# discontinuity

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## 1—10 of 19 matching pages

##### 1: 1.4 Calculus of One Variable

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*simple discontinuity*of $f(x)$ at $x=c$ occurs when $f(c+)$ and $f(c-)$ exist, but $f(c+)\ne f(c-)$. 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 … ►###### Stieltjes Measure with $\alpha (x)$ Discontinuous

…##### 2: 10.25 Definitions

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►In particular, the

*principal branch*of ${I}_{\nu}\left(z\right)$ is defined in a similar way: it corresponds to the principal value of ${(\frac{1}{2}z)}^{\nu}$, is analytic in $\u2102\setminus (-\mathrm{\infty},0]$, and two-valued and discontinuous on the cut $\mathrm{ph}z=\pm \pi $. … ►The*principal branch*corresponds to the principal value of the square root in (10.25.3), is analytic in $\u2102\setminus (-\mathrm{\infty},0]$, and two-valued and discontinuous on the cut $\mathrm{ph}z=\pm \pi $. …##### 3: Bibliography W

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Smoothing of Stokes’s discontinuity for the generalized Bessel function. II.
Proc. Roy. Soc. London Ser. A 455, pp. 3065–3084.
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Smoothing of Stokes’s discontinuity for the generalized Bessel function.
Proc. Roy. Soc. London Ser. A 455, pp. 1381–1400.
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##### 4: 10.2 Definitions

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►Except in the case of ${J}_{\pm n}\left(z\right)$, the principal branches of ${J}_{\nu}\left(z\right)$ and ${Y}_{\nu}\left(z\right)$ are two-valued and discontinuous on the cut $\mathrm{ph}z=\pm \pi $; compare §4.2(i).
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►The principal branches of ${H}_{\nu}^{(1)}\left(z\right)$ and ${H}_{\nu}^{(2)}\left(z\right)$ are two-valued and discontinuous on the cut $\mathrm{ph}z=\pm \pi $.
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##### 5: 4.15 Graphics

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##### 6: 4.24 Inverse Trigonometric Functions: Further Properties

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##### 7: 4.2 Definitions

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►Consequently $\mathrm{ln}z$ is two-valued on the cut, and discontinuous across the cut.
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►This is an analytic function of $z$ on $\u2102\setminus (-\mathrm{\infty},0]$, and is two-valued and discontinuous on the cut shown in Figure 4.2.1, unless $a\in \mathbb{Z}$.
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##### 8: 18.40 Methods of Computation

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►Results similar to these appear in Langhoff et al. (1976) in methods developed for physics applications, and which includes treatments of systems with discontinuities in $\mu (x)$, using what is referred to as the

*Stieltjes derivative*which may be traced back to Stieltjes, as discussed by Deltour (1968, Eq. 12). …##### 9: 28.12 Definitions and Basic Properties

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►As a function of $\nu $ with fixed $q$ ($\ne 0$), ${\lambda}_{\nu}\left(q\right)$ is discontinuous at $\nu =\pm 1,\pm 2,\mathrm{\dots}$.
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##### 10: Bibliography D

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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.
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