Complex Analysis - Spring Semester 2020
Part of the Geometric Function Theory Course
This course is an introduction to the study of complex analysis in one complex variable.
Faculty:
Part 1 (10/02 - 21/02/2020):
Daniele Angella (Firenze, Italy)
Francesco Pediconi (Tutor, Firenze, italy)
Part 2 (23/03 - 17/04/2020):
Alberto Abbondandolo (Bochum, Germany)
Gabriele Benedetti (Heidelberg, Germany)
The main textbook for the course is
[How] John M. Howie, Complex analysis, Springer Undergraduate Mathematics Series, Springer-Verlag, 2003.
Prerequisites: The fields of rational, real, (optional: complex) numbers. Elementary functions of one real variable. Sequences and series of real numbers. Topology of the Euclidean plane. Continuity and differentiability for functions of one real variable. Continuity and differentiability for functions of two real variables. Integration in one real variable, and the Fundamental Theorem of Calculus. (See, for example, [How, Chapters 1–3].)
Part 1
Detailed Syllabus:
1. Introduction.
Introduction and motivation to holomorphic functions. Some differences between real and complex analysis.
2. The field of complex numbers.
The field of rational numbers is not complete. The field of real numbers is the unique field, endowed with a total order compatible with the operations, and being complete; but in general we can not solve quadratic equations. The field of complex numbers as an extension of the field of real numbers; it does not admit a total order compatible with the operations, but the fundamental theorem of algebra allows to solve any polynomial equation of positive degree. The representation of complex numbers in the Gauss plane: real and imaginary part, conjugate, norm. Trigonometric representation of complex numbers: norm and argument. Root of unit. The Riemann sphere.
3. Holomorphic functions.
Summary on differentiability in one and several real variables. The structures of C as and R, respectively C-vector space. Complex-differentiability of a complex functions; holomorphic functions and entire functions. The Cauchy-Riemann equations: holomorphic functions satisfy the Cauchy-Riemann equations, and also the converse holds true under the assumptions of the real and imaginary having continuous derivatives. Examples of holomorphic funtions: polynomials are holomorphic. The “formal” change of variable from (x, y) to (z = x+iy, z ̄ = x−iy): the operators ∂ and ∂ , holomorphic functions f are characterized by ∂ z ∂ z ̄∂f = 0. Harmonicity: real and imaginary part of a holomorphic functions ∂ z ̄are harmonic; the converse is true on specific domains, an example. Some applications of the Cauchy-Riemann equations: a holomorphic function f satisfying ∂f = 0 is constant; a holomorphic function with constant norm is ∂z constant. Series of complex numbers: convergence, absolute convergence, criteria for absolute convergence; the geometric series. Power series: con- vergence of power series, and their radius of convergence. Differentiating a power series term by term, we get another power series with the same radius of convergence: power series are holomorphic. Exponential function, trigonometric functions, hyperbolic trigonometric functions; logarithms.
4. Cauchy theorem.
Parametrized (piecewise-smooth, simple, regular) curves, closed parametrized curves. Length of a rectifiable curve. Integration of a complex function along a parametrized curve. Example: integration of zn along the anti-clockwise unit circle, for n ∈ Z. Independence of the integral on the parametrization, up to sign. The fundamental theorem of calculus: the integral of a derivative along a closed parametrized curve is zero. The Cauchy theorem: the integral of a holomorphic function along a closed curve is zero; proof in the case of triangular countours, polygonal contours, convex curves. Applications: homotopy independence of the integral. Cauchy representation formula; the derivative of a holomorphic function is holomorphic. Morera theorem. Liouville theorem, and a proof of the Fun- damental Theorem of Algebra. Lemma on uniform convergence allowing to interchange the series and the integral signs. Taylor series expansion of holomorphic functions. A non-zero holomorphic function has isolated zeroes, and its behaviour around the zeroes. Open Mapping Theorem and Maximum Modulus Principle. Inverse Function Theorem. Introdution to singularities.
Lecture Notes:
Course Materials:
Final Exam Preparation Materials:
Timetable:
Daily Schedule:
Part 2
Update: Due to the global state of emergency caused by COVID 19 , the IMM has suspended on-site lessons until further notice and will complete the rest of the semester through online lessons. This includes the Complex Analysis II course, which will be taught through a combination of pre-recorded video lectures and interactive teacher-to-students Q&A/exercise sessions through the Zoom video conferencing tool. These Zoom interactive sessions will also be recorded and made available to any students who miss a session.
Introduction to the Complex Analysis II course
The second part of the Complex Analysis course will run for 4 weeks. The first and third weeks video lectures will be given by Prof Abbondandolo and the exercise sessions will be run by Professor Benedetti, and the second and fourth weeks the video lectures will be given by Professor Benedetti and the exercise classes run by Professor Abbonandolo.
