13th European Intensive Course on Complex Analysis  

and applications to partial differential equations

Departamento de Matemática, Universidade de Coimbra, Portugal

Departamento de Matemática, Universidade de Aveiro, Portugal

  June 18 to 29, 2007

Goal of the Course

This intensive course follows the twelve held at the Universities of Coimbra and Aveiro from 1995 to 2006 (1995, 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006) and there are plans for intensive courses in the following years. The lecture notes of some of the courses have been published in Coimbra and others are in print.

This intensive course will have a total of 40 hours of lectures and is at postgraduate level. Lecturers will have time available to discuss with the students. Successfully participating students will get a certificate. This course is organized by the Universities of Coimbra and Aveiro with the same goals as the ones organized under the Socrates/Erasmus Intensive Program of Higher Education, and is opened to all young mathematicians interested in Complex Analysis and its applications.

There will be a Workshop on "Applications and Generalizations of Complex Analysis" on the 23rd of June 2007. 

First Week


June 18 (room 2.4)

June 19 (room 2.4)

June 20 (room 2.4)

June 21 (room 2.4)

June 22 (room 2.4)

Opening session


Eduardo Godoy 

10h - 12h 30m

10h - 12h 30m

10h - 13h



Erick Lehman

14h30m - 17h

14h30m - 17h


14h30m - 18h


Antonio Durán 



14h30m - 17h

10h - 12h30m

10h - 13h

Second Week


June 25 (room ...)

June 26 (room ...)

June 27 (room ...)

June 28 (room ...)

June 29 (room ...)

Richard Delanghe

10h - 12h

10h - 12h


10h - 12h

10h - 12h

Wolfgang Sprössig

14h - 16h

14h - 16h


14h - 15h

Paula Cerejeiras    


15h - 16h

14h - 16h

Social Program    




Author: Eduardo Godoy, Vigo University, Spain

first week

 Summary: While orthogonal polynomials in one variable already have numerous and varied applications in many fields of science, the theory of orthogonal polynomials in two and more variables is applied insufficiently widely. The study of orthogonal polynomials of several variables goes back to C. Hermite ( 1865). Later on, books of P.Appell (1881), P. Appell & K. de Feriet (1926) and papers of D. Jackson (1938) and T. Koornwinder (1975) introduced a considerable number of results on this theory. With the intention of making this seminar useful to a wide audience, we shall introduce standard matrix notation in order to present general properties of orthogonal polynomials of several variables and specially of two variables over a domain with arbitrary weight. Following mainly the monographs of P.K. Suetin (1988) and Ch.F. Dunkl & Y. Xu (2001), a systematic exposition and detailed discussion of many important results, examples and applications on the theory of orthogonal polynomials in two (continuous and discrete) variables is given.
 Author: Antonio Durán, Sevilla University, Spain

first week


The subject of orthogonal polynomials cuts across a large piece of mathematics and its applications. Two notable examples are mathematical physics in the 19th and 20th centuries, as well as the theory of spherical functions for symmetric spaces. Matrix orthogonality on the real line has been sporadically studied during the last half century since Krein devoted some papers to the subject in 1949. In the last decade this study has been made more systematic with the consequence that many basic results of scalar orthogonality have been extended to the matrix case. The most recent of these results is the discovery of important examples of orthogonal matrix polynomials: many families of orthogonal matrix polynomials have been found that (as the classical families of Hermite, Laguerre and Jacobi in the scalar case) satisfy second order differential equations with coefficients independent of the degree the polynomials. The aim of these talks is to give an overview of the techniques that have led to these examples and a small sample of the examples themselves; in particular we will discuss some of the many differences among the matrix and the scalar case, such as the (non) uniqueness of the differential equation or the role of the Rodrigues' formula.
Author: Eric Lehman,  Université de Caen, France

first week

 Summary: Abstract and practical definitions of Clifford Algebras Cl_{p,q,r}=R_{p,q,r}. Rotation in a euclidean plane, in a 3-dimensional euclidean space: C=R_{2,0,0}^+ and H=R_{3,0,0}^+ and in 4-dimensional spaces. The Dirac equation and spinors. Monogenic, hypermonogenic, Clifford holomorphic and Clifford analytical functions in R_{0,n,0}. Translation of Clifford Analysis from R_{0,n,0} to R_{n+1,0,0}.
Author: Richard Delanghe, University of Ghent, Belgium

second week

1. The real Clifford algebra R0,m+1 
2. Monogenic functions versus self-conjugate differential forms 
3. Conjugate harmonicity, harmonic primitives and monogenic primitives of mono- 
genic functions. 
4. Monogenic r-forms versus harmonic r-forms 
5. Bases for the space of monogenic homogeneous vector-(or para-)vector valued 
6. Cauchy transforms and conjugate harmonicity 
Author: Wolfgang Sprössig, Technical University of Freiberg, Germany

second week

1.	Lecture: About the notion of holomorphy
2.	Lecture: Quaternionic operator calculus - Hodge type decompositions
3.	Lecture: Fluid flow problems
4.	Lecture: problems in Elasticity
Author: Paula Cerejeiras, Aveiro University, Portugal

second week

1. Heat equation and related topics
2. Scattering


Living expenses can be partially covered for some students if they do not have support from their own institution and if there is enough money available.

REGISTRATION FORM (please copy into email)

I intend to participate in the Intensive Course: Yes/No
I intend to participate in the Workshop and present a paper ___ / I will not present a paper __
Tentative title:
I need an invitation letter: _____; for that purpose, contact me via the e-mail:____________or by fax:_______________________
I am a Postgraduate/PhD student and I need financial support ______
I need help with accommodation in Coimbra or Aveiro:_______;
Please send as soon as possible a short abstract of your paper (in Latex, at most 10 lines).

List of Participants


Rafael Hernández Heredero, Universidad Politécnica de Madrid
Odete Ribeiro, Departamento de Matemática, Instituto Politécnico de Viseu
Márcio Sacramento, Departamento de Matemática, Instituto Politécnico de Viseu
Ulises Fidalgo Prieto, Universidade de Aveiro
Luis Garza, Universidad Carlos III de Madrid
Norman Gürlebeck, Friedrich Schiller UniversityJena, Germany
Luís Cotrim , Instituto Politécnico de Leiria
Anabela Monteiro Paiva, Universidade da Beira Interior
Maria da Neves Vieiro Rebocho, Univ. Beira Interior/Univ. Coimbra
Ana Isabel Gonçalves Mendes, Instituto Politécnico de Leiria
Herbert Dueñas Ruiz, Universidad Carlos III de Madrid
Judit Mínguez Ceniceros, Universidad de La Rioja
Sven Ebert, University of Freiberg
Frank Dierich, University of Freiberg
Matti Schneider, University of Freiberg
André Schlichting, University of Freiberg
Juliane Mueller , University of Freiberg
Jaime Carvalho e Silva (Departamento de Matemática Universidade de Coimbra)
Ana Foulquié (Departamento de Matemática da Universidade de Aveiro)
Isabel Cação (Departamento de Matemática da Universidade de Aveiro)
With support from CMUC (Centro de Matemática da Universidade de Coimbra), UI&D "Matemática e Aplicações" da Universidade de Aveiro, and the Socrates programme