removable discontinuity

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1: 1.4 Calculus of One Variable

… ► ►A simple discontinuity of

f(x)

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 … ►

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 ${{}_{0}\phi_{0}},{{}_{1}\phi_{0}},{{}_{1}\phi_{1}}$ Functions

… ►

17.5.1

{{}_{0}\phi_{0}}\left(-;-;q,z\right)=\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{% \genfrac{(}{)}{0.0pt}{}{n}{2}}z^{n}}{\left(q;q\right)_{n}}=\left(z;q\right)_{% \infty};

ⓘ

Symbols:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : 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 $|z|<1$ is not necessary and was removed.
See also:: Annotations for §17.5, §17.5 and Ch.17

… ►

17.5.5

{{}_{1}\phi_{1}}\left({a\atop c};q,c/a\right)=\frac{\left(c/a;q\right)_{\infty% }}{\left(c;q\right)_{\infty}}.

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : 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 $|c|<|a|$ 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

(\tfrac{1}{2}z)^{\nu}

, is analytic in

\mathbb{C}\setminus(-\infty,0]

, and two-valued and discontinuous on the cut

\operatorname{ph}z=\pm\pi

. … ►The principal branch corresponds to the principal value of the square root in (10.25.3), is analytic in

\mathbb{C}\setminus(-\infty,0]

, and two-valued and discontinuous on the cut

\operatorname{ph}z=\pm\pi

. …

6: Bibliography W

… ►

R. Wong and Y.-Q. Zhao (1999a) 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

►

R. Wong and Y.-Q. Zhao (1999b) 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

n<0

, and in this case the Laurent series becomes the Taylor series. …Lastly, if

a_{n}\not=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

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

\mathbb{C}

by removing the real axis from

1

\infty

and from

-1

-\infty

; see Figure 1.10.1. …

8: 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)

(t,k)

t=1,2,\ldots,n

B\setminus(j,k)

denotes

B

with the element

(j,k)

removed. …

9: 23.18 Modular Transformations

… ►

23.18.7

{s(d,c)=\sum_{r=1}^{c-1}\frac{r}{c}\left(\frac{dr}{c}-\left\lfloor\frac{dr}{c}% \right\rfloor-\frac{1}{2}\right),}

c>0

ⓘ

Symbols:: $\left\lfloor\NVar{x}\right\rfloor$ : floor of $x$ , $\left(\NVar{m},\NVar{n}\right)$ : 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

…

10: 10.2 Definitions

… ►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

\operatorname{ph}z=\pm\pi

; compare §4.2(i). … ►The principal branches of

{H^{(1)}_{\nu}}\left(z\right)

and

{H^{(2)}_{\nu}}\left(z\right)