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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: 34.11 Higher-Order $3nj$ Symbols

§34.11 Higher-Order $3nj$ Symbols

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4: 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). … … ►

5: Nico M. Temme

… ► 1940 in Grootebroek, The Netherlands) is Emeritus Researcher in the Centrum Wiskunde & Informatica (CWI), Amsterdam. … ►On the occasion of his retirement in 2005 he was awarded the decoration Knight in the Order of the Dutch Lion, issued by the King of the Netherlands. …

6: 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 … ► … ►

7: 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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8: 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:

BibTeX

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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:

BibTeX

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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:

BibTeX

►

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:

BibTeX

►

Eigenfunction expansions associated with second-order differential equations. Part I. Second edition, Clarendon Press, Oxford.

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External Links:

MathReview Entry

Referenced by:

(1.18.59), §1.18(ix), §1.18(v), §1.18(vi), §1.18(x), (10.22.78), §10.22(v), §18.39(ii).

Encodings:

BibTeX

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9: 10.76 Approximations

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Real Variable and Order $:$ Functions

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Real Variable and Order $:$ Zeros

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Real Variable and Order $:$ Integrals

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Complex Variable; Real Order

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Real Variable; Imaginary Order

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10: 10.77 Software

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§10.77(ii) Bessel Functions–Real Argument and Integer or Half-Integer Order (including Spherical Bessel Functions)

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§10.77(iii) Bessel Functions–Real Order and Argument

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§10.77(vi) Bessel Functions–Imaginary Order and Real Argument

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§10.77(vii) Bessel Functions–Complex Order and Real Argument

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§10.77(viii) Bessel Functions–Complex Order and Argument

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