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Borrowing from complex analysis, this is sometimes called an essential singularity. The possible cases at a given value for the argument are as follows. A point of continuity is a value of for which () = = (+), as one expects for a smooth function. All the values must be finite.
Point a is an ordinary point when functions p 1 (x) and p 0 (x) are analytic at x = a. Point a is a regular singular point if p 1 (x) has a pole up to order 1 at x = a and p 0 has a pole of order up to 2 at x = a. Otherwise point a is an irregular singular point.
In complex analysis, a branch of mathematics, an isolated singularity is one that has no other singularities close to it. In other words, a complex number z 0 is an isolated singularity of a function f if there exists an open disk D centered at z 0 such that f is holomorphic on D \ {z 0}, that is, on the set obtained from D by taking z 0 out.
In complex analysis, an essential singularity of a function is a "severe" singularity near which the function exhibits striking behavior. The category essential singularity is a "left-over" or default group of isolated singularities that are especially unmanageable: by definition they fit into neither of the other two categories of singularity ...
The complex plane extended by a point at infinity is called the Riemann sphere. If f is a function that is meromorphic on the whole Riemann sphere, then it has a finite number of zeros and poles, and the sum of the orders of its poles equals the sum of the orders of its zeros.
When discussing mathematical analysis in general, or more specifically real analysis or complex analysis or differential equations, it is common for a function which contains a mathematical singularity to be referred to as a 'singular function'. This is especially true when referring to functions which diverge to infinity at a point or on a ...
Points of V that are not singular are called non-singular or regular. It is always true that almost all points are non-singular, in the sense that the non-singular points form a set that is both open and dense in the variety (for the Zariski topology, as well as for the usual topology, in the case of varieties defined over the complex numbers). [1]
A singular point p of X is called isolated (or that p is an isolated singularity of X) if X(p) = 0 and there exists an open neighbourhood U ⊆ R n, containing p, such that X(q) ≠ 0 for all q in U, different from p. An isolated singularity of X is called algebraically isolated if, when considered over the complex domain, it remains isolated ...