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MATH 617/618 materials (Nataliya Goncharuk, 2023/2024)

MATH 617

Quizzes and tests:

Quiz, Midterm, Final exam,

Homeworks:

HW1, HW2, HW3, HW4, HW5, HW6, HW7, HW8, HW9, HW10, HW11, HW12,

MATH 618

Quizzes and tests:

Quiz 1, Midterm, Quiz 2, Final exam, Exam program

Homeworks:

HW1, HW2, HW3, HW4, HW5, HW6, HW7, HW8, HW9, HW10,

 

Course journal MATH 617:

On Monday, we discussed that algebraically, C is a field; discussed the polar representation and de Moivre's formula; introduced the notation Re z, Im z, |z|, \bar z, arg z, and e^z. On Wednesday, we plotted sets on the complex plane, discussed roots of unity (and of other complex numbers), validity of various algebraic formulas for C, and I stated that C is an algebraically closed field. On Friday, I introduced the notion of the metric space and described the Riemann sphere (the extended complex plane \bar C) as a metric space, using the stereographic projection. I also briefly introduced the notion of an abstract manifold (we will not need this till the next semester).

See http://www.dimensions-math.org/Dim_tour_E.htm for a math film by J.Leys, E.Ghys, and  A.Alvarez; I used part of its Sec 1 to illustrate the stereographic projection, but you might also enjoy Sections 5 and 6 on complex numbers. 

Week 2, Aug 28-Sept 1    Ch 2, §1-3: Complex numbers as a metric space and a topological space

On Monday, I recalled the definition of the metric space and introduced notions of open, closed sets, boundary, interior, and closure. We proved that an open ball is open; unions and finite intersections of open sets are open; boundaries of the set and its complement coincide; interior is open. Finally, I gave a definition of a limit of a sequence and proved that closed sets are closed under passing to limits. On Wednesday, I introduced the notion of dense sets. Then I defined  Cauchy sequences and complete metric spaces, and we showed that C and \bar C are complete. Also, we had our first presentation -- on Funny Metric Spaces. On Friday, I gave two definitions of a compact set (sequential and through open covers), proved Heine-Borel theorem, introduced two definitions of continuity (eps-delta and sequential), and showed that a continuous function on a compact set attains max/min of modulus. 

Presentations: Wesley gave a presentation on funny metric spaces. 

On Wednesday, I discussed theorems related to limits and continuity: arithmetic properties, three definitions of continuity (eps-delta, sequential, and via inverse images of open sets), and proved that continuous functions take compacts to compacts. Also, I gave a definition of connected set and proved the Intermediate Value Theorem. On Friday, I gave a definition of uniform continuity, proved that continuous functions on compacts are uniformly continuous, and explained why uniformly continuous functions extend to the complement. I gave a definition of a uniformly convergent sequence of functions and formulated the result that uniformly convergent sequence of continuous functions has a continuous limit.

Presentations: Alex told us about Gauss numbers and their applications to number theory; Levi explained  how we can use complex numbers to prove trigonometric identitites. 

On Monday, I proved that a uniform limit of continuous functions is continuous, proved the Weierstrass M-test (if you can estimate a functional series by a convergent number sequence, then the series converges uniformly), introduced power series and started to prove the formula for the radius of convergence. On Wednesday, I proved the formula for the radius of convergence, discussed operations with power series (addition, multiplication, substitution), defined analytic functions, and introduced complex differentiation. On Friday, I discussed Cauchy-Riemann conditions for complex differentiability. Also, I formulated the theorem on differentiability of power series and discussed the relation between power series and Taylor series. 

Presentations: Matthew and Vishwam told us about the Mandelbrot set.

On Monday, I proved the theorem on differentiability of power series, introduced the complex logarithm and its principal branch Log, and obtained the power series expansion of log(1+x). In particular, I showed that if a C-differentiable function has a zero derivative in an open connected set, then it is constant. This required a lemma on staircase-like paths in open connected sets from Sec. 2.2. On Wednesday, I proved that C-differentiable functions with nonzero derivatives preserve angles. Then I discussed geometry of exp, log, z^a for real a, and started to talk about Moebius maps z -> (az+b)/(cz+d). On Friday, I showed colorful visualizations of complex functions. Also, I proved the following on Moebius maps: They are 1-to-1 in the extended complex plane \bar C, and conformal; there is exactly one Moebius map that takes given three points of \bar C to given three points; Moebius maps take circles and lines to circles and lines.  

