  The present report, has been inspired by the need of the author and its
colleagues to understand the underlying theory of Wirtinger's Calculus and to
further extend it to include the kernel case. The aim of the present manuscript
is twofold: a) it endeavors to provide a more rigorous presentation of the
related material, focusing on aspects that the author finds more insightful and
b) it extends the notions of Wirtinger's calculus on general Hilbert spaces
(such as Reproducing Hilbert Kernel Spaces).
