Complex Analysis


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

General characterization





Responsible teacher

João Pedro Bizarro Cabral


Weekly - 5

Total - 70

Teaching language



Working knowledge of real analysis (one and several variables), analytic geometry of the plane and the usual topology of R2.


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

L. V. Ahlfors, Complex Analysis, McGraw-Hill (1979)

M. A. Carreira e M. S. Nápoles, Variável complexa - teoria elementar e exercícios resolvidos, McGraw-Hill (1998)

S. Lang, Complex Analysis, Springer (1999), ISBN 0-387-98592-1

J. E. Marsden and M. J. Hoffman, Basic Complex Analysis - Third Edition, Freeman (1999), ISBN 0-7167-2877-X 

Teaching method

The theory is explained and illustrated with examples. Main results are proved. The students are given the opportunity of working in a list of problems, with the instructor´s support if needed, and the instructor´s comments on relevant results highlighted in the problems.

Evaluation method


Frequency will be granted to any student who does not unjustifiably miss more than 1/3 of the practical classes taught and 1/3 of theory classes. Students who have obtained it in the previous  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. It is forbidden the use of electronic devices during evaluations.

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, 23/10, (T1): all students enrolled in the course may present themselves to the 1st test.

2nd Test, 20/12, (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 roudend to an integer:

CT=(T1 + T2) / 2 

The student is approved in the course if CT is greater than or equal to 10 values. If CT is less or equal to 17, the final grade of the course will be CT. If CT is strictly greater than  17 values, the student can choose between obtaining a final grade of 17 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, 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 rounded to an integer is greater than or equal to 10, the student is approved with the grade obtained if it is less or equal to 17. If the grade is strictly greater to 17 values, the student can choose between obtaining a final grade of 17 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. Complex Functions. Algebra of complex numbers.Definition of the elementary complex functions. Limits and continuity. Differentiability - analytic functions. Harmonic functions. Differentiability of the elementary functions. Conformal mappings; fractional linear transformations

2. Complex integration - Cauchy’s Theorem and applications.Complex integration. Cauchy’s Theorem. Cauchy’s Integral Formula. Fundamental theorems: Morera’s theorem, Cauchy’s inequalities, Liouville’s theorem, Fundamental Theorem of Algebra,  maximum principle. 

3. Power series; Laurent series.  Pointwise and uniform convergence of function sequences and series. Power series.Taylor’s Theorem; analyticity. Singularities – Laurent series. Isolated singularities; classification of isolated singularities

4. Residues. Calculation of residues. Residue theorem. Evaluation of definite integrals.

5. Conformal Mapping. Exemples and applications.


Programs where the course is taught: