The purpose of this book is to develop the foundations of the theory of holomorphicity on the
ring of bicomplex numbers. Accordingly the main focus is on expressing the similarities with
and differences from the classical theory of one complex variable. The result is an elementary
yet comprehensive introduction to the algebra geometry and analysis of bicomplex numbers.
Around the middle of the nineteenth century several mathematicians (the best known being Sir
William Hamilton and Arthur Cayley) became interested in studying number systems that extended
the field of complex numbers. Hamilton famously introduced the quaternions a skew field in
real-dimension four while almost simultaneously James Cockle introduced a commutative
four-dimensional real algebra which was rediscovered in 1892 by Corrado Segre who referred to
his elements as bicomplex numbers. The advantages of commutativity were accompanied by the
introduction of zero divisors something that for a while dampened interest in this subject. In
recent years due largely to the work of G.B. Price there has been a resurgence of interest in
the study of these numbers and more importantly in the study of functions defined on the ring
of bicomplex numbers which mimic the behavior of holomorphic functions of a complex variable.
While the algebra of bicomplex numbers is a four-dimensional real algebra it is useful to
think of it as a complexification of the field of complex numbers from this perspective the
bicomplex algebra possesses the properties of a one-dimensional theory inside four real
dimensions. Its rich analysis and innovative geometry provide new ideas and potential
applications in relativity and quantum mechanics alike. The book will appeal to researchers in
the fields of complex hypercomplex and functional analysis as well as undergraduate and
graduate students with an interest in one- or multidimensional complex analysis.
