Mathematical Analysis III B
Objectives
Acquire knowledge and skills that enable to:
-Understand the concept of an infinite sum and the definition of convergence.
-Determine when a series is convergent by applying convergence criteria.
-Understand the concept of sequences and series of functions.
-Grasp the difference between pointwise and uniform convergence.
-Determine intervals of convergence and understand their implications for the convergence of the power series and the possibility of differentiating and integrating term by term.
-Understand and be proficient in the algebra of complex numbers, particularly their polar form.
-Understand the concept of a complex function of a complex variable, especially understanding elementary functions as generalizations of the real case.
-Understand the concept of continuity and differentiability for functions of a complex variable and their relationship with the Cauchy-Riemann equations.
-Grasp the concept of the integral of a complex function of a complex variable along a path and its properties.
-Understand Cauchy''s integral formula and be able to apply it in practical cases.
-Understand the concept of Taylor and Laurent series.
-Understand the concept of residue, in its various forms, and apply it to the calculation of integrals.
General characterization
Code
5005
Credits
6.0
Responsible teacher
Rogério Ferreira Martins
Hours
Weekly - 4
Total - 59
Teaching language
Português
Prerequisites
Basic calculus concepts of one and several variables, taught in the courses Mathematical Analysis I and II.
Bibliography
The syllabus of this course unit is typical in engineering curricula, so there is an extensive list of references that can be used. Below are some examples.
ANTON, H., BIVENS, I. e DAVIS, S., Cálculo II; 10ª Edição, Bookman, 2014 (Ou outra edição do mesmo livro).
CAMPOS FERREIRA, J., Introdução à Análise Matemática, Fundação Calouste Gulbenkian, 1982.
SAFF, E. B. e SNIDER, A. D., Fundamentals of Complex Analysis with Applications to Engineering and Science - 3rd Edition, Pearson Education, 2003.
KREYSZIG, E., Advanced Engineering Methods, 8ª edição, John Wiley and Sons, Inc., 1999.
AGARWAL, R., PERERA, K. e PINELAS, S., An Introduction to Complex Analysis, 2011, Springer.
CARREIRA, M. A. e NÁPOLES, M. S., Variável complexa - Teoria elementar e exercícios resolvidos, McGraw-Hill.
Teaching method
Teaching Method is based on conferences and problems solving sessions with the support of a personal attending schedule.
Evaluation method
Theoretical-Practical Assessment: Two written tests, the first one with a duration of 1 hour and the second one with the duration of 2h, to be held during the semester. The first test will be graded between 0 and 6 points and the second one between 0 and 10 points.
Summative Assessment: Each week, a few exercises (around 4 or 5) will be selected for the student to solve before each practical class. At the beginning of the practical class, a sheet will be passed around where each student indicates the exercises they solved that week. The final grade for this component, ranging from 0 to 4 points, will be given by the teacher based on the student''s work. In this assessment component, more than just a correct solution, the student''s effort will be valued. The exercises can be solved with reference materials or even in groups, although each student must bring their own solution to class and be prepared to solve the exercise on the board without consulting any materials.
A student who meets the attendance requirement will have a final grade for Continuous Assessment equal to T1 + T2 + AS. Where T1 and T2 are the final grades of the first and second test, respectively, and AS is the Summative Assessment grade.
Any student, working student, who already has attendance from a previous edition, or another student who, for some reason, cannot or does not want to attend practical classes, can ask the teacher to be assessed without the Summative Assessment component. To do this, they just need to send an email to the teacher explaining why they wish to waive this assessment component. In this case, the final Continuous Assessment grade will be (T1 + T2) * 5/4. Attention: this request must be made by the end of the second week of classes; otherwise, the student will be compulsorily assessed with the Summative Assessment component.
Attendance: To obtain attendance, a student must complete all activities in the web platform and must attend at least 6 practical classes. Students who requested not to have the Summative Assessment component are exempt from attending the 6 classes. Only students who have obtained attendance in this edition or the previous edition will receive a final grade in the curricular unit.
Appeal Exam: Students who have not passed through Continuous Assessment and have obtained attendance for the curricular unit may take an Appeal Exam. This exam, lasting 3 hours, assesses all the content taught in the curricular unit and will be graded with a score between 0 and 16 points. The final grade will be E + AS or E * 5/4, whichever is more favorable to the student, where E is the exam score.
Grade Improvements: Students who passed the curricular unit can request, following all the regulations imposed by FCT NOVA, Grade Improvement by taking the Appeal Exam, being able to use the AS grade for the final grade. The final grade will be E + AS or E * 5/4, whichever is more favorable.
There will be pre-registration for Tests and Appeal Exam within a deadline to be set later. On the day of the test, students must bring a blank exam booklet, writing materials, and an official identification document. All tests and exams must be taken without consultation and without the use of any computing materials for calculations.
In any unclear situation, the FCTUNL Knowledge Assessment Regulations, revised in May 2024, will apply.
Subject matter
Series:
Numerical series.
Convergence of numerical series. Necessary conditions for convergence. Telescoping series. Geometric series.
Series of non-negative terms. Integral criterion. Dirichlet series. Comparison criteria. Ratio Test. D''Alembert''s Criterion. Root Test. Cauchy''s root criterion. Kummer''s criterion. Raabe''s criterion.
Simple and absolute convergence. Alternating series and Leibniz''s criterion.
Multiplication of series.
Series of Functions:
Sequences of functions. Pointwise and uniform convergence.
Series of functions. Pointwise and uniform convergence. Weierstrass Criterion. Continuity, term-by-term integrability and differentiability.
Power series. Radius of convergence. Interval of convergence. Uniform convergence. Term-by-term integrability and differentiability.
Taylor and MacLaurin series.
Complex analysis:
Overview of the field of complex numbers. Conjugate, modulus, and argument. Polar form of a complex number. n''th roots. De Moivre''s formulas.
Polynomial functions of a complex variable. Exponential, hyperbolic and trigonometric functions. Principal branch of the logarithm, and inverse trigonometric functions.
Limits and continuity of complex functions of a complex variable.
Holomorphic functions. Cauchy-Riemann equations.
Integral of a complex function of a complex variable along a piecewise regular curve.
Cauchy''s Theorem. Cauchy''s integral formulas.
Analytic functions. Taylor series. Relationship with holomorphic functions.
Essential singularities, poles, and removable singularities. Laurent series.
Residue Theorem. Applications to the calculation of improper integrals.