A removable singularity in complex analysis refers to a point in the complex plane where a function is undefined or has a value that is not finite. However, this singularity can be removed by defining a new value for the function at that point, making it continuous and well-defined throughout the complex plane.
A removable singularity can be identified by looking at the behavior of the function near the point in question. If the function approaches a finite value as the point is approached from all directions, then it is a removable singularity. This is in contrast to essential singularities, where the function approaches infinity or oscillates as the point is approached.
Removable singularities are significant because they allow us to extend the domain of a function to include points where it was previously undefined. This allows for a more complete understanding and analysis of the function, as well as the ability to use techniques such as differentiation and integration.
Yes, a removable singularity can be removed without changing the function by defining a new value for the function at the singularity point. This new value should ensure that the function is continuous and well-defined at that point, while also not altering the behavior of the function at other points in the complex plane.
Analytic functions are functions that are differentiable at every point within their domain. Removable singularities are a type of singularity that can be removed by defining a new value for the function, thereby making it differentiable at the singularity point. Therefore, analytic functions cannot have removable singularities within their domain.