## The Casorati Weierstrass Theorem

If is a complex valued function on the complex plane and has an essential singularity at a point then as approaches strange things start to happen. starts to change wildly, approaching arbitrarily close every point in the complex plane. More precisely,

The Casorati Weierstrass Theorem

Suppose that a complex function has an essential singularity at Let be any complex number and let and be positive real numbers. There exists such that and  Proof

Assume the theorem is false. Then there exists and positive real numbers and such that the function is analytic on the punctured open disc and the last line does not hold, so that for Since for the function is analytic. Moreover, for and so has a removable singularity at so by defining appropriately, we can make analytic on Now for and so for If then would have a removable singularity at which could be removed by letting If then for some positive integer where is analytic at with Thus There is a stronger theorem called Picard's Theorem which states that takes on all values in except possible one, for  