The tool to visualize complex functions: https://samuelj.li/complex-function-plotter/#z%5E2*(z-t)*(1%2Bt*z)

         Take-home term test is due on Mon Oct 2. It covers Weeks 1-5. 

On Monday, I defined the cross-ratio and proved that it is preserved under Moebius maps. I used this to prove that the cross-ratio is real if and only if the four points are on the same circle or line, and that Moebius maps preserve symmetry with respect to circles and lines. As a result, we obtained a useful formula for Moebius maps that preserve the unit circle. On Wednesday, I defined the Riemann-Stieltjes integral for piecewise smooth curves and proved its properties (linearity, independence on parametrization, FTC), then defined the Riemann-Stieltjes integral in the general case for rectifiable curves. On Friday, I was dealing with the Riemann-Stiltjies integral for continuous rectifiable curves. I proved that it gives the same value for  piecewise smooth curves as the Riemann integral of f * (gamma' ) and outlined ideas of the proof of integrabiity of continuous functions and on approximating rectifiable curves by piecewise continuous curves. Also, I formulated the Cauchy's theorem.

 Another tool to visualize complex mappings: https://mabotkin.github.io/complex/

On Monday, I outlined the proof of the Green's theorem and used it to prove Cauchy's theorem (assumptions: f is continuously complex differentiable on some open set, our domain is inside this set and is bounded by several piecewise smooth curves). On Wednesday, I proved Cauchy's integral formula (same assumptions) and derived that any continuously complex differentiable function is analytic (expands in power series) and obtained the integral formula for its derivatives. On Friday, I showed that we can integrate termwise uniformly convergent series (we used this last time). After that, I derived Cauchy estimates, proved Lioville's theorem and obtained the Fundamental Theorem of Algebra as a corollary. Also, I showed that the value of a holomorphic function in the center of a circle is the average of its values over the circle, and briefly explained how this implies the maximum principle.

Presentations: Kevin and Preston gave a talk on modular forms.  

On Wednesday, I discussed the notion of multiplicity for a zero of a holomorphic function and proved the theorem on non-accumulation of zeros for a holomorphic function. On Friday, I proved the Maximum Modulus principle and discussed its corollaries including the Schwartz' lemma and  proved Morera's theorem. 

Week 9, Oct 16-20          Ch 4, §4-6: Indices of planar curves.  Homotopic version of Cauchy's theorem and Cauchy's integral formula. 

On Monday, I discussed homotopy of the curves with fixed endpoints, introduced the group pi_1 (first homotopy group), and proved the homotopic version of the Cauchy integral formula (if two curves are homotopic curves in the domain where f is holomorphic, then the integrals over these curves are equal). Also, I introduced indices of the curves with respect to points. On Wednesday, I expressed the integral of a holomorphic function over the closed curve in a domain with holes via the integrals around the holes, using indices of the curve with respect to the holes, and explained the relation to the group H_1 of the punctured plane. Also, I proved that any holomorphic function in a simply connected domain has an antiderivative. Finally, I explained how the log map can be used to prove that curves with the same indices in C\{0} are homotopic.   

On Monday, Yuri proved Goursat's theorem. On Wednesday, he discussed a way to count zeros of the function f in the domain using the integral of f'/f along the boundary gamma of the domain  (this is the number of turns of f(gamma) around zero). On Friday, he used this to re-prove the Fundamental theorem of Algebra and to prove the Open Mapping theorem (images of open sets under holomorphic functions are open). Also, he proved Rouche's  theorem  (see Sec.3.8) for the number of turns of f(gamma). 

On Monday, we discussed several examples on the Rouche's theorem, and I proved the stronger version of the Rouche's theorem (if |f-g|<|f|+|g| on the curve gamma, then f(gamma) and g(gamma) make same number of turns around zero). Also, I introduced the classification of isolated singularities (removable, poles, and essential) and showed colorful visualizations. On Wednesday, I provided the classification of singularities of holomorphic functions and proved the Casorati-Weierstrass theorem. On Friday, I showed that any holomorphic function in the annulus uniquely expands in Laurent series. 

Presentations: David gave a presentation on Esher's tilings.

On Monday, I proved the Residue theorem. On Wednesday, we considered many examples of its applications, including to functions with essential singularities and to improper integrals.  On Friday, I proved the Argument principle (including the weighted version) and derived the integral formula for  the inverse function; also, I discussed examples, showing how to determine the number of solutions of the equation f(z)=c in the domain D looking at the image of the boundary of this domain f(d D ). 

