The purpose of this book is to provide an integrated course in real and complex analysis for
those who have already taken a preliminary course in real analysis. It particularly emphasises
the interplay between analysis and topology. Beginning with the theory of the Riemann integral
(and its improper extension) on the real line the fundamentals of metric spaces are then
developed with special attention being paid to connectedness simple connectedness and various
forms of homotopy. The final chapter develops the theory of complex analysis in which emphasis
is placed on the argument the winding number and a general (homology) version of Cauchy's
theorem which is proved using the approach due to Dixon. Special features are the inclusion of
proofs of Montel's theorem the Riemann mapping theorem and the Jordan curve theorem that arise
naturally from the earlier development. Extensive exercises are included in each of the
chapters detailed solutions of the majority of which are given at the end. From Real to
Complex Analysis is aimed at senior undergraduates and beginning graduate students in
mathematics. It offers a sound grounding in analysis in particular it gives a solid base in
complex analysis from which progress to more advanced topics may be made.
