Search results
Results From The WOW.Com Content Network
The function in example 1, a removable discontinuity. Consider the piecewise function = {< = >. The point = is a removable discontinuity.For this kind of discontinuity: The one-sided limit from the negative direction: = and the one-sided limit from the positive direction: + = + at both exist, are finite, and are equal to = = +.
6.1 Example 1. 6.2 Example 2. ... Similarly, the Chebyshev polynomials of the second kind ... At a discontinuity, the series will converge to the average of the right ...
The self-similar solution of the second kind also appears in different contexts such as in boundary-layer problems subjected to small perturbations, [8] as was identified by Keith Stewartson, [9] Paul A. Libby and Herbert Fox. [10] Moffatt eddies are also a self-similar solution of the second kind.
Let be a real-valued monotone function defined on an interval. Then the set of discontinuities of the first kind is at most countable.. One can prove [5] [3] that all points of discontinuity of a monotone real-valued function defined on an interval are jump discontinuities and hence, by our definition, of the first kind.
The following table gives an overview of Green's functions of frequently appearing differential operators, where = + +, = +, is the Heaviside step function, () is a Bessel function, () is a modified Bessel function of the first kind, and () is a modified Bessel function of the second kind. [2]
n (x) with n = −0.5 in the complex plane from −2 − 2i to 2 + 2i Plot of the Hankel function of the second kind H (2) n (x) with n = −0.5 in the complex plane from −2 − 2i to 2 + 2i. Another important formulation of the two linearly independent solutions to Bessel's equation are the Hankel functions of the first and second kind, H (1 ...
Again, f(z) = 1 is more than 1 ⁄ 2 away from f(y) = 0. Its restrictions to the set of rational numbers and to the set of irrational numbers are constants and therefore continuous. The Dirichlet function is an archetypal example of the Blumberg theorem .
In magnetohydrodynamics (MHD), shocks and discontinuities are transition layers where properties of a plasma change from one equilibrium state to another. The relation between the plasma properties on both sides of a shock or a discontinuity can be obtained from the conservative form of the MHD equations, assuming conservation of mass, momentum, energy and of .