Mathematical Analysis III B


The student should understand the basic concepts and be able to compute the quantities presented in the exercises.

General characterization





Responsible teacher

Maria de Serpa Salema Reis de Orey


Weekly - 5

Total - 70

Teaching language



The student should know the basic concepts of Calculus in one and several variables that he learnt  in the disciplines of Mathematical Analysis I and II.


AHLFORS, L. V., Complex Analysis, McGraw-Hill, 1979.

AGARWAL, Ravi, PERERA, Kanisshka e PINELAS, Sandra - An Introduction to Complex Analysis, 2011, Springer

ANTON, H.; BIVENS, I.; DAVIS, S. -  Cálculo II; 8ª Edição, Bookman, 2007. 

CAMPOS FERREIRA, J. - Introdução à Análise Matemática, Fundação Calouste Gulbenkian, 1982. 

CARREIRA, M. A. e NÁPOLES, M. S., Variável complexa - Teoria elementar e exercícios resolvidos, McGraw-Hill. 

DIAS AGUDO, F. R. - Análise Real, 2ª edição, Livraria Escolar Editora, 1994.  

MARSDEN, J., e HOFFMAN, M. J., Basic Complex Analysis, 3ª edição, Freeman, 1999.

MARSDEN, J. e WEINSTEIN, A. - Calculus III; Springer, 2ªEdição, 1984.

SAFF, E. B.; SNIDER, A. D. - Fundamentals of Complex Analysis with Applications to Engineering and Science - 3rd Edition, Pearson Education, 2003.

SÁ, A. e LOURO, B. - Sucessões e Séries, Teoria e Prática. Escolar Editora, 2009.

Teaching method

Teaching  Method  is based on conferences and problems solving sessions with the support of a personal attending schedule.

Evaluation method



Frequency will be granted to any student who does not unjustifiably miss more than 1/3 of the practical classes taught. Students who have obtained it in one of the semesters of the 2018/2019 school year or who have any of the special statutes provided by law are exempt from attendance.

The evaluation is carried out through Continuous evaluation or Exam evaluation.

Continuous evaluation

During the semester two tests will be carried out with a duration of 1 hour 30 minutes. Each test is rated up to a maximum of 20 values.

 1st Test (t1): all students enrolled in the course may present themselves to the 1st test.

2nd Test (t2): all the students enrolled in the course that have obtained a frequency or have a special status may submit to the 2nd test.

The classification of continuous evaluation (CA) is obtained by the following formula:

AC =(t1 + t2) / 2 

The student is approved in the course if AC is greater than or equal to 9.5 values. If AC is less than 16.5, the final grade of the course will be AC. If AC is greater than or equal to 16.5 values, the student can choose between obtaining a final grade of 16 values ​​or performing a supplementary examination.


All the students enrolled in the course that have obtained Frequency or have special status may submit to the Exam.

At the date and time scheduled for the Exam in January 2020, any student enrolled in the course that has obtained Frequency or has special status and who has not obtained approval in the Continuous Evaluation can take the exam for 3 hours or can opt to repeat one of the1 hour 30 minutes . If the student chooses to repeat one of the tests, the classification is calculated as in the case of Continuous Evaluation.

If the student performs the Exam and his or her classification is greater than or equal to 9.5, the student is approved with the grade obtained if it is less than 16.5. If the grade is greater than or equal to 16.5 values, the student can choose between obtaining a final grade of 16 values ​​or performing a supplementary examination.

Grade improvement

Students have the right to improve grade by enrollment within the established deadlines, at the time of the Exam. In this case, they may take the 3-hour Exam or repeat one of the 1-hour 30-minute tests as described in the previous paragraph. In the case that a student wants to improve his grade, having obtained approval in a previous semester, he can only take the 3-hour Exam.



Only those students who carry an official identification document with a photograph (for example, Citizen''s Card, Identity Card, Passport, some versions of Student Cards) and a blank examination notebook may take any of the tests.

Final considerations

In all that this Regulation is missing, the FCT-UNL General Regulations are valid.

Subject matter

1. Series

1.1 Number series

1.1.1 Convergence of Numeric Series. Telescopic Series. Geometric Series.

1.1.2 Series of non-negative terms. Criterion of integral. Dirichlet series. Criteria for comparison. Ratio Criterion. Criterion of d''''''''Alembert. Root Criterion. Cauchy Root Criterion. Kummer Criterion. Raabe Criterion.

1.1.3 Simple and absolute convergence. Alternate series and Leibniz criteria. Multiplication of series.

1.2 Series of Functions

1.2.1. Sequences of functions. Point convergence and uniform convergence.

1.2.2 Function series: point convergence and uniform convergence. Weierstrass criterion. Continuity. Integrability and differentiability term by term.

1.2.3 Power series. Radius of convergence. Interval of convergence. Uniform convergence. Integrability and differentiability term by term.

1.2.4 Series of Taylor and MacLaurin.

2. Complex analysis

2.1 Generalities about the field of complex numbers; conjugate, module and argument; polar form of a complex number. N-th roots of complex numbers. Formulas of De Moivre.
2.2 Complex variable polynomial functions. Exponential function, circular and hyperbolic trigonometric functions, main branch of the logarithm and inverse trigonometric functions.

2.3 Limits and continuity of complex functions of complex variable.

2.4 Holomorphic Functions. Cauchy-Riemann equations.

2.5 Integral of a complex complex variable function along a sectionally regular curve.

2.6 Cauchy''''''''s theorem. Cauchy integral formulas.

2.7 Analytical functions. Taylor series. Relation with holomorphic functions.

2.8 Essential singularities, poles and removable singularities. Laurent series.

2.9 Residue Theorem. Applications to the calculation of improper integrals.