# simple discontinuity

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## 4 matching pages

##### 1: 1.4 Calculus of One Variable
A simple discontinuity of $f(x)$ at $x=c$ occurs when $f(c+)$ and $f(c-)$ exist, but $f(c+)\not=f(c-)$. If $f(x)$ is continuous on an interval $I$ save for a finite number of simple discontinuities, then $f(x)$ is piecewise (or sectionally) continuous on $I$. For an example, see Figure 1.4.1
##### 2: 28.12 Definitions and Basic Properties
When $q=0$ Equation (28.2.16) has simple roots, given by … As in §28.7 values of $q$ for which (28.2.16) has simple roots $\lambda$ are called normal values with respect to $\nu$. …As a function of $\nu$ with fixed $q$ ($\neq 0$), $\lambda_{\nu}\left(q\right)$ is discontinuous at $\nu=\pm 1,\pm 2,\dots$. …
##### 3: Bibliography D
• A. M. Din (1981) A simple sum formula for Clebsch-Gordan coefficients. Lett. Math. Phys. 5 (3), pp. 207–211.
• B. I. Dunlap and B. R. Judd (1975) Novel identities for simple $n$-$j$ symbols. J. Mathematical Phys. 16, pp. 318–319.
• T. M. Dunster (1996b) Asymptotics of the generalized exponential integral, and error bounds in the uniform asymptotic smoothing of its Stokes discontinuities. Proc. Roy. Soc. London Ser. A 452, pp. 1351–1367.
• T. M. Dunster (2001a) Convergent expansions for solutions of linear ordinary differential equations having a simple turning point, with an application to Bessel functions. Stud. Appl. Math. 107 (3), pp. 293–323.
• T. M. Dunster (2014) Olver’s error bound methods applied to linear ordinary differential equations having a simple turning point. Anal. Appl. (Singap.) 12 (4), pp. 385–402.
• ##### 4: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
Often circumstances allow rather stronger statements, such as uniform convergence, or pointwise convergence at points where $f(x)$ is continuous, with convergence to $(f(x_{0}-)+f(x_{0}+))/2$ if $x_{0}$ is an isolated point of discontinuity. …
###### §1.18(vi) Continuous Spectra and Eigenfunction Expansions: Simple Cases
this being a matrix element of the resolvent $F(T)=\left(z-T\right)^{-1}$, this being a key quantity in many parts of physics and applied math, quantum scattering theory being a simple example, see Newton (2002, Ch. 7). …This is the discontinuity across the branch cut in (1.18.52) $\boldsymbol{\sigma}_{c}\subset\mathbb{R}$, from $z$ below to above the cut, divided by $2\pi\mathrm{i}$. …