Difference between revisions of "Liouville's Theorem (complex analysis)"
(expanded) |
Hashtagmath (talk | contribs) (→See also) |
||
Line 56: | Line 56: | ||
[[Category:Complex analysis]] | [[Category:Complex analysis]] | ||
+ | [[Category:Theorems]] |
Revision as of 12:19, 30 May 2019
In complex analysis, Liouville's Theorem states that a bounded holomorphic function on the entire complex plane must be constant. It is named after Joseph Liouville. Picard's Little Theorem is a stronger result.
Contents
Statement
Let be a holomorphic function.
Suppose there exists some real number
such that
for all
. Then
is a
constant function.
Proof
We use Cauchy's Integral Formula.
Pick some ; let
denote the simple
counterclockwise circle of radius
centered at
. Then
Since
is holomorphic on the entire complex plane,
can
be arbitrarily large. It follows that
, for every
point
. Now for any two complex numbers
and
,
so
is constant, as desired.
Extensions
It follows from Liouville's theorem if is a non-constant entire
function, then the image of
is dense in
; that is, for every
, there exists
some
that is arbitrarily close to
.
Proof
Suppose on the other hand that there is some not in the
image of
, and that there is a positive real
such
that
has no point within
of
. Then
the function
is holomorphic on the entire complex plane, and it is bounded by
. It is therefore constant. Therefore
is constant.
Picard's Little Theorem offers the stronger result that if
avoids two points in the plane, then it is constant. It is
possible for an entire function to avoid a single point, as
avoids 0.