removable%20singularity

(0.002 seconds)

11—20 of 196 matching pages

11: 25.2 Definition and Expansions

… ►It is a meromorphic function whose only singularity in

\mathbb{C}

is a simple pole at

s=1

, with residue 1. … ►

25.2.4

\zeta\left(s\right)=\frac{1}{s-1}+\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}\gamma% _{n}(s-1)^{n},

ⓘ

Symbols:: $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\gamma_{\NVar{n}}$ : Stieltjes constants, $!$ : factorial (as in $n!$ ), $\Re$ : real part, $n$ : nonnegative integer and $s$ : complex variable
Keywords:: infinite series, Laurent series
Sources:: Hardy (1912, p. 216); mistakenly missing factor of $(-1)^{n}/n!$ ; Ivić (1985, (1.11), p. 4)
A&S Ref:: 23.2.5 (with unnecessary constraint removed)
Referenced by:: Erratum (V1.0.17) for Equation (25.2.4), Erratum (V1.0.23) for Subsection 25.2(ii) Other Infinite Series
Permalink:: http://dlmf.nist.gov/25.2.E4
Encodings:: pMML, png, TeX
Clarification (effective with 1.0.17):: Originally this equation had the constraint $\Re s>0$ . This constraint is unnecessary because, as stated after (25.2.1), $\zeta\left(s\right)$ is meromorphic with a simple pole at $s=1$ .
Suggested 2017-12-05 by John Harper
See also:: Annotations for §25.2(ii), §25.2 and Ch.25

…

12: Errata

… ►

Subsection 14.3(iv)

A sentence was added at the end of this subsection indicating that from (15.9.15), it follows that $1-2\mu=0,-1,-2,\dots$ and $\nu+\mu+1=0,-1,-2,\dots$ are removable singularities.

… ►

Equation (5.11.14)

The previous constraint $\Re\left(b-a\right)>0$ was removed, see Fields (1966, (3)).

… ►

Section 36.1 Special Notation

The entry for $\ast$ to represent complex conjugation was removed (see Version 1.0.19).

… ►

Equation (25.2.4)

The original constraint, $\Re s>0$ , was removed because, as stated after (25.2.1), $\zeta\left(s\right)$ is meromorphic with a simple pole at $s=1$ , and therefore $\zeta\left(s\right)-(s-1)^{-1}$ is an entire function.

Suggested by John Harper.

… ►

References

Bibliographic citations and clarifications have been added, removed, or modified in §§5.6(i), 5.10, 7.8, 7.25(iii), and 32.16.

…

13: 11.6 Asymptotic Expansions

… ►

c_{3}(\lambda)=20\lambda^{-6}-4\lambda^{-4},

…

14: 5.10 Continued Fractions

…

15: Bibliography O

… ►

A. B. Olde Daalhuis and F. W. J. Olver (1994) Exponentially improved asymptotic solutions of ordinary differential equations. II Irregular singularities of rank one. Proc. Roy. Soc. London Ser. A 445, pp. 39–56.

ⓘ

External Links:: ISSN 0962-8444, FullText, MathReview (Hussain S. Nur), ZentralBlatt
Referenced by:: §13.4(i), §2.11(v), §2.7(ii).
Encodings:: BibTeX

… ►

A. B. Olde Daalhuis (1998a) Hyperasymptotic solutions of higher order linear differential equations with a singularity of rank one. Proc. Roy. Soc. London Ser. A 454, pp. 1–29.

ⓘ

External Links:: ISSN 1364-5021, Document, MathReview (Donald A. Lutz), ZentralBlatt
Referenced by:: §2.11(v), §2.7(ii).
Encodings:: BibTeX

… ►

F. W. J. Olver and F. Stenger (1965) Error bounds for asymptotic solutions of second-order differential equations having an irregular singularity of arbitrary rank. J. Soc. Indust. Appl. Math. Ser. B Numer. Anal. 2 (2), pp. 244–249.

ⓘ

External Links:: Document, MathReview, ZentralBlatt
Referenced by:: §2.7(ii).
Encodings:: BibTeX

… ►

F. W. J. Olver (1965) On the asymptotic solution of second-order differential equations having an irregular singularity of rank one, with an application to Whittaker functions. J. Soc. Indust. Appl. Math. Ser. B Numer. Anal. 2 (2), pp. 225–243.

ⓘ

External Links:: Document, MathReview (T. E. Hull), ZentralBlatt
Referenced by:: §13.19, §13.7(ii).
Encodings:: BibTeX

… ►

F. W. J. Olver (1997a) Asymptotic solutions of linear ordinary differential equations at an irregular singularity of rank unity. Methods Appl. Anal. 4 (4), pp. 375–403.

