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The singularities of solutions of these equations are The point , and; The point 0 for types III, V and VI, and; The point 1 for type VI, and; Possibly some movable poles; For type I, the singularities are (movable) double poles of residue 0, and the solutions all have an infinite number of such poles in the complex plane.
Essential singularities approach no limit, not even if valid answers are extended to include . In real analysis, a singularity or discontinuity is a property of a function alone. Any singularities that may exist in the derivative of a function are considered as belonging to the derivative, not to the original function.
Null singularities: These singularities occur on light-like or null surfaces. An example might be found in certain types of black hole interiors, such as the Cauchy horizon of a charged (Reissner–Nordström) or rotating black hole. A singularity can be either strong or weak:
Singularities of this kind include caustics, very familiar as the light patterns at the bottom of a swimming pool. Other ways in which singularities occur is by degeneration of manifold structure. The presence of symmetry can be good cause to consider orbifolds , which are manifolds that have acquired "corners" in a process of folding up ...
In fact more is true: Schwarz's list underlies all second-order equations with regular singularities on compact Riemann surfaces having finite monodromy, by a pullback from the hypergeometric equation on the Riemann sphere by a complex analytic mapping, of degree computable from the equation's data. [1] [2]
Resolution of singularities in characteristic 0 in all dimensions was first proved by Hironaka (1964). He proved that it was possible to resolve singularities of varieties over fields of characteristic 0 by repeatedly blowing up along non-singular subvarieties, using a very complicated argument by induction on the dimension.
The attributes of singularities include the following in various degrees, according to context: Instability: because singularities tend to produce effects out of proportion to the size of initial causes. System relatedness: the effects of a singularity are characteristic of the system.
In algebraic geometry, a Du Val singularity, also called simple surface singularity, Kleinian singularity, or rational double point, is an isolated singularity of a complex surface which is modeled on a double branched cover of the plane, with minimal resolution obtained by replacing the singular point with a tree of smooth rational curves, with intersection pattern dual to a Dynkin diagram of ...