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Algebraic curves in the plane may be defined as the set of points (x, y) satisfying an equation of the form (,) =, where f is a polynomial function :. If f is expanded as = + + + + + + If the origin (0, 0) is on the curve then a 0 = 0.
In mathematics, a cusp, sometimes called spinode in old texts, is a point on a curve where a moving point must reverse direction. A typical example is given in the figure. A cusp is thus a type of singular point of a curve. For a plane curve defined by an analytic, parametric equation
A plane curve defined by an implicit equation (,) =,where F is a smooth function is said to be singular at a point if the Taylor series of F has order at least 2 at this point.. The reason for this is that, in differential calculus, the tangent at the point (x 0, y 0) of such a curve is defined by the equation
The simplest example of singularities are curves that cross themselves. But there are other types of singularities, like cusps. For example, the equation y 2 − x 3 = 0 defines a curve that has a cusp at the origin x = y = 0. One could define the x-axis as a tangent at this point, but this definition can not be the same as the definition at ...
An algebraic curve in the Euclidean plane is the set of the points whose coordinates are the solutions of a bivariate polynomial equation p(x, y) = 0.This equation is often called the implicit equation of the curve, in contrast to the curves that are the graph of a function defining explicitly y as a function of x.
It was noticed in the formulation of Bézout's theorem that such singular points must be counted with multiplicity (2 for a double point, 3 for a cusp), in accounting for intersections of curves. It was then a short step to define the general notion of a singular point of an algebraic variety; that is, to allow higher dimensions.
A singular solution y s (x) of an ordinary differential equation is a solution that is singular or one for which the initial value problem (also called the Cauchy problem by some authors) fails to have a unique solution at some point on the solution. The set on which a solution is singular may be as small as a single point or as large as the ...
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.