ⓘ

External Links:: ISSN 1073-2772, MathReview (A. D. Wood), ZentralBlatt
Referenced by:: §2.7(ii).
Encodings:: BibTeX

…

16: 14.3 Definitions and Hypergeometric Representations

… ►From (15.9.15) it follows that

1-2\mu=0,-1,-2,\dots

and

\nu+\mu+1=0,-1,-2,\dots

are removable singularities of the right-hand sides of (14.3.21) and (14.3.22).

17: Bibliography

… ►

A. R. Ahmadi and S. E. Widnall (1985) Unsteady lifting-line theory as a singular-perturbation problem. J. Fluid Mech 153, pp. 59–81.

ⓘ

External Links:: Document, ZentralBlatt
Referenced by:: §11.12.
Encodings:: BibTeX

… ►

H. H. Aly, H. J. W. Müller-Kirsten, and N. Vahedi-Faridi (1975) Scattering by singular potentials with a perturbation – Theoretical introduction to Mathieu functions. J. Mathematical Phys. 16, pp. 961–970.

ⓘ

Notes:: Erratum: same journal 17 (1975), no. 7, p. 1361
External Links:: Document, MathReview (K. Veselic), ZentralBlatt
Referenced by:: 4th item.
Encodings:: BibTeX

… ►

V. I. Arnol’d, S. M. Guseĭn-Zade, and A. N. Varchenko (1988) Singularities of Differentiable Maps. Vol. II. Birkhäuser, Boston-Berlin.

ⓘ

Notes:: Monodromy and asymptotics of integrals, Translated from the Russian by Hugh Porteous, Translation revised by the authors and James Montaldi
External Links:: ISBN 0-8176-3185-2, MathReview, ZentralBlatt
Referenced by:: §36.12(iii).
Encodings:: BibTeX

►

V. I. Arnol’d (1972) Normal forms of functions near degenerate critical points, the Weyl groups

{A}_{k},{D}_{k},{E}_{k}

and Lagrangian singularities. Funkcional. Anal. i Priložen. 6 (4), pp. 3–25 (Russian).

ⓘ

Notes:: In Russian. English translation: Functional Anal. Appl., 6(1973), pp. 254–272.
External Links:: MathReview (G. R. Belickii), ZentralBlatt
Referenced by:: §36.2(i).
Encodings:: BibTeX

… ►

J. Avron and B. Simon (1982) Singular Continuous Spectrum for a Class of Almost Periodic Jacobi Matrices. Bulletin of the American Mathematical Society 6 (1), pp. 81–85.

ⓘ

External Links:: MathReview Entry
Referenced by:: §1.18(vii).
Encodings:: BibTeX

…

18: 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

19: Bibliography F

… ►

FDLIBM (free C library)

ⓘ

Notes:: The “Freely Distributable LIBM” package provides implementations of standard elementary functions plus a few higher functions, e.g. gamma. Double precision, maximum accuracy 20S. Developed by Sun Microsystems.
External Links:: GAMS (package FDLIBM)
Referenced by:: § ‣ Software Cross Index, § ‣ Software Cross Index.
Encodings:: BibTeX

… ►

S. Fempl (1960) Sur certaines sommes des intégral-cosinus. Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).

ⓘ

External Links:: MathReview (L. Carlitz), ZentralBlatt
Referenced by:: §6.15.
Encodings:: BibTeX

… ►

F. Feuillebois (1991) Numerical calculation of singular integrals related to Hankel transform. Comput. Math. Appl. 21 (2-3), pp. 87–94.

ⓘ

Notes:: Fortran, minimum precision 6D.
External Links:: ISSN 0898-1221, Document, MathReview, ZentralBlatt
Referenced by:: §10.77(ix).
Encodings:: BibTeX

… ►

P. Flajolet and A. Odlyzko (1990) Singularity analysis of generating functions. SIAM J. Discrete Math. 3 (2), pp. 216–240.

ⓘ

External Links:: ISSN 0895-4801, Document, MathReview (E. Rodney Canfield), ZentralBlatt
Referenced by:: §2.10(iv).
Encodings:: BibTeX

… ►

G. Freud (1969) On weighted polynomial approximation on the whole real axis. Acta Math. Acad. Sci. Hungar. 20, pp. 223–225.

ⓘ

External Links:: ISSN 0001-5954, Document, Link, MathReview (A. K. Varma)
Referenced by:: §18.32.
Encodings:: BibTeX

…

20: 31.12 Confluent Forms of Heun’s Equation

… ►Confluent forms of Heun’s differential equation (31.2.1) arise when two or more of the regular singularities merge to form an irregular singularity. … ►This has regular singularities at

z=0

and

1

, and an irregular singularity of rank 1 at

z=\infty

. … ►This has irregular singularities at

z=0

and

\infty

, each of rank

1

. … ►This has a regular singularity at

z=0

, and an irregular singularity at

\infty

of rank

2

. … ►This has one singularity, an irregular singularity of rank

3

z=\infty

. …