regular%20singularity

SearchAdvancedHelp

(0.001 seconds)

11—20 of 197 matching pages

11: 30.2 Differential Equations

… ►This equation has regular singularities at $z=\pm 1$ with exponents $\pm \frac{1}{2}\mu$ and an irregular singularity of rank 1 at $z=\mathrm{\infty }$ (if $\gamma \ne 0$). … …

12: 33.8 Continued Fractions

… ►

33.8.1 $$\frac{F_{\mathrm{\ell }}^{\prime }}{F_{\mathrm{\ell }}}=S_{\mathrm{\ell }+1}-\frac{R_{\mathrm{\ell }+1}^2}{T_{\mathrm{\ell }+1}-}\frac{R_{\mathrm{\ell }+2}^2}{T_{\mathrm{\ell }+2}-}\mathrm{\cdots }.$$

ⓘ

Symbols:

$F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\mathrm{\ell }$: nonnegative integer, $R_{\mathrm{\ell }}$: factor, $S_{\mathrm{\ell }}$: factor and $T_{\mathrm{\ell }}$: factor

Referenced by:

§33.8, §33.8

Permalink:

http://dlmf.nist.gov/33.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §33.8 and Ch.33

… ►If we denote $u=F_{\mathrm{\ell }}^{\prime }/F_{\mathrm{\ell }}$ and $p+\mathrm{i}q=H_{\mathrm{\ell }}^{+}{}_{}{}^{\prime }/H_{\mathrm{\ell }}^{+}$, then ►

$F_{\mathrm{\ell }}=\pm {(q^{-1}{(u-p)}^2+q)}^{-1/2},$

►

$F_{\mathrm{\ell }}^{\prime }=u{F}_{\mathrm{\ell }},$

►

$G_{\mathrm{\ell }}=q^{-1}(u-p)F_{\mathrm{\ell }},$

…

13: 2.7 Differential Equations

… ►

§2.7(i) Regular Singularities: Fuchs–Frobenius Theory

… ►All solutions are analytic at an ordinary point, and their Taylor-series expansions are found by equating coefficients. … ►In a punctured neighborhood $𝐍$ of a regular singularity $z_0$ … ►Thus a regular singularity has rank 0. … ►The transformed differential equation either has a regular singularity at $t=\mathrm{\infty }$, or its characteristic equation has unequal roots. …

14: 15.11 Riemann’s Differential Equation

… ►

§15.11(i) Equations with Three Singularities

►The importance of (15.10.1) is that any homogeneous linear differential equation of the second order with at most three distinct singularities, all regular, in the extended plane can be transformed into (15.10.1). … ►Cases in which there are fewer than three singularities are included automatically by allowing the choice $\{0,1\}$ for exponent pairs. … ►The reduction of a general homogeneous linear differential equation of the second order with at most three regular singularities to the hypergeometric differential equation is given by …

15: Bibliography S

… ►

A new Fortran 90 program to compute regular and irregular associated Legendre functions. Comput. Phys. Comm. 181 (12), pp. 2091–2097.

ⓘ

External Links:

ISSN 0010-4655, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

8th item, §14.34(ii).

Encodings:

BibTeX

… ►

Euler-Maclaurin expansions for integrals with endpoint singularities: A new perspective. Numer. Math. 98 (2), pp. 371–387.

ⓘ

External Links:

ISSN 0029-599X, Document, MathReview (Iulian Coroian), ZentralBlatt

Referenced by:

§2.10(i).

Encodings:

BibTeX

►

A simple approach to asymptotic expansions for Fourier integrals of singular functions. Appl. Math. Comput. 216 (11), pp. 3378–3385.

ⓘ

External Links:

ISSN 0096-3003, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§2.3(iv).

Encodings:

BibTeX

… ►

Euler-Maclaurin expansions for integrals with arbitrary algebraic endpoint singularities. Math. Comp. 81 (280), pp. 2159–2173.

ⓘ

External Links:

ISSN 0025-5718, Document, Link, MathReview (Vittoria Demichelis), ZentralBlatt

Referenced by:

§2.10(i).

Encodings:

BibTeX

… ►

Operators with Singular Continuous Spectrum: I. General Operators. Annals of Mathematics 141 (1), pp. 131–145.