 Presentations: Michail gave a talk on the Riemann zeta function.

On Monday, I presented a proof of the Maximum Modulus theorem using the Open Mapping theorem and proved the version of the Maximum Modulus principle for unbounded domains. Also, I formulated and proved the Schwartz's inequality.  On Wednesday, I proved the Schwartz's inequality and deduced that the only one-to-one holomorphic mappings of the open unit disc are Moebius maps. On Friday, I used the Riemann mapping theorem to extend the Schwartz's lemma to other open simply connected domains. Also, i defined the Poincare (hyperbolic) metric in such domains, and proved the Schwartz's- Ahlfors-Pick lemma (holomorphic self-maps of open simply connected domains contract in the hyperbolic metric).

Presentations:  Try gave a presentation on the hyperbolic geometry.

On Monday, I proved that any harmonic function is a real part of a holomorphic function in a simply connected domain. On Wednesday, I obtained the Poisson formula in the unit disc using the Cauchy integral formula, and explained how to prove that it provides the solution of the Dirichlet problem in the unit disc. On Friday, I gave a geometric interpretation for the behavior of the Poisson kernel on the unit circle and explained how we can use conformal mappings to solve Dirichlet boundary problems in other domains. As an illustration, I used a Moebius map that takes the disc into the upper half-plane to solve the Dirichlet problem in the unit disc with boundary conditions u(upper semicircle)=0 and u(lower semicircle)=1, and gave the geometric interpretation of the result.  

Presentations: Mike gave a presentation on Generating functions and Catalan numbers.

MATH618

On Friday, I gave a very brief sketch of the proof of the Riemann mapping theorem. Then I introduced the uniform convergence on compact subsets, and proved that the uniform limit of analytic functions is analytic and derivatives converge.

On Monday, I discussed the metric in the space of continuous functions on an open set G that corresponds to uniform convergence on compact subsets of G. Also, I started proving the Arcela-Ascoli theorem. On Wednesday, Prof. Straube completed the proof of the Arcela-Ascoli theorem and discussed Hurwitz's theorem. On Friday, Prof. Kudryashov discussed Montel's theorem. 

On Monday, I proved the Riemann Mapping theorem. On Wednesday, I formulated the Caratheodori theorem (its proof is not covered in the course) and covered most of the proof of the Weierstrass theorem for entire functions. On Friday, I completed the proof, and proved a version for an arbitrary domain.

On Monday, I discussed corollaries of the Weierstrass theorem: a representation of a meromorphic function as f=g/h where g, h are holomorphic and the factorization of the sine function. Also, I formulated Runge's theorem. On Wednesday, I proved Runge's theorem. I also proved Pompeiu's formula, to be used in an alternative functional analysis proof of Runge's theorem.

Presentations: Wesley gave a talk on the Hilbert transform.

On Monday, I presented an alternative ``functional analysis'' proof of Runge's theorem, via Pompeiu's formula, Hahn-Banach theorem, and Riesz-Markov representation theorem. 

Sources for an alternative proof: Rudin's Real and complex analysis; lecture notes https://www.math.wustl.edu/~jabbari/SCV.pdf

On Wednesday, I proved Mittag-Leffler's theorem, provided examples (including Weierstrass function), and outlined an application of Runge's approximation theorem to the solution of the d-bar problem (see Rudin's Real and complex analysis for more details).

On Friday, I discussed two notions of convergence that are used for functions with singularities: uniform convergence of compact subsets that do not contain singularities (here I discussed the last items in the quiz and proved an analogue to Hurwitz's theorem for the difference between the number of zeros and the number of poles) and uniform convergence in the spherical metric (space M(G) in the textbook). For this type of convergence, I showed that the limit of a sequence of holomorphic/meromorphic functions is either holomorphic/meromorphic or constantly equal to infinity.

On Monday, I proved Marty's theorem (criterion of normality in M(G)) and formulated Montel-Caratheodori theorem. Also, I proved Schwartz reflection principle and gave an example related to the elliptic sine. On Wednesday, I provided a formula for Schwartz-Christoffel mappings and related it to the elliptic sine, and then discussed analytic continuation along the paths. Friday class was cancelled because of power outage in Blocker.

For more of Schwartz-Christoffel mappings, see e.g. L. Ahlfors, Complex Analysis, Sec.6.2.2.

 Take-home term test is due on Week 7. It covers Week 1-5.  

