proof of extended Liouville’s theorem
This is a proof of the second, more general, form of Liouville’s theorem given in the parent (http://planetmath.org/LiouvillesTheorem2) article.
Let be a holomorphic function such that
for some and for with sufficiently large. Consider
Since is holomorphic, is as well, and by the bound on , we have
again for sufficiently large.
By induction, is a polynomial of degree at most , and thus is a polynomial of degree at most .
|Title||proof of extended Liouville’s theorem|
|Date of creation||2013-03-22 16:18:31|
|Last modified on||2013-03-22 16:18:31|
|Last modified by||rm50 (10146)|