Basic Complex Analysis - Computer Graphics and Geometric Modeling - page 906

E.5.2. Theorem. (The Fundamental Theorem of Algebra) Every complex polyno-
mial p( z ) of degree one or greater has a root.

Proof. If p( z ) has no zero, then the function f( z ) = 1/p( z ) is an analytic function on
the whole complex plane. Since it is easy to show that f( z ) approaches 0 as | z | goes
to infinity, it follows that f( z ) is bounded and must therefore be constant by Liouville's
theorem. This is impossible and so p( z ) must have a zero.

E.5.3. Corollary. Every complex polynomial of degree n, n ≥ 1, factors into n linear
factors and has n roots counted with their multiplicity.

E.5.4. Corollary. Every real polynomial of degree n, n ≥ 1, has at most n roots
counted with their multiplicity.

Liouville's theorem also implies

E.5.5. Theorem. Every function meromorphic on the extended complex plane is a

rational function.

Proof. See [SakZ71].

E.5.6. Theorem. (The Maximum Principle) If f( z ) is an analytic function on a closed
and bounded set, then the maximum of |f( z )| occurs on the boundary of the set.

Proof. See [Ahlf66].

E.5.7. Theorem. (Fundamental Theorem of Conformal Mappings) Every simply
connected Riemann surface can be mapped in a biholomorphic way (the map and its
inverse are holomorphic) onto either the closed plane, the plane, or the interior of the
unit disk.

Proof. See [BehS62].

E.5.8. Corollary. (The Riemann Mapping Theorem) Let U be a simply connected
open subset of the complex plane that is not the entire plane and let z 0 ΠU . Then
there exists a unique analytic function f( z ) on U that satisfies f( z 0 ) = 0, f¢ ( z 0 ) > 0, and
that maps U in a one-to-one fashion onto the open unit disk.

Proof. See [Ahlf66].