On Monday, I commented on applications of  Schwartz reflection principle to Schwartz-Christoffel mappings. Then,  I proved that analytic continuation along paths is well-defined, and gives same answers for close paths. On Wednesday, I proved Monodromy theorem and defined monodromy along paths for multivalued functions. I discussed examples, in particular, I explained why the graph of   w = sqrt{z^3 + az+b} is a torus and how it can be parametrized by $(\wp, \wp')$. On Friday, I gave more examples for monodromy along paths (indefinite integrals and inverse functions) and introduced Puiseaux series. 

On Monday, I showed equivalence of multiple definitions of simple connectedness for open subsets of the plane. (see Ch. 8.2 of the textbook). 

On Wednesday, I gave a definition of a topological space, and I gave a definition and examples of analytic complex manifolds. On Friday, I gave more examples of complex manifolds that are quotient spaces by a group action (including a sphere with 3 punctures as a quotient space of a disc), and formulated the classification theorem for Riemann surfaces.  

On Monday, I discussed analysis on complex manifolds. Then I defined covering maps and explained how to lift analytic structures to covering spaces.  On Wednesday, I explained how to lift paths, homotopies, and analytic functions to the covering spaces, and proved the Little Picard Theorem. On Friday (2 hours), I proved the Montel-Caratheodory theorem using the covering map from D to C\{0,1} and the Schwartz-Pick lemma. Also, I introduced Poincare metric in C\{0,1} and explained an alternative proof (via contraction in the Poincare metric in C\{0,1} which implies equicontinuity).  I used the same argument to prove Schottky's theorem. (The textbook derived Schottky's theorem in a different way and obtains Montel-Caratheodori theorem from Schottky's). Finally, I explained how universal covers are constructed, which implies classification of Riemann surfaces as having D, C, or \bar C as a universal cover. 

On Monday, I provided several examples on Montel-Caratheodori theorem, proved the Great Picard theorem (using Montel-Caratheodori theorem), and applied it in several problems. 

On Monday, I discussed quiz problems 1 and 3. Also, I discussed complete analytic functions of local inverses and their Riemann surfaces. 

On Wednesday, I introduced the sheaf of germs of analytic functions, introduced Riemann surfaces of multivalued functions in the general case, and explained the relation between the monodromy theorem for analytic continuations and the monodromy theorem for covering maps. 

On Friday, I listed main properties of harmonic functions: a harmonic function is locally a real part of an analytic function; harmonic function stays harmonic under analytic change of variable; mean value property and its equivalence to harmonicity; maximum/minimum principle; Poisson formula in a disc. Proofs  of the max/min principle and the Poisson formula will be next time.  

Presentations: David gave a talk on Fatou and Julia sets. 

On Monday, I proved the Maximum principle. Also, I discussed several ways to think about the Poisson formula (Moebius change of variable in the mean value property; Fourier method for solving PDEs; relation to Cauchy formula) and proved it. On Wednesday, I discussed corollaries of the Poisson formula: solution of the Dirichlet problem in bounded simply connected domains, the fact that the mean value property implies harmonicity, and Harnack's inequality.  On Friday, I proved existence and uniqueness of the Green's function in bounded simply connected domains with a good boundary. Also, I explained how the Green's function is used to solve the nonhomogeneous Laplace equation \Delta u = f with zero boundary conditions. I included the Green's identity, the Green's representation formula, the definition of the Green's function via the delta-function, and explained why the leading term of the Green's function should be a logarithm.   

Presentations:  Kevin and Preston gave a talk on tilings.

On Monday, I proved Harnack's theorem and introduced the notions of order and rank of an entire function, with some examples. On Wednesday, I proved Jensen and Poisson-Jensen's formulas. On Friday, I showed that the function of a finite genus mu has a finite order lambda<=mu+1,  and proved that a function of a finite order lambda has a finite rank p<=lambda. This is a part of the Hadamard factorization formula.   

On Monday, I completed the proof of the Hadamard factorization theorem. On Wednesday, I discussed its applications (easy derivation of the product formula for the sine, zeros of entire functions of finite order). Also, I started the review, namely, I discussed analytic continuation along the path, Monodromy theorem, and Little Picard theorem. On Friday, I discussed Montel-Caratheodori and the Great Picard theorem. On Monday Apr 29, I discussed Runge theorem.

Presentations: Try gave a talk on the Sokhotsky - Plemelj theorem.

Take-home exam due: May 7, 10 pm.