Lectures
Lecture 1: Laurent Series
Lecture 2: Isolated Singularities
Lecture 3: Residues
Lecture 4: Integrals and Residues
Lecture 5: Special Contours
Lecture 6a: Holomorphic Functions as Conformal Mappings
Lecture 6b: Möbius Transformations
Complement to the Lecture Big O and Little o Notation for Laurent Series
Lecture 7a: Möbius Circles
Lecture 7b: Biholomorphisms
Lecture 8: Series of Meromorphic Functions
Lecture 9, Part 1: Infinite Products of Holomorphic Functions
Lecture 9, Part 2: Infinite Products of Holomorphic Functions
Worksheet Videos
Worksheet 1, Exercise 4
Worksheet 1, Exercise 5
Worksheet 2, Exercise 1
Worksheet 2, Exercise 3
Worksheet 2, Exercise 4
Worksheet 2, Exercise 6
Worksheet 2, Exercise 7
Worksheets:
Mid-term Exam Solutions
Exercise 1
Exercise 2
Exercise 3
Exercise 4
Exercise 5
Final Exam Preparation Materials:
Timetable:
Week 1:
Monday, March 23rd:
morning - students watch Lecture 1 on YouTube
14:30 PM (Pakistan time) - interactive lesson with Professor Alberto on Zoom
Tuesday, March 24th:
morning - students work individually
14:30 PM (Pakistan time) - interactive lesson with Professor Gabriele on Zoom, doing exercise 1
Wednesday, March 25th:
morning - students watch Lecture 2 on YouTube
14:30 PM (Pakistan time) - interactive lesson with Professor Alberto on Zoom
Thursday, March 26th:
morning - students work individually
14:30 PM (Pakistan time) - interactive lesson with Professor Gabriele on Zoom, doing exercise 2
Friday, March 27th: students watch Lecture 3 on YouTube
Week 2:
Monday, March 30th:
morning - students work individually;
14:30 PM (Pakistan time) - interactive lesson with Professor Gabriele on Zoom, doing exercises 3
Tuesday, March 31st:
morning - students watch Lecture 4 on YouTube;
14:30 PM (Pakistan time) - interactive lesson with Professor Alberto on Zoom
Wednesday, April 1st:
morning - students work individually;
14:30 PM (Pakistan time) - interactive lesson with Professor Gabriele on Zoom, doing exercises 4
Thursday, April 2nd:
morning - students watch Lecture 5 on YouTube;
14:30 PM (Pakistan time) - interactive lesson with Professor Alberto on Zoom
Friday, April 3rd: morning - written test
Week 3:
Monday, April 6th:
morning - students watch Lecture 6 on YouTube
14:30 PM (Pakistan time) - interactive lesson with Professor Gabriele on Zoom
Tuesday, April 7th:
morning - students work individually
14:30 PM (Pakistan time) - interactive lesson with Professor Alberto on Zoom
Wednesday, April 8th:
morning - students watch Lecture 7 on YouTube
14:30 PM (Pakistan time) - interactive lesson with Professor Gabriele on Zoom about lecture 7a: Möbius circles
Thursday, April 9th:
morning - students work individually
14:30 PM (Pakistan time) - interactive lesson with Professor Alberto on Zoom about Worksheet 6
Friday, April 10th: students watch Lecture 7b on YouTube and solve Worksheet 7
Week 4:
Monday, April 13th:
morning - students work individually;
14:30 PM-17:30 (Pakistan time) - individual meetings on Zoom, solving Worksheet 7
Tuesday, April 14th:
morning - students work individually;
14:30 PM (Pakistan time) - discussion of Worksheet 7 with Professor Alberto on Zoom
Wednesday, April 15th:
morning - students watch Lecture 8 on YouTube;
14:30 PM (Pakistan time) - discussion of Lecture 8 with Professor Gabriele on Zoom
Thursday, April 16th:
morning - students work individually;
14:30 PM (Pakistan time) - discussion of Worksheet 8 with Professor Alberto on Zoom
Related Materials
ICTP Diploma course in Complex Analysis (video playlist, 2016) by Professor Fabio Vlacci
Visualizing the Riemann hypothesis and analytic continuation
Why is pi here? And why is it squared? A geometric answer to the Basel problem
Additional reading:
Lars V. Ahlfors, Complex analysis. An introduction to the theory of analytic functions of one complex variable, Third edition, International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., 1978
Wolfgang Fischer, Ingo Lieb, A course in complex analysis. From basic results to advanced topics, Textbook, Wiesbaden: Vieweg Teubner, 2012
Course Gallery
MTH632 - Geometric Function Theory
Course ContentsRiemann mapping theorem, conformal mappings and their properties, univalent functions and their subclasses, Functions with positive real part, Herglotz Formula, Some basic properties of univalent and multivalent functions.
Books
A.W Goodman, Univalent Functions, Vol. I, II, Marina Publishing Company Florida 1983.
W.K. Hayman, Multivalent Functions, Cambridge University Press, UK.
St. Ruscheweyh, Convolution in Geometric Function Theory, Sem. Math . Sup. Les Presses De L’Universite De Montreal, Canada 1982