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 - 52

Teaching language



Consistent knowledge of  Mathematical Analysis 1 and Mathematical Analysis 2.


Support text of the curricular unit.

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

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

Analyse Complexe; Eric Amar et Étienne Matheron; Cassini. (2004 - 2 edition 2020)

Elementary Theory of Analytic Functions of One or Several Complex Variables; Henri Cartan (1904 - 1995 edition). 

Teaching method

Theoretical/Solving Problems sessions complementes by a an individual support schedule. 

Evaluation method

Continuous evaluation: Three tests of one hour, uniformly distributes along the semester, with grades T1,T2 and T3.

The final grade of continuous evaluation (AC) is determined by 3/10T1+3/10T2+4/10T3. The student is aproved if AC greater or equal to 9,5.



Exam evaluation: Final exam, of 3h, with grade NE. The student is aproved if NE is greater to 9.5. 

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.