Definition. A Schur function
is defined as a complex function analytic and of modulus not exceeding
unity in
.
Theorem. The function
Proof.
Suppose is positive real. Then for
,
re
re
is PR. Consequently,
is minimum phase which implies all roots of
lie in the unit circle.
Thus
is analytic in
. Also,
Conversely, suppose is Schur. Solving Eq.
(5) for
gives
If
is constant, then
is PR. If
is not
constant, then by the maximum principle,
for
. By
Rouche's theorem applied on a circle of radius
,
, on
which
, the function
has the same number of
zeros as the function
in
. Hence,
is
minimum phase which implies
is analytic for
. Thus
is PR.