Lets consider the function 
defined over the 
plane minus the
real semi-axis 
. Since this function is analytic in its
definition domain, it permits a power series expansion around z=1.
For convenience we set 
. It is clear that
the family of FIR filters, whose transfer functions 
are
given in equation (3), is exactly a LIF family since each
transfer function is a partial series expansion for 
around
z=1 and thus verifies the maximally flat condition
(eq. (2)). These are new expressions for LIFs.
