SM/0104 - Advanced Mathematical Analysis 1
Academic Year 2021/2022
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CLAUDIA ANEDDA (Tit.)
- Teaching style
- Lingua Insegnamento
|[60/65] MATHEMATICS||[65/00 - Ord. 2020] PERCORSO COMUNE||9||72|
KNOWLEDGE AND UNDERSTANDING: Know and master concepts relating to complex functions (in particular definitions, properties and characteristics of olomorphic functions and of some specific functions: complex polynomials, complex exponential, complex logarithm, complex trigonometric functions); complex integrals; conformal maps; primitive functions; Cauchy theorems; analyticity of holomorphic functions and consequences; Laurent series; residual theorem.
Know the basic notions of functional analysis: Banach space, linear functional and linear operator and their properties; Helly-Hahn-Banach theorem; Banach-Steinhaus theorem; weak topologies; reflexive spaces; separable spaces; uniformly convex spaces. Know the most commonly used functional spaces (in particular the L^p spaces and their characteristics).
Know the Fourier and Laplace transforms and their main properties: differentiation, convolution, inverse transform and the connection between them.
APPLYING KNOWLEDGE: Know how to recognize olomorphic functions, calculate derivatives and integrals of complex functions; Know how to apply Cauchy theorems, identify isolated singularities and zeros of a holomorphic function; know how to calculate the Laurent series associated with particular functions (referable to known series),
know how to calculate integrals along complex curves and know how to apply the residue theorem and Jordan Lemma.
MAKING JUDGEMENT: Know how to link topics discussed in different contexts; Know how to search and locate contents in different books and identify examples and applications on theoretical contents.
IN COMMUNICATION SKILLS: Know how to expose studied topics in a precise, appropriate and rigorous language.
TO LEARN SKILLS: Acquire an independent method of study to know how to orientate in scientific literature and to be able to understand, interpret and deepen topics and issues that will be encountered during university and/or work career.
Differential and integral calculus in one and more variables; topology;
linear differential forms; functions sequences and functions series; powers and Fourier series; real analytic functions; ordinary differential equations; Lebesgue measure and integration.
Complex analysis: short references on complex numbers; complex functions: complex functions of complex variable; limits of complex functions; olomorphic functions; the Cauchy-Riemann condition; some particular complex functions including, in particular: complex polynomials; complex exponential; complex trigonometric functions; complex logarithm; n-th complex root; harmonic functions; complex curves; conformal mapping; complex integration; primitive functions; Morera theorem; Cauchy integral theorem and Cauchy integral formulas; short references on power series and analyticity of olomorphic functions; Liouville Theorem; Identity Theorem; analytic continuation; maximum modulus principle; Laurent series; zeros and isolated singularities of olomorphic functions; meromorphic functions; Casorati's Theorem; Picard's theorem; residue Theorem; argument Theorem; Rouché Theorem; Jordan Lemma. Some types of integrals that can be solved through complex integration.
Functional analysis elements: short references on normed spaces and Banach spaces; linear operators and linear functionals; duality; Helly-Hahn-Banach Theorem; Banach-Steinhaus Theorem; weak topologies; reflexive spaces; separable spaces; uniformly convex spaces; Lebesgue spaces: definition and main properties of L^p spaces; Holder inequality; Fischer-Riesz Theorem; reflexivity, separability and duality; Riesz representation theorem.
Fourier transform: Fourier transform in L^1(R); definition and properties; Fourier transform and differentiation; Fourier transform and convolution; inverse Fourier transform; Fourier transform in L^2(R); Plancherel Theorem; some applications.
Laplace transform: definition and properties; Laplace transform and differentiation; Laplace transform and convolution; inverse Laplace transform; connection with Fourier transform; some applications.
The course consists of 72 hours of lectures.
Teaching will be delivered mainly in the presence, as foreseen in the Manifesto degli Studi for the A.Y. 2021/22, compatibly with the continuation of the COVID-19 health emergency; the tools used for the lectures will be both the blackboard and the tablet, with projection system via screen in the classroom and via internet in streaming.
