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51: Bibliography M

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On the Representation of Meijer’s $G$-Function in the Vicinity of Singular Unity. In Complex Analysis and Applications ’81 (Varna, 1981), pp. 383–398.

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External Links:

MathReview (B. D. Agrawal), ZentralBlatt

Referenced by:

§16.21.

Encodings:

BibTeX

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On the convergence of the Chebyshev series for functions possessing a singularity in the range of representation. SIAM J. Numer. Anal. 3 (3), pp. 390–409.

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External Links:

ISSN 1095-7170, Document, MathReview (Y. L. Luke), ZentralBlatt

Referenced by:

§3.11(ii).

Encodings:

BibTeX

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Singular integrals whose kernels involve certain Sturm-Liouville functions. I. J. Math. Mech. 19 (10), pp. 855–873.

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External Links:

Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§9.10(ii).

Encodings:

BibTeX

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Hyperasymptotic solutions of second-order ordinary differential equations with a singularity of arbitrary integer rank. Methods Appl. Anal. 4 (3), pp. 250–260.

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External Links:

ISSN 1073-2772, MathReview (W. Balser), ZentralBlatt

Referenced by:

§2.11(v).

Encodings:

BibTeX

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52: 28.2 Definitions and Basic Properties

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28.2.2 $$\zeta (1-\zeta )w^{\prime \prime }+\frac{1}{2}(1-2\zeta )w^{\prime }+\frac{1}{4}(a-2q(1-2\zeta ))w=0.$$

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Symbols:

$q=h^2$: parameter, $a$: parameter, $w(z)$: Mathieu’s equation solution and $\zeta$: change of variable

Permalink:

http://dlmf.nist.gov/28.2.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §28.2(i), §28.2 and Ch.28

►This equation has regular singularities at 0 and 1, both with exponents 0 and $\frac{1}{2}$, and an irregular singular point at $\mathrm{\infty }$. … ►Since (28.2.1) has no finite singularities its solutions are entire functions of $z$. …

53: 14.2 Differential Equations

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§14.2(iii) Numerically Satisfactory Solutions

►Equation (14.2.2) has regular singularities at $x=1$, $-1$, and $\mathrm{\infty }$, with exponent pairs $\{-\frac{1}{2}\mu ,\frac{1}{2}\mu \}$, $\{-\frac{1}{2}\mu ,\frac{1}{2}\mu \}$, and $\{\nu +1,-\nu \}$, respectively; compare §2.7(i). …

54: 4.40 Integrals

… ►The results in §§4.40(ii) and 4.40(iv) can be extended to the complex plane by using continuous branches and avoiding singularities. …

55: 13.2 Definitions and Basic Properties

… ►This equation has a regular singularity at the origin with indices $0$ and $1-b$, and an irregular singularity at infinity of rank one. …In effect, the regular singularities of the hypergeometric differential equation at $b$ and $\mathrm{\infty }$ coalesce into an irregular singularity at $\mathrm{\infty }$. …

56: 19.12 Asymptotic Approximations

… ►With $\psi \left(x\right)$ denoting the digamma function (§5.2(i)) in this subsection, the asymptotic behavior of $K\left(k\right)$ and $E\left(k\right)$ near the singularity at $k=1$ is given by the following convergent series: …

57: 31.7 Relations to Other Functions

… ►Similar specializations of formulas in §31.3(ii) yield solutions in the neighborhoods of the singularities $\zeta =K$, $K+\mathrm{i}{K}^{\prime }$, and $\mathrm{i}{K}^{\prime }$, where $K$ and $K^{\prime }$ are related to $k$ as in §19.2(ii).

58: 31.9 Orthogonality

… ►and the integration paths ${ℒ}_1$, ${ℒ}_2$ are Pochhammer double-loop contours encircling distinct pairs of singularities $\{0,1\}$, $\{0,a\}$, $\{1,a\}$. …

59: 31.16 Mathematical Applications

… ► thesis “Inversion problem for a second-order linear differential equation with four singular points”. …

60: Bibliography D

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Oscillatory integrals, Lagrange immersions and unfolding of singularities. Comm. Pure Appl. Math. 27, pp. 207–281.

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External Links:

MathReview (Alan Weinstein), ZentralBlatt

Referenced by:

§36.12(iii).

Encodings:

BibTeX

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Error bounds for exponentially improved asymptotic solutions of ordinary differential equations having irregular singularities of rank one. Methods Appl. Anal. 3 (1), pp. 109–134.

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External Links:

ISSN 1073-2772, Pdf, MathReview (Roderick Wong), ZentralBlatt

Referenced by:

§2.11(v).

Encodings:

BibTeX

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