Complex Analysis


By the end of the semester, the student should be proficient on the following subjects:

1) Functions of  complex variable and fundamental examples. Differentiability conditions (Cauchy Riemann conditions), conformal map.

2) Path Integration of functions of complex variable and corresponding fundamental theorems: Cauchy''s Theorem, Cauchy''s integral formula, Morera''s Theorem, Cauchy inequality, Algebra''s Fundamental Theorem,Maximum Modulus Theorem.

3) Concepts of Taylor Series and Laurent Series. Classification of isolated singularities.

4) Residue theorem and it''s importance in the calculus of closed pathes integrals (Residue Theorem). Compute residues using pre-established formulas.


General characterization





Responsible teacher

José Maria Nunes de Almeida Gonçalves Gomes


Weekly - 4

Total - 70

Teaching language



Consistent knowledge of  Mathematical Analysis 1 and Mathematical Analysis 2.


Basic Complex Analysis; J. Marsden and M. Hoffman; W.H. Freeman and company. (1996)

Complex Analysis; L. Ahlfors; McGraw-Hill International Editions. (1979)

Teaching method

Our model will be of a balanced theoretical exposition with immediate practical exercices, as well as discussing challenging problems. Our method relies on dialog and interaction with and within students, treby providing an high-level of understanding of the subject as well as a complement to the evalutaion of students.

Evaluation method

Continuous Evaluation:

Two tests of 1h30min. The student gets aproval if the average grade is greater or equal to 9,5.

In the date of the exam, the student may choose to improve one of the tests (whether or nos he is already aproved at the disciplin).


Evaluation by Final Exam.

3h00 exam. The student is aproved if the grade is greater or equal to 9,5. 

In any unspecified circunstance, the Evaluating Rulement of FCT is applied. Recomendations of the Pedagogical Council will be followed in case of  students with ENEP status.


Subject matter

1. Complex Functions. Algebra of complex numbers. Elementary complex functions. Limits and continuity. Differentiability - analytic functions, conformal mappings. Differentiability of the elementary functions. Basic properties of the complex derivative. Harmonic functions. Power series.

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. Examples and applications.