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1: 28.12 Definitions and Basic Properties

… ►The introduction to the eigenvalues and the functions of general order proceeds as in §§28.2(i), 28.2(ii), and 28.2(iii), except that we now restrict $\widehat{\nu }\ne 0,1$; equivalently $\nu \ne n$. … ►

§28.12(ii) Eigenfunctions ${\mathrm{me}}_{\nu }(z,q)$

… ►For $q=0$, … ► … ►

2: 28.2 Definitions and Basic Properties

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§28.2(vi) Eigenfunctions

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3: Robb J. Muirhead

… … ► 1946 in Adelaide, South Australia) is Senior Director, Statistical Research and Consulting Center, Pfizer Global R&D, New London, Connecticut. …

4: 34.11 Higher-Order $3nj$ Symbols

§34.11 Higher-Order $3nj$ Symbols

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5: 10.24 Functions of Imaginary Order

§10.24 Functions of Imaginary Order

… ►and ${\tilde{J}}_{\nu }\left(x\right)$, ${\tilde{Y}}_{\nu }\left(x\right)$ are linearly independent solutions of (10.24.1): … ►In consequence of (10.24.6), when $x$ is large ${\tilde{J}}_{\nu }\left(x\right)$ and ${\tilde{Y}}_{\nu }\left(x\right)$ comprise a numerically satisfactory pair of solutions of (10.24.1); compare §2.7(iv). … … ►

6: Bibliography T

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Steepest descent paths for integrals defining the modified Bessel functions of imaginary order. Methods Appl. Anal. 1 (1), pp. 14–24.

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External Links:

ISSN 1073-2772, FullText, MathReview (Anatoly Kilbas), ZentralBlatt

Referenced by:

§10.74(viii).

Encodings:

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Erratum to “COULCC: A continued-fraction algorithm for Coulomb functions of complex order with complex arguments”. Comput. Phys. Comm. 159 (3), pp. 241–242.

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External Links:

Document, Code, ZentralBlatt

Referenced by:

§33.23(vi), I. J. Thompson and A. R. Barnett (1985).

Encodings:

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Eigenfunction Expansions Associated with Second-Order Differential Equations. Clarendon Press, Oxford.

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External Links:

MathReview (H. Pollard)

Referenced by:

§1.18(x).

Encodings:

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Eigenfunction Expansions Associated with Second Order Differential Equations, Part 2, Partial Differential Equations. Clarendon Press, Oxford.

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External Links:

ISBN 0198533187, MathReview Entry

Referenced by:

§1.18(x).

Encodings:

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Introduction to the Theory of Fourier Integrals. Third edition, Chelsea Publishing Co., New York.

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Notes:

Original edition was published in 1937 by Oxford University Press. Expanded bibliography and some improvements were made for the second and third editions.

External Links:

ISBN 0-8284-0324-4, MathReview, ZentralBlatt

Referenced by:

§1.14(i), §1.14(ii), §1.14(iv), §1.14(v), §1.14(vii), §1.8(iv), §10.22(v), §10.32(iii), §5.13.

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7: 10.45 Functions of Imaginary Order

§10.45 Functions of Imaginary Order

… ►and ${\tilde{I}}_{\nu }\left(x\right)$, ${\tilde{K}}_{\nu }\left(x\right)$ are real and linearly independent solutions of (10.45.1): … ►The corresponding result for ${\tilde{K}}_{\nu }\left(x\right)$ is given by … ► … ►

8: Bibliography D

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History of the Theory of Numbers (3 volumes). Carnegie Institution of Washington, Washington, D.C..

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Notes:

Reprinted by Chelsea Publishing Co., New York, 1966.

External Links:

MathReview, ZentralBlatt

Referenced by:

§27.21.

Encodings:

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A simplified algorithm for the second-order sound fields. J. Acoust. Soc. Amer. 108 (6), pp. 2759–2764.

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External Links:

Document

Referenced by:

§8.24(iii).

Encodings:

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Über die Bestimmung der mittleren Werthe in der Zahlentheorie. Abhandlungen der Königlich Preussischen Akademie der Wissenschaften von 1849, pp. 69–83 (German).

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Notes:

Reprinted in G. Lejeune Dirichlet’s Werke, Band II, pp. 49–66, Reimer, Berlin, 1897 and with corrections in G. Lejeune Dirichlet’s Werke, AMS Chelsea Publishing Series, Providence, RI, 1969.

External Links:

ISBN 0-8284-0225-6, Link, MathReview

Referenced by:

§27.11.

Encodings:

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Uniform asymptotic expansions for associated Legendre functions of large order. Proc. Roy. Soc. Edinburgh Sect. A 133 (4), pp. 807–827.

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External Links:

ISSN 0308-2105, Document, MathReview, ZentralBlatt

Referenced by:

§14.15(i), §14.15(i), §14.15(ii), §14.15(ii), §14.26.

Encodings:

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Conical functions of purely imaginary order and argument. Proc. Roy. Soc. Edinburgh Sect. A 143 (5), pp. 929–955.

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External Links:

ISSN 0308-2105, Document, Link, MathReview (Eugenia N. Petropoulou), ZentralBlatt

Referenced by:

§14.20(ix), §14.20(ix).

Encodings:

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9: Bibliography N

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The resurgence properties of the large order asymptotics of the Anger-Weber function I. J. Class. Anal. 4 (1), pp. 1–39.

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External Links:

ISSN 1848-5979, Document, Link, MathReview (José Luis López), ZentralBlatt

Referenced by:

(11.11.10), (11.11.8), §11.11(iii).

Encodings:

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The resurgence properties of the large order asymptotics of the Anger-Weber function II. J. Class. Anal. 4 (2), pp. 121–147.

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External Links:

ISSN 1848-5979, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

(11.11.10), (11.11.8), §11.11(iii).

Encodings:

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Uniform Asymptotic Approximations of Solutions of Second-order Linear Differential Equations, with a Coalescing Simple Turning Point and Simple Pole. Ph.D. Thesis, University of Maryland, College Park, MD.

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Referenced by:

§2.8(vi).

Encodings:

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Die Gammafunktion. Band I. Handbuch der Theorie der Gammafunktion. Band II. Theorie des Integrallogarithmus und verwandter Transzendenten. Chelsea Publishing Co., New York (German).

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Notes:

Both books are textually unaltered except for correction of errata.

External Links:

MathReview

Referenced by:

N. Nielsen (1906a), N. Nielsen (1906b).

Encodings:

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Vorlesungen über Differenzenrechnung. Springer-Verlag, Berlin (German).

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Notes:

Reprinted 1954 by Chelsea Publishing Company, New York

External Links:

ZentralBlatt

Referenced by:

§24.1, §24.1, §24.13(i), §24.13(ii), §24.16(i), §24.17(i), §24.17(i), §24.2(i), §24.2(ii), §24.4(i), §24.4(ii), §24.4(iii), §24.4(v), §24.5(i), Ch.24, (25.6.6).

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10: 10.26 Graphics

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§10.26(i) Real Order and Variable

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§10.26(ii) Real Order, Complex Variable

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§10.26(iii) Imaginary Order, Real Variable

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Figure 10.26.7: ${\tilde{I}}_{1/2}\left(x\right)$, ${\tilde{K}}_{1/2}\left(x\right)$, $0.01\le x\le 3$. Magnify

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Figure 10.26.8: ${\tilde{I}}_1\left(x\right)$, ${\tilde{K}}_1\left(x\right)$, $0.01\le x\le 3$. Magnify

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