This book presents complex analysis of several variables from the point of view of the
Cauchy-Riemann equations and integral representations. A more detailed description of our
methods and main results can be found in the introduction. Here we only make some remarks on
our aims and on the required background knowledge. Integral representation methods serve a
twofold purpose: 1° they yield regularity results not easily obtained by other methods and 2°
along the way they lead to a fairly simple development of parts of the classical theory of
several complex variables. We try to reach both aims. Thus the first three to four chapters
if complemented by an elementary chapter on holomorphic functions can be used by a lecturer as
an introductory course to com plex analysis. They contain standard applications of the
Bochner-Martinelli-Koppelman integral representation a complete presentation of
Cauchy-Fantappie forms giving also the numerical constants of the theory and a direct study of
the Cauchy-Riemann com plex on strictly pseudoconvex domains leading among other things to a
rather elementary solution of Levi's problem in complex number space en. Chapter IV carries the
theory from domains in en to strictly pseudoconvex subdomains of arbitrary - not necessarily
Stein - manifolds. We develop this theory taking as a model classical Hodge theory on compact
Riemannian manifolds the relation between a parametrix for the real Laplacian and the
generalised Bochner-Martinelli-Koppelman formula is crucial for the success of the method.
