Complex Analysis

General characterization

Prerequisites

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

Bibliography

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

Grade In Class (AEA)

Each week, a list of proposed exercises will be provided in advance for the students to work on before the practical classes. Students will be randomly chosen to present their work on the board. At the end of the semester, a grade of 0, 1, or 2 points will be assigned to each student. If a student is chosen to present their work on the board and is not present, they will receive a grade of zero unless the absence is justified by an official document.

Continuous evaluation

It is forbidden to use graphical calculating machines, or any calculation support instruments during evaluation moments.

During the semester, two tests will be carried out with a duration of at most 2 hours. Each test is rated up to a maximum of 20 values.

Let CT be the simple arithmetic mean of the two tests rounded up to the units, and NF the minimum between 20 and CT + AEA. The student obtains approval in the course if NF≥10  with the final classification NF.

Exam

It is forbidden to use graphical calculating machines, or any calculation support instruments during evaluation moments.

All students enrolled in the disciplinewho have not obtained approval in the Continuous Evaluation, can take the appeal exam. Let ER be the exam classification, rounded to the nearest integer, and NR the minimum between 20 and ER+ AEA. The student obtains approval in the course if NR≥10 with the final classification NR.

Grade improvement

It is forbidden to use graphical calculating machines, or any calculation support instruments during evaluation moments.

Students have the right to improve their grades, upon enrollment within the deadlines, at the time of exam, according to the rules of the Exam

 Logistics

Only students who, at the time of the exam, carry an official identification document, containing a photograph (for example, Citizen Card, Identity Card, Passport, some versions of Student Card) and blank exam notebook.

 Final considerations

In any omitted situation, the Knowledge Assessment Regulation of the FCT-UNL applies.

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

Programs where the course is taught: