Singulärpunkte

Singulärpunkte, translated as singular points, are locations in mathematics where a function or a geometric object behaves in an unusual or undefined way. These points are exceptions to the general rules that govern the behavior of the function or object elsewhere. In the context of complex analysis, a singular point of a function is a point where the function is not analytic, meaning it is not differentiable in the complex sense. These singular points can be further classified into removable singularities, poles, and essential singularities. A removable singularity occurs when the function can be redefined at that point to become analytic. A pole is a singularity where the function's magnitude approaches infinity. Essential singularities are the most complex, where the function's behavior near the point is highly erratic.

In algebraic geometry, singular points refer to points on an algebraic curve or surface where the tangent space is not well-defined. This typically happens at points where the curve or surface self-intersects or has a cusp. These points are of great interest as they often reveal important structural properties of the geometric object. For example, the nature of the singularity at a point can help classify the overall shape of the curve or surface. The study of singular points is crucial for understanding the complete behavior of mathematical objects, as they often represent critical transitions or degenerate cases.