ⓘ

External Links:

MathReview Entry

Referenced by:

§1.18(vii).

Encodings:

BibTeX

…

16: Bibliography L

… ►

Algorithm 917: complex double-precision evaluation of the Wright $\omega$ function. ACM Trans. Math. Software 38 (3), pp. Art. 20, 17.

ⓘ

External Links:

ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§4.13, 2nd item, §4.48(iv).

Encodings:

BibTeX

… ►

Heun’s equation with nearby singularities. Proc. Roy. Soc. London Ser. A 455, pp. 4347–4361.

ⓘ

External Links:

ISSN 1364-5021, Document, MathReview, ZentralBlatt

Referenced by:

§31.13, §31.17(ii).

Encodings:

BibTeX

… ►

An asymptotic estimate for the Bernoulli and Euler numbers. Canad. Math. Bull. 20 (1), pp. 109–111.

ⓘ

External Links:

MathReview (D. H. Lehmer), ZentralBlatt

Referenced by:

§24.11.

Encodings:

BibTeX

… ►

Evaluation of Bessel function integrals with algebraic singularities. J. Comput. Appl. Math. 37 (1-3), pp. 101–112.

ⓘ

External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

… ►

Generalized Riemann $\zeta$-function regularization and Casimir energy for a piecewise uniform string. Phys. Rev. D 44 (2), pp. 560–562.

ⓘ

External Links:

Document

Referenced by:

§24.18.

Encodings:

BibTeX

…

17: 25.17 Physical Applications

… ►Quantum field theory often encounters formally divergent sums that need to be evaluated by a process of regularization: for example, the energy of the electromagnetic vacuum in a confined space (Casimir–Polder effect). It has been found possible to perform such regularizations by equating the divergent sums to zeta functions and associated functions (Elizalde (1995)).

18: 33.23 Methods of Computation

… ►The power-series expansions of §§33.6 and 33.19 converge for all finite values of the radii $\rho$ and $r$, respectively, and may be used to compute the regular and irregular solutions. … ►Thus the regular solutions can be computed from the power-series expansions (§§33.6, 33.19) for small values of the radii and then integrated in the direction of increasing values of the radii. … ►This implies decreasing $\mathrm{\ell }$ for the regular solutions and increasing $\mathrm{\ell }$ for the irregular solutions of §§33.2(iii) and 33.14(iii). … ►§33.8 supplies continued fractions for $F_{\mathrm{\ell }}^{\prime }/F_{\mathrm{\ell }}$ and $H_{\mathrm{\ell }}^{\pm }{}_{}{}^{\prime }/H_{\mathrm{\ell }}^{\pm }$. Combined with the Wronskians (33.2.12), the values of $F_{\mathrm{\ell }}$, $G_{\mathrm{\ell }}$, and their derivatives can be extracted. …

19: 15.10 Hypergeometric Differential Equation

… ►It has regular singularities at $z=0,1,\mathrm{\infty }$, with corresponding exponent pairs $\{0,1-c\}$, $\{0,c-a-b\}$, $\{a,b\}$, respectively. …They are also numerically satisfactory (§2.7(iv)) in the neighborhood of the corresponding singularity. ►

Singularity $z=0$

… ►

Singularity $z=1$

… ►

Singularity $z=\mathrm{\infty }$

…

20: 33.5 Limiting Forms for Small $\rho$, Small $|\eta |$, or Large $\mathrm{\ell }$

… ►

$F_{\mathrm{\ell }}(\eta ,\rho )\sim C_{\mathrm{\ell }}\left(\eta \right){\rho }^{\mathrm{\ell }+1},$

►

$F_{\mathrm{\ell }}^{\prime }(\eta ,\rho )\sim (\mathrm{\ell }+1)C_{\mathrm{\ell }}\left(\eta \right){\rho }^{\mathrm{\ell }}.$

… ►

$F_{\mathrm{\ell }}(0,\rho )=\rho {𝗃}_{\mathrm{\ell }}\left(\rho \right),$

… ►

$F_0(0,\rho )=\sin \rho ,$

… ►

$F_{\mathrm{\ell }}(\eta ,\rho )\sim C_{\mathrm{\ell }}\left(\eta \right){\rho }^{\mathrm{\ell }+1},$

…