, q$ are real polynomials, with $\deg{q} \ge \deg{p}+2$ to ensure convergence), by reducing them to a contour integral and evaluating this using the residue theorem. Then we also examined integrals of the form $\Ds \int_{-\infty}^\infty \frac{p(x)}{q(x)}\cos(ax)\,dx$, with $a>0$. We saw that to make an analogous reduction we have to compute, instead, the integral $$ \int_{-\infty}^\infty \frac{p(x)}{q(x)}e^{i a x}\,dx, $$ whose real part is the integral we started with. We did not cover § 59 beyond equation (7).

Problems: No 20 ( / )

Thu, 12 Dec. 2019:

[CB, Ch. 6, § 59, 60, 64, 65] We talked about how to compute integrals of the form $\Ds \int_{-\infty}^\infty \frac{p(x)}{q(x)} \cos x\,dx$, where $p(x), q(x)$ are real polynomials, $q$ has no real roots, and $\deg{q} = \deg{p}+1$. Such an integral does not converge absolutely (the degree of $q$ should be at least 2 more than the degree of $p$ for this to happen) but it does converge. We use an inequality of Jordan (inequality (7) in § 59) in order to evaluate our integral. In this manner we compute the Principal Value $$ PV \int_{-\infty}^\infty \cdots = \lim_{M \to +\infty} \int_{-M}^M \cdots $$ of the integral. But the integral is actually convergent and we proved this using integration by parts. We also showed how to use summation by parts in order to show that certain infinite series that do not converge absolutely actually converge. We applied this method to the simple series $\Ds\sum_{n=1}^\infty \frac{(-1)^n}{n}$.

We then covered integrals of the form $\int_0^{2\pi} F(\sin\theta, \cos\theta)\,d\theta$ and we saw how to compute them by reducing them to a contour integral along the unit circle (§ 60).

We then started talking (again) about simple transformations of the form $w = f(z)$, where $f(z)$ is a very simple function and saw that the transformation $w = \frac{1}{z}$ transforms any circle or line again to a shape that is a circle or line.

Problems: No 21 ( / )

Tue, 17 Dec. 2019:

[CB, Ch. 6, § 64-66] We proved and saw several examples of mapping properties of the function $w=1/z$. We saw how we include the point $\infty$ into the complex numbers and how se use it in order to distinguish several situations in mapping problems. We defined Möbius transformations and we saw that they too map circles or lines to circles or lines and that they form a group of transformations under composition. We also saw that for any three distinct points $z_1, z_2, z_3$ and for any choice of distinct $w_1, w_2, w_3$ we can always find a Möbius transformation that maps $z_j$ to $w_j$, for $j=1,2,3$.

Problems: No 22 ( / )

Thu, 19 Dec. 2019:

[CB, Ch. 6, § 64-66] We saw several examples of Möbius transformations and how we handle them. We saw how we find such transformations which map given domains to some other domains and how to find the image of a domain under such a transformation. We saw that Möbius transformations are a spectial case of conformal mappings (analytic maps which are 1-1 and onto from one domain to another) and proved that any conformal map preserves angles between curves. As an example of a conformal mapping which is not a Möbius transformation we saw how to map a semidisk to a half-plane by appropriately using the function $z^2$.

Problems: No 23 ( / )

Complex Analysis

Fall 2019-20

Department of Mathematics and Applied Mathematics

University of Crete