removable discontinuity

SearchAdvancedHelp

(0.002 seconds)

1—10 of 52 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 $f(c+)\ne f(c-)$. 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: Viewing DLMF Interactive 3D Graphics

… ►Any installed VRML or X3D browser should be removed before installing a new one. …

3: 5.10 Continued Fractions

…

4: 17.5 ${}_0{}^{}\varphi _0^{},{}_1{}^{}\varphi _0^{},{}_1{}^{}\varphi _1^{}$ Functions

… ►

17.5.1 $${}_0{}^{}\varphi _0^{}(-;-;q,z)=\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n{q}^{\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)}{z}^n}{{(q;q)}_n}={(z;q)}_{\mathrm{\infty }};$$

ⓘ

Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function, $q$: complex base, $n$: nonnegative integer and $z$: complex variable

Referenced by:

§17.2(iii), Erratum (V1.1.11) for Equation (17.5.1)

Permalink:

http://dlmf.nist.gov/17.5.E1

Encodings:

pMML, png, TeX

Correction (effective with 1.1.11):

The constraint is not necessary and was removed.

See also:

Annotations for §17.5, §17.5 and Ch.17

… ►

17.5.5 $${}_1{}^{}\varphi _1^{}(\genfrac{}{}{0pt}{}{a}{c};q,c/a)=\frac{{(c/a;q)}_{\mathrm{\infty }}}{{(c;q)}_{\mathrm{\infty }}}.$$

ⓘ

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function and $q$: complex base

Referenced by:

Erratum (V1.1.10) for Equation (17.5.5)

Permalink:

http://dlmf.nist.gov/17.5.E5

Encodings:

pMML, png, TeX

Correction (effective with 1.1.10):

The constraint is not necessary and was removed.

See also:

Annotations for §17.5, §17.5 and Ch.17

5: 10.25 Definitions

… ►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 $ℂ\setminus (-\mathrm{\infty },0]$, 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 $ℂ\setminus (-\mathrm{\infty },0]$, and two-valued and discontinuous on the cut $\mathrm{ph}z=\pm \pi$. …

6: Bibliography W

… ►

Smoothing of Stokes’s discontinuity for the generalized Bessel function. II. Proc. Roy. Soc. London Ser. A 455, pp. 3065–3084.

ⓘ

External Links:

ISSN 1364-5021, Document, MathReview, ZentralBlatt

Referenced by:

§10.46.

Encodings:

BibTeX

►

Smoothing of Stokes’s discontinuity for the generalized Bessel function. Proc. Roy. Soc. London Ser. A 455, pp. 1381–1400.

ⓘ

External Links:

ISSN 1364-5021, Document, MathReview, ZentralBlatt

Referenced by:

§10.46.

Encodings:

BibTeX

…

7: 1.10 Functions of a Complex Variable

… ►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 $(\sin z)/z$ 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 $ℂ$ by removing the real axis from $1$ to $\mathrm{\infty }$ and from $-1$ to $-\mathrm{\infty }$; see Figure 1.10.1. …

8: 25.14 Lerch’s Transcendent

… ►

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

ⓘ

Defines:

$\mathrm{\Phi }(z,s,a)$: Lerch’s transcendent

Symbols:

$\equiv$: equals by definition, $\mathrm{\Re }$: real part, $n$: nonnegative integer, $a$: real or complex parameter, $s$: complex variable and $z$: complex variable

Keywords:

definition, infinite series, series representation

Source:

Erdélyi et al. (1953a, (1.11.1), p. 27)

Referenced by:

(25.14.2), (25.14.3), §25.14(ii), §25.14, Erratum (V1.0.21) for Equation (25.14.1)

Permalink:

http://dlmf.nist.gov/25.14.E1

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.21):

The previous constraint $a\ne 0,-1,-2,\mathrm{\dots },$ was removed. A clarification regarding the correct constraints for Lerch’s transcendent $\mathrm{\Phi }(z,s,a)$ has been added in the text immediately below.

See also:

Annotations for §25.14(i), §25.14 and Ch.25

…

9: 26.15 Permutations: Matrix Notation

… ►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. …

10: 23.18 Modular Transformations

… ►

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

ⓘ

Symbols:

$\lfloor x\rfloor$: floor of $x$ , $\left(m,n\right)$: greatest common divisor (gcd), $c$: integer and $d$: integer

Referenced by:

§23.18, Erratum (V1.0.11) for Equation (23.18.7)

Permalink:

http://dlmf.nist.gov/23.18.E7

Encodings:

pMML, png, TeX

Errata (effective with 1.0.11):

Originally the sum ${\sum }_{r=1}^{c-1}$ was written with an additional constraint on the summation, that $\left(r,c\right)=1$. This additional condition was incorrect and has been removed.

Reported 2015-10-05 by Howard Cohl and Tanay Wakhare

See also:

Annotations for §23.18, §23.18 and Ch.23

…