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In mathematics, a singularity is a point at which a given mathematical object is not defined, or a point where the mathematical object ceases to be well-behaved in some particular way, such as by lacking differentiability or analyticity.
May 28, 2023 · Removable singularity. When every \(b_n\) is zero, so that \(\begin{eqnarray}\label{residue003} f(z)=\sum_{n=0}^{\infty} a_n(z-z_0)^n,\quad (0\lt |z-z_0| \lt R_2). \end{eqnarray}\) In this case, \(z_0\) is known as a removable singular point. Note that the residue at a removable singular point is always zero.
Jul 13, 2024 · In general, a singularity is a point at which an equation, surface, etc., blows up or becomes degenerate. Singularities are often also called singular points. Singularities are extremely important in complex analysis, where they characterize the possible behaviors of analytic functions.
singularity. The embedding dimension is equal to the smallest dimension of any smooth germ (M; q) such that . embed. M is smooth. As TpX TpM, and the dimension of M is equal to the dimension of TpM, it is clear that the dimension of M .
Nov 18, 2023 · In the field of mathematics, singularity refers to a point in a function or equation where it behaves in an unusual way, deviating from the expected patterns. This concept has...
Singularity theory is a field of intensive study in modern mathematics with fascinating relations to algebraic geometry, complex analysis, commutative algebra, representation theory, theory of Lie groups, topology, dynamical systems, and many more, and with numerous applications in the natural and technical sciences.
For a function \(f(z)\), the singularity \(z_0\) is an isolated singularity if \(f\) is analytic on the deleted disk \(0 < |z - z_0| < r\) for some \(r > 0\). Example \(\PageIndex{1}\) \(f(z) = \dfrac{}{}\) has isolated singularities at \(z = 0\), \(\pm i\).
In mathematics, singularity theory studies spaces that are almost manifolds, but not quite. A string can serve as an example of a one-dimensional manifold, if one neglects its thickness. A singularity can be made by balling it up, dropping it on the floor, and flattening it.
Singularity, of a function of the complex variable z is a point at which it is not analytic (that is, the function cannot be expressed as an infinite series in powers of z) although, at points arbitrarily close to the singularity, the function may be analytic, in which case it is called an isolated.
This volume consists of ten articles which provide an in-depth and reader-friendly survey of some of the foundational aspects of singularity theory. Authored by world experts, the various contributions deal with both classical material and modern developments, covering a wide range of topics.
