Singulärstelle

Singulärstelle is a German term that translates to singular point or singularity in English. In mathematics, particularly in complex analysis, a singular point of a function is a point where the function ceases to be analytic. This means that at a singular point, the function's derivative does not exist or behaves in a way that violates the conditions for analyticity.

There are several types of singular points. Isolated singularities are points that have a neighborhood containing no other singularities. These can be further classified as removable singularities, poles, and essential singularities. A removable singularity is a point where the function can be redefined to be analytic. A pole is a singularity where the function's magnitude approaches infinity. An essential singularity is the most complex type, where the function's behavior near the point is highly erratic.

Non-isolated singularities occur in sets where every neighborhood of the singularity contains other singularities. Examples include branch points, which are common in functions involving roots or logarithms, where the function's value is not uniquely defined over a continuous path.

The study of singular points is crucial for understanding the behavior of complex functions and for techniques like contour integration, residue calculus, and the classification of differential equations. Identifying and characterizing singularities helps in determining the convergence of series expansions and the overall structure of a mathematical object.