This book evolved out of a graduate course given at the University of New Orleans in 1997. The
class consisted of students from applied mathematics andengineering.
Theyhadthebackgroundofatleastafirstcourseincomplex
analysiswithemphasisonconformalmappingandSchwarz-Christoffeltrans formation a firstcourse in
numerical analysis and good to excellent working knowledgeofMathematica
withadditionalknowledgeofsomeprogramming languages. Sincetheclasshad nobackground
inIntegralEquations thechap tersinvolvingintegralequationformulations werenotcoveredindetail
except for Symm's integral equation which appealed to a subsetofstudents who had some training
in boundary element methods. Mathematica was mostly used for computations. In fact it
simplified numerical integration and other oper ations very significantly which would have
otherwise involved programming inFortran C orotherlanguageofchoice ifclassical
numericalmethods were attempted. Overview Exact solutions of boundary value problems for simple
regions such as cir cles squares or annuli can be determined with relative ease even where
the boundaryconditionsarerathercomplicated. Green'sfunctionsforsuchsimple regions are known.
However for regions with complex structure the solution ofa boundary value problem often
becomes more difficult even for a simple problemsuchastheDirichletproblem.
Oneapproachtosolvingthesedifficult problems is to conformally transform a given multiply
connected region onto Mathematica is a registered trade mark of Wolfram Research Inc. ix x
PREFACE simpler canonical regions. This will however result in change not only in the region
and the associated boundary conditions but also in the governing differential equation. As
compared to the simply connected regions confor mal mapping ofmultiply connected regions
suffers from severe limitations one of which is the fact that equal connectivity ofregions is
not a sufficient condition to effect a reciprocally connected map ofone region onto another.
