Residue
Singularity: A singularity is a point at which a function becomes undefined, infinite, or behaves in a way that prevents the function from being smooth or continuous. Isolated Singularity: An isolated singularity is a singularity that is distinct and separated from other singularities. In other words, it is a point where the function misbehaves, but if you look in a small neighborhood around that point, the function is well-defined and analytic (smooth). Types of Isolated Singularities: Removable Singularity: The function can be defined at the singular point in such a way that it becomes analytic in the entire neighborhood. The singularity is "removed" by defining the function appropriately. Pole: The function approaches infinity as you get closer to the singular point. Essential Singularity: The function has a more complicated behavior at the singular point, and it cannot be "smoothed out" or made analytic in the neighborhood. Examples: For the function � ( � ) =