proof of Casorati-Weierstrass theorem
Assume that is an essential singularity of . Let be a punctured neighborhood of , and let . We have to show that is a limit point of . Suppose it is not, then there is an such that for all , and the function
for , and is either a removable singularity of (if ) or a pole of order (if has a zero of order at ). This contradicts our assumption that is an essential singularity, which means that must be a limit point of . The argument holds for all , so is dense in for any punctured neighborhood of .
To prove the converse, assume that is dense in for any punctured neighborhood of . If is a removable singularity, then is bounded near , and if is a pole, as . Either of these possibilities contradicts the assumption that the image of any punctured neighborhood of under is dense in , so must be an essential singularity of .
|Title||proof of Casorati-Weierstrass theorem|
|Date of creation||2013-03-22 13:32:40|
|Last modified on||2013-03-22 13:32:40|
|Last modified by||pbruin (1001)|