Complex Analysis

Code:

M2008

Acronym:

M2008

Level:

200

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2016/2017 - 2S

Active?	Yes
Responsible unit:	Department of Mathematics
Course/CS Responsible:	Bachelor in Mathematics

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L:B	10	Official Study Plan	3	-	6	56	162
L:M	82	Official Study Plan	2	-	6	56	162
L:Q	1	study plan from 2016/17	3	-	6	56	162

Teaching language

Portuguese

Objectives

Introduction to the theory of holmorphic functions of one complex varible using amethods from Weierstarss approach via  analytic functions and Cauchy's theory via path integrals and topological arguments.

Learning outcomes and competences

Become competent in the use of classical  fundamental theorems of the theory of holomorphic functions in obe complex variable.

Working method

Presencial

Pre-requirements (prior knowledge) and co-requirements (common knowledge)

Calculus in one and several real variables.

Program

I  Review of elementary notions from the algebra and topology of real and  complex numbers and : liminf and limsup of sequences of real numbers.  field of complex numbets, complex numbers as complete metric space isomorphic to the euclidean plane, open, closed, path connected and compact subset of C. Sequences and series of real and complex numbers. Simple, absolute and commutative convergence of numeric series. Convergence criteria for numeric series: ratio, root, Leibniz and Abel . Sequences and series of  complex functions of complex variable. Simple, uniform and normal convergence.

II. Complex derivative of  functions of complex variable. Operations with complex derivatives. Holomorfic functions. Complex derivatives in terms of real derivative. Cauchy-Riemann equations. Harmonic functions.

III. Complex Analytic Functions: radius of convergence, sum, product and quotien (when definied) of power series with complex coefficients. holomorphicity of convergent power series. Analytic functions. Principle of analytic continuation, principle of isolated zeros,  analitycity of convergent power series. Exponential function, branches of logarithmic, trigonometric and hyperbolic functions.

IV. Fundamental theorems about holomorphic functions:   Cauchy tehorm in a  disc. analiticity of holomorfic functions, Cauchy inequalities, Liouville theorem, d'Alembert-Gauss theorem (fundamental theorem of Algebra), Maximum modulus theorem, open mapping theorm.

V. Cauchy theory of holomorphic functions: paths and loops in an open subset of C, homotopy of paths and homotopy of loops, simply connected open subsets. Inegrals along paths. Theorems of Goursat and  Morera, Cauchy integral formula, Primitivizability of holomorphic functions on a simply connected domain, index of a point relative to a loop. 

VI Singular points and Meromorphic functions: Cauchy's theorem on a ring, Laurent's series, Laurent's theorem, classification of isolated  singular points, Riemann's removable singulaity theorem.  The Riemann sphere.  Meromorphic functions. Essential Singulaities. Casorati-Weierstrass's theorem.

VII Residues's Theorem and applications: Residue of a meromorphic function at a pole, the theorem of residues, the argument princple, Rouché's theorem. Calculus of integrals of real variables using the  theorem of residues: trigonometrc integrals,  integrals of rational fractions, inegrals of Fourier transform  type.

 

Mandatory literature

Carlos Menezes;; Apontamentos de Análise Complexa, 2016

Complementary Bibliography

Matos Aníbal Coimbra A. de; Curso de análise complexa. ISBN: 9789725921159
Cartan Henri; Théorie élémentaire des fonctions analytiques d.une ou plusieurs variables complexes
Remmert Reinhold; Theory of complex functions. ISBN: 0-387-97195-5

Teaching methods and learning activities

Explanation of the subject in lectures, exemplification of concepts , methods and solution of problems,  solution of problems in the practical classes, and appontemnt  hours to receive students to help them on eventual doubts on the subjects.

Evaluation Type

Evaluation with final exam

Assessment Components

designation	Weight (%)
Exame	100,00
Total:	100,00

Calculation formula of final grade

The final classification is the mark obtained by the student in the final exam.