# removable discontinuity

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## 1—10 of 43 matching pages

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

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►A

*removable singularity*of $f(x)$ at $x=c$ occurs when $f(c+)=f(c-)$ but $f(c)$ is undefined. … ►A*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 …##### 2: Viewing DLMF Interactive 3D Graphics

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##### 3: 5.10 Continued Fractions

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##### 4: 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 $. …##### 5: 1.10 Functions of a Complex Variable

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►This singularity is

*removable*if ${a}_{n}=0$ for all $$, and in this case the Laurent series becomes the Taylor series. …Lastly, if ${a}_{n}\ne 0$ for infinitely many negative $n$, then ${z}_{0}$ is an*isolated essential singularity*. … ►An isolated singularity ${z}_{0}$ is always removable when ${lim}_{z\to {z}_{0}}f(z)$ exists, for example $(\mathrm{sin}z)/z$ at $z=0$. … ►A*cut domain*is one from which the points on finitely many nonintersecting simple contours (§1.9(iii)) have been removed. … ►Branches of $F(z)$ can be defined, for example, in the cut plane $D$ obtained from $\u2102$ by removing the real axis from $1$ to $\mathrm{\infty}$ and from $-1$ to $-\mathrm{\infty}$; see Figure 1.10.1. …##### 6: 25.14 Lerch’s Transcendent

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25.14.1
$$\mathrm{\Phi}(z,s,a)\equiv \sum _{n=0}^{\mathrm{\infty}}\frac{{z}^{n}}{{(a+n)}^{s}},$$
$$; $\mathrm{\Re}s>1,|z|=1$.

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##### 7: 26.15 Permutations: Matrix Notation

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►For $(j,k)\in B$, $B\setminus [j,k]$ denotes $B$ after removal of all elements of the form $(j,t)$ or $(t,k)$, $t=1,2,\mathrm{\dots},n$.
$B\setminus (j,k)$ denotes $B$ with the element $(j,k)$
removed.
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##### 8: 23.18 Modular Transformations

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23.18.7
$$s(d,c)=\sum _{r=1}^{c-1}\frac{r}{c}\left(\frac{dr}{c}-\lfloor \frac{dr}{c}\rfloor -\frac{1}{2}\right),$$
$c>0$.

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##### 9: 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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##### 10: 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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