Bickley function

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1: 10.76 Approximations

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Bickley Functions

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2: 10.73 Physical Applications

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§10.73(iv) Bickley Functions

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3: 10.43 Integrals

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§10.