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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.
The equation has two linearly independent solutions. At each of the three singular points 0, 1, ∞, there are usually two special solutions of the form x s times a holomorphic function of x, where s is one of the two roots of the indicial equation and x is a local variable vanishing at a regular singular point. This gives 3 × 2 = 6 special ...
For affine and projective varieties, the singularities are the points where the Jacobian matrix has a rank which is lower than at other points of the variety. An equivalent definition in terms of commutative algebra may be given, which extends to abstract varieties and schemes : A point is singular if the local ring at this point is not a ...
A curve with a triple point at the origin: x(t) = sin(2t) + cos(t), y(t) = sin(t) + cos(2t) In general, if all the terms of degree less than k are 0, and at least one term of degree k is not 0 in f, then curve is said to have a multiple point of order k or a k-ple point.
Euler's identity is a direct result of Euler's formula, published in his monumental 1748 work of mathematical analysis, Introductio in analysin infinitorum, [16] but it is questionable whether the particular concept of linking five fundamental constants in a compact form can be attributed to Euler himself, as he may never have expressed it.
This equation, stated by Euler in 1758, [2] is known as Euler's polyhedron formula. [3] It corresponds to the Euler characteristic of the sphere (i.e. χ = 2 {\displaystyle \ \chi =2\ } ), and applies identically to spherical polyhedra .
Some solutions of a differential equation having a regular singular point with indicial roots = and .. In mathematics, the method of Frobenius, named after Ferdinand Georg Frobenius, is a way to find an infinite series solution for a linear second-order ordinary differential equation of the form ″ + ′ + = with ′ and ″.
Then for an ideal gas the compressible Euler equations can be simply expressed in the mechanical or primitive variables specific volume, flow velocity and pressure, by taking the set of the equations for a thermodynamic system and modifying the energy equation into a pressure equation through this mechanical equation of state. At last, in ...