and,

Moreover, and is invertible iff .

It is the subclass of holomorphic mappings of satisfying the complexified Cauchy-Riemann equations.

where and . This representation is very useful because: addition, multiplication and division can be done term-by-term. Also, an element will be noninvertible iff or . The notion of holomorphicity can also be see with this kind of notation. For this we need to define the functions as and . Also, we need the following definition:

is -holomorphic on the domain

We note here that and will also be domains of .

with and . Secondly, the polynomials of degree n, with bicomplex multiplication, defined as:

with for and . We note that the derivative will be what we expect for polynomials.