Hille–Hardy formula

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1: Bibliography H

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Asymptotic formulae in combinatory analysis. Proc. London Math. Soc. (2) 17, pp. 75–115.

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Notes:

Reprinted in Ramanujan (1962, pp. 276–309).

External Links:

Document, MathReview Entry, ZentralBlatt

Referenced by:

§27.14(iii).

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Über die Integration logarithmisch-rationaler Differentiale. J. Reine Angew. Math. 3, pp. 101–159.

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External Links:

ZentralBlatt

Referenced by:

§25.12(i).

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Algorithm 442: Normal deviate. Comm. ACM 16 (1), pp. 51–52.

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Notes:

Includes Algol program, minimum accuracy 24D.

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ISSN 0001-0782, Document

Referenced by:

§7.25(ii).

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Algorithm AS66: The normal integral. Appl. Statist. 22 (3), pp. 424–427.

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Notes:

Double-precision Fortran, maximum accuracy 12D.

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FullText, Code

Referenced by:

5th item.

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Note on some hypergeometric series of higher order. J. London Math. Soc. 4, pp. 50–54.

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External Links:

Document, ZentralBlatt

Referenced by:

§16.9.

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2: Bibliography Z

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Integrali di funzioni di Anger, Weber ed Airy-Hardy. Rend. Sem. Mat. Univ. Padova 58, pp. 275–285 (Italian).

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External Links:

FullText, MathReview (M. E. Muldoon), ZentralBlatt

Referenced by:

§11.10(x).

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Distribution Theory and Transform Analysis, An Introduction and Generalized Functions with Applications. Dover, New York.

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Notes:

Reprint of 1965 McGraw-Hill Publication

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ISBN 0-486-65479-6, MathReview Entry

Referenced by:

§1.16(ix).

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Tables and formulae for the spherical functions $P_{-1/2+i\tau }^m(z)$ . Translated by E. L. Albasiny, Pergamon Press, Oxford.

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Notes:

Translation of Žurina and Karmazina, Tablitsy i formuly dlya sfericheskikh funktsii $P_{-1/2+i\tau }^m(z)$ , Vyčisl. Centr Akad Nauk SSSR, Moscow. 1962

External Links:

MathReview, ZentralBlatt

Referenced by:

§14.20(vii).

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3: Bibliography R

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Collected Papers of Srinivasa Ramanujan. Chelsea Publishing Co., New York.

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Notes:

Reprinted with commentary by Bruce Berndt, AMS Chelsea Publishing Series, American Mathematical Society (2000).

External Links:

ISBN 0-8218-2076-1

Referenced by:

§8.11(v), G. H. Hardy and S. Ramanujan (1918).

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Diffraction of plane radio waves by a parabolic cylinder. Calculation of shadows behind hills. Bell System Tech. J. 33, pp. 417–504.

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External Links:

MathReview (J. Shmoys)

Referenced by:

§12.17.

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Real and complex analysis. McGraw-Hill Book Co., New York-Toronto, Ont.-London.

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External Links:

MathReview (R. E. Edwards)

Referenced by:

§1.2(v), §1.4(v), §1.5(v).

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Functional Analysis. McGraw-Hill Book Co., New York.

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Notes:

McGraw-Hill Series in Higher Mathematics

External Links:

MathReview (F. Smithies), ZentralBlatt

Referenced by:

§1.18(x), §2.6(i).

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Principles of Mathematical Analysis. 3rd edition, McGraw-Hill Book Co., New York.

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Notes:

International Series in Pure and Applied Mathematics

External Links:

MathReview, ZentralBlatt

Referenced by:

§1.4(iii).

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4: 28 Mathieu Functions and Hill’s Equation

Chapter 28 Mathieu Functions and Hill’s Equation

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5: 18.18 Sums

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§18.18(iv) Connection and Inversion Formulas

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§18.18(v) Linearization Formulas

… ►Formula (18.18.27) is known as the Hille–Hardy formula. … ►Formula (18.18.28) is known as the Mehler formula. …

6: 27.20 Methods of Computation: Other Number-Theoretic Functions

… ►The recursion formulas (27.14.6) and (27.14.7) can be used to calculate the partition function $p\left(n\right)$ for . …To compute a particular value $p\left(n\right)$ it is better to use the Hardy–Ramanujan–Rademacher series (27.14.9). … ►A recursion formula obtained by differentiating (27.14.18) can be used to calculate Ramanujan’s function $\tau \left(n\right)$, and the values can be checked by the congruence (27.14.20). …

7: Bibliography J

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Tafeln höherer Funktionen (Tables of Higher Functions). 7th edition, B. G. Teubner, Stuttgart (Bilingual).

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Notes:

Reprint of sixth edition (McGraw-Hill, New York, 1960) with corrections and updated bibliography. The sixth edition is a completely revised and enlarged version of Jahnke and Emde (1945). In German and English.

External Links:

MathReview

Referenced by:

§19.1, §28.3(i).

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Tables of Functions with Formulae and Curves. 4th edition, Dover Publications, New York.

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Notes:

Contains corrections and enlargements of earlier editions. Originally published by B. G. Teubner, Leipzig, in 1933. In German and English.

External Links:

MathReview, ZentralBlatt

Referenced by:

2nd item, §20.15, §5.1, Notations P, E. Jahnke, F. Emde, and F. Lösch (1966).

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Asymptotic formulas for the zeros of the Meixner polynomials. J. Approx. Theory 96 (2), pp. 281–300.

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ISSN 0021-9045, Document, MathReview (N. Hayek Calil), ZentralBlatt

Referenced by:

§18.24.

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Efficient implementation of the Hardy-Ramanujan-Rademacher formula. LMS J. Comput. Math. 15, pp. 341–359.

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ISSN 1461-1570, Document, Link, MathReview (Tim Huber), ZentralBlatt

Referenced by:

§27.20.

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8: Bibliography K

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On an integral of Hardy and Littlewood. Kyushu J. Math. 52 (1), pp. 249–263.

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ISSN 1340-6116, Document, MathReview (Fritz Haslinger), ZentralBlatt

Referenced by:

§9.13(ii).

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Connection formulae for the first Painlevé transcendent in the complex domain. Lett. Math. Phys. 27 (4), pp. 243–252.

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ISSN 0377-9017, Document, MathReview (B. L. J. Braaksma), ZentralBlatt

Referenced by:

§32.11(i), §32.12(i).

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On formulas involving both the Bernoulli and Fibonacci numbers. Scripta Math. 23, pp. 27–35.

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External Links:

MathReview (L. Carlitz), ZentralBlatt

Referenced by:

§24.15(iv).

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Electromagnetic Fields and Relativistic Particles. International Series in Pure and Applied Physics, McGraw-Hill Book Co., New York.

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ISBN 0-07-035264-X, MathReview (George Zipfel, Jr.)

Referenced by:

§1.17(ii), §10.73(ii).

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The addition formula for Laguerre polynomials. SIAM J. Math. Anal. 8 (3), pp. 535–540.

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External Links:

Document, MathReview (M. Mikolas), ZentralBlatt

Referenced by:

(18.14.8), §18.14(i), §18.17(ii), §18.18(ii).

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9: 28.29 Definitions and Basic Properties

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§28.29(i) Hill’s Equation

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§28.29(ii) Floquet’s Theorem and the Characteristic Exponent

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10: Simon Ruijsenaars

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28 Mathieu Functions and Hill’s Equation