Removable singularity

The Riemannian Hebbarkeitssatz ( after Bernhard Riemann ) is a fundamental result of the mathematical theory of functions. The theorem states that a singularity (ie a point at which a holomorphic function is not defined ) if and only removed ( " fixed " ) may be, if an area exists around the singularity, where the holomorphic function is limited. Such a singularity is called liftable.

The set

It is a point of the domain, is a holomorphic function on. Is on a punctured neighborhood of limited, so there is a holomorphic function on all of

You can then continue thus holomorphic in the point itself and thus " cancel " the "gap" in the domain of definition of.

Generalizations

A simple generalization is to the effect to weaken the condition of boundedness that only

Applies. It follows easily from the above formulation by applying it to the limited in an environment of function.

Example of use: non-existence of a holomorphic square root function

Claim: There is no on holomorphic function that matches for met and for positive real arguments with the usual root function.

Proof by contradiction: Suppose that in order to be in a punctured neighborhood of zero is limited, according to the Riemannian Hebbarkeitssatz therefore holomorphic on all continuable. This would also be the derivative of at 0 is locally bounded. On the other hand, is unbounded for positive real: From this contradiction it follows that the original claim must be true.