# Complex Analysis

## Objectives

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

## General characterization

##### Code

7813

##### Credits

6.0

##### Responsible teacher

João Pedro Bizarro Cabral

##### Hours

Weekly - 5

Total - 70

##### Teaching language

Português

### Prerequisites

Working knowledge of real analysis (one and several variables), analytic geometry of the plane and the usual topology of **R**^{2}.

### 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

Knowledge assessment is carried out through Continuous Evaluation or Exam Evaluation Examination, presential. The Continuous Assessment consists of two tests and a grade in class.

**Grade In Class (AEA)**

For each week, a list of proposed exercises for students to work on before practical classes will be made available in advance. In practical classes, students make presentations based on this work, and at the end of the semester a grade of 0, 1 or 2 is attributed to each student.

**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 1 hour 30 minutes. 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. If NF≤16, the student is approved with the final classification NF. If NF≥17, the student can choose to stay with the final classification of 16 values or take a complementary test to defend his grade.

**Exam**

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

All students enrolled in the discipline who have obtained Frequency or are exempt from it, and who have not obtained approval in the Continuous Evaluation, can take the appeal exam. Students can choose to repeat one of the 1 hour and 30 minute tests. If the student chooses to repeat one of the tests, the classification is calculated as in the case of Continuous Evaluation. If the student takes 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. If NR≤16, the student is approved with the final classification NR. If NR≥17, the student can choose to stay with the final classification of 16 values or take a complementary test to defend his grade.

**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. In that case, they can take the 3 hour Exam or repeat one of the 1 hour and 30 minute tests as described in the previous paragraph. In the event that a student makes a grade improvement having obtained approval in a previous semester, he can only take the 3 hour exam.

**Logistics**

In order to rationalize the resources of FCT (facilities, teaching staff and non-teaching staff), only students who register for the purpose through CLIP, during the period stipulated therein, may take any of the tests.

If, at the time of the exam, the student chooses to repeat one of the tests, he / she must register for this test, otherwise he / she will take an appeal exam.

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.