The lessons, in addition to the theoretical content, also provide illustration of resolution procedures pertaining to the parts of the program that contemplate them (examples aimed at achieving the objectives relating to the application skills) are done on the blackboard and they involve active participation of the students especially in applying knowledge. During the lessons, the teacher also urges the students to attend with clarification questions as well as with any possible deepening and connection remarks and/or comments on the topics discussed. Some proofs, not requested in the oral test, are left to the students as a deepening: the meetings with the teacher for clarifications and explanations are therefore considered an integral part of the training activity. On the teacher's website https://www.unica.it/unica/it/ateneo_s07_ss01.page?contentId=SHD30623 the handouts of the updated course are gradually provided.
Verification of learning
The goal of the exam is to check the level of achievement of the above-mentioned training objectives.
Compatibly with the indications of the University on how to carry out the exams according to the evolution of the COVID-19 emergency, the exams could be held in the presence or online.
The learning verification takes place through a written and an oral test.
The written test involves the resolution of a complex analysis exercise (about this, see the training objective "applying knowledge”), similar to the practices made during the course and an open-ended question, concerning the contents of the program related to the complex analysis, which may require the proof of a theorem and/or the exposition of concepts and/or the exhibition of an example; the duration of the written test is two and a half hours.
The oral test (which must be given within three months of the written test) consists of of a talk during which the student must prove to know the contents of the course about functional analysis and transformations: the teacher asks three questions concerning the topics discussed during the lectures; to the three main questions can be added, if necessary, other clarifications, aimed at ascertaining that the student has mastery of the contents (the number of questions of the teacher also depends, of course, on the student's level of training: if it is sufficiently exhaustive, further questions will be reduced); during the test the teacher will consider the ability to connect the topics covered, to independently (without explicit requested) cite examples and/or applications about the explained theory. The duration of the test depends on the level of preparation of the student; on average it is 45 minutes.
The grade is on a scale of thirty in both tests, and the final grade is obtained by the average of the two. If the oral test is not passed, the student has the opportunity to take it once more while maintaining the validity of the written test mark (if the three-month deadline has not been exceeded). In the written test the score assigned to each part is specified.
The level of knowledge of the discussed content, the ability to relate the topics, the clarity and propriety of exposition and the ability to apply knowledge about the exercises of complex analysis are evaluated.
To pass the exam the student must attest a basic knowledge of all the topics covered in the course.The “lode” will acknowledge specially brilliant performances. In particular,
the final grade after the oral interview is assigned according to the following Docimological Table.
Insufficient: the student proves he did not know and/or understand many of the basic Knowledge of the course.
18-24: the student knows almost all the arguments submitted during the examination; demonstrates that has understood and assimilated the arguments sufficiently.
25-28: the student knows all the arguments submitted during the examination; demonstrates that has understood and assimilated well the arguments; the student has a good mastery of language and a clear presentation of contents.
29-30: the student knows very well all the arguments submitted during the examination; demonstrates that has understood and assimilated very well the arguments; the student has a very good mastery of language and a very clear presentation of contents; is able to explain the concepts learned.
30 e lode: the student knows perfectly all the arguments submitted during the examination; demonstrates that has understood and assimilated in depth all the topics; the student has a very good mastery of language and a very clear presentation contents; is able to explain the concepts learned.
The dates of written test are fixed (6 in an academic year), but for the oral test the student can agree with the teacher (personally, by telephone or email) for a different date at least one week in advance.
The main texts used during the course are the following (for arguments: complex analysis, functional analysis, transform):
L.V. Ahlfors, Complex analysis: an introduction to the theory of analytic functions of one complex variable, third edition, McGraw-Hill (1979).
H. Brezis, Functional analysis, Sobolev Spaces and partial differential equations, Springer, New York (2011)
G.C. Barozzi, Matematica per l’ingegneria dell’informazione, Zanichelli (2001).
During the lessons of the course any other texts and/or notes of the teacher (which will be available on the website) will be provided.
The teacher receives the students, during the COVID-19 emergency, through the Microsoft Teams or Zoom system. There is not a prefixed reception time: it is agreed upon by mail (email@example.com), or by telephone (070 675 8510), from time to time.
Lectures notes and other useful information (among which previous exams tests) are available on the teacher's website
The teacher does not have a prefixed reception time:
At the end of the course, the lesson register and the detailed program will be published with precise information on the various topics covered and the proofs (w.p. = with proof) required for the oral test.
Our University provides support for students with specific learning disability (SLD). Those interested can find more information at this link: http://corsi.unica.it/matematica/info-dsa/