What is a singular function in math?
In mathematics, a real-valued function f on the interval [a, b] is said to be singular if it has the following properties: f is continuous on [a, b]. (**) there exists a set N of measure 0 such that for all x outside of N the derivative f ′(x) exists and is zero, that is, the derivative of f vanishes almost everywhere.
What is singularity in differential equation?
In the theory of ordinary differential equations, a movable singularity is a point where the solution of the equation behaves badly and which is “movable” in the sense that its location depends on the initial conditions of the differential equation.
Do singularities exist?
A singularity is a point in space where there is a mass with infinite density. Singularities are predicted to exist in black holes by Einstein’s theory of general relativity, which is a theory that has done remarkably well at matching experimental results.
Is Infinity a singularity?
Definition (Isolated Singularity at Infinity): The point at infinity z = ∞ is called an isolated singularity of f(z) if f(z) is holomorphic in the exterior of a disk {z ∈ C : |z| > R}.
What is numerical singularity?
If the model is used in a static simulation with no boundary conditions (only applied forces), this small net force would cause unlimited rigid body motion of the model. Such rigid body motion is known mathematically as a numerical singularity.
Why is it called singularity?
The concept and the term “singularity” were popularized by Vernor Vinge in his 1993 essay The Coming Technological Singularity, in which he wrote that it would signal the end of the human era, as the new superintelligence would continue to upgrade itself and would advance technologically at an incomprehensible rate.
What are points of singularity?
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. fails to be analytic. Isolated singularities may be classified as poles, essential singularities, logarithmic singularities, or removable singularities.
What is ordinary and singular point?
In mathematics, in the theory of ordinary differential equations in the complex plane , the points of. are classified into ordinary points, at which the equation’s coefficients are analytic functions, and singular points, at which some coefficient has a singularity.
What is analytic point?
Definition: A function f is called analytic at a point z0 ∈ C if there exist r > 0 such that f is differentiable at every point z ∈ B(z0, r). Analyticity =⇒ Differentiability, where as Differentiability =⇒ Analyticity.