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1: 31.12 Confluent Forms of Heun’s Equation

… ►This has regular singularities at $z=0$ and $1$, and an irregular singularity of rank 1 at $z=\mathrm{\infty }$. … ►This has irregular singularities at $z=0$ and $\mathrm{\infty }$, each of rank $1$. … ►This has a regular singularity at $z=0$, and an irregular singularity at $\mathrm{\infty }$ of rank $2$. … ►This has one singularity, an irregular singularity of rank $3$ at $z=\mathrm{\infty }$. …

2: 22.9 Cyclic Identities

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§22.9(ii) Typical Identities of Rank 2

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§22.9(iii) Typical Identities of Rank 3

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§22.9(iv) Typical Identities of Rank 4

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§22.9(v) Identities of Higher Rank

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3: Charles W. Clark

… ►Clark received the R&D 100 Award, Distinguished Presidential Rank Award of the U. …

4: Bibliography O

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Exponentially improved asymptotic solutions of ordinary differential equations. II Irregular singularities of rank one. Proc. Roy. Soc. London Ser. A 445, pp. 39–56.

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External Links:

ISSN 0962-8444, FullText, MathReview (Hussain S. Nur), ZentralBlatt

Referenced by:

§13.4(i), §2.11(v), §2.7(ii).

Encodings:

BibTeX

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

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External Links:

ISSN 1364-5021, Document, MathReview (Donald A. Lutz), ZentralBlatt

Referenced by:

§2.11(v), §2.7(ii).

Encodings:

BibTeX

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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.

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External Links:

Document, MathReview, ZentralBlatt

Referenced by:

§2.7(ii).

Encodings:

BibTeX

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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.

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External Links:

Document, MathReview (T. E. Hull), ZentralBlatt

Referenced by:

§13.19, §13.7(ii).

Encodings:

BibTeX

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

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External Links:

ISSN 1073-2772, MathReview (A. D. Wood), ZentralBlatt

Referenced by:

§2.7(ii).

Encodings:

BibTeX

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5: 15.17 Mathematical Applications

… ►First, as spherical functions on noncompact Riemannian symmetric spaces of rank one, but also as associated spherical functions, intertwining functions, matrix elements of SL$(2,ℝ)$, and spherical functions on certain nonsymmetric Gelfand pairs. …

6: 2.7 Differential Equations

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§2.7(ii) Irregular Singularities of Rank 1

… ►Thus a regular singularity has rank 0. The most common type of irregular singularity for special functions has rank 1 and is located at infinity. … ►For extensions to singularities of higher rank see Olver and Stenger (1965). … ►For irregular singularities of nonclassifiable rank, a powerful tool for finding the asymptotic behavior of solutions, complete with error bounds, is as follows: …

7: Publications

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Methods of Relevance Ranking and Hit-content Generation in Math Search, Proceedings of Mathematical Knowledge Management (MKM2007), RISC, Hagenberg, Austria, June 27–30, 2007.

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8: Philip J. Davis

… ►At CalTech, John Todd dedicated himself to the training of new researchers in numerical analysis, and Olga Taussky, who had been a full-time NBS consultant influential in establishing the field of matrix theory, became the first woman at CalTech to attain the academic rank of full professor. …

9: 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 $\gamma \ne 0$). …

10: 2.9 Difference Equations

… ►This situation is analogous to second-order homogeneous linear differential equations with an irregular singularity of rank 1 at infinity (§2.7(ii)). …