The inside-outside theorem states that if 
and 
are univariate polynomials
in a complex variable 
whose polynomial degrees
satisfy 
,
then
 |  |  |
(1)
|
 |  |  |
(2)
|
where 
is a simple closed clockwise-oriented contour on which
has no roots, 
is the set of roots
of 
inside of 
,
and 
is the set of roots of
outside of 
.
The first equality is the residue theorem with clockwise orientation. On the Riemann
sphere, the simple closed contour 
splits the sphere into two
regions. The polynomial degree hypothesis ensures
that the meromorphic one-form 
has no pole at 
. Because the sum of all residues
on the Riemann sphere is zero,
the sum of the inside residues
is the negative of the sum of
the outside residues, giving the second equality.
On the Riemann sphere, the roles of inside and outside switch between a view centered at 
and a view centered at 
.
The theorem also follows from taking the contour integral at infinity, i.e., a circle
of large radius 
. The hypothesis on the polynomial
degree says that this integral tends to zero.
Hence it must actually be zero, because at some point
the circle contains all of the poles
of 
.
This is a special case of the fact that on a compact Riemann surface, in this case the Riemann
sphere, the sum of the complex
residues of a meromorphic one-form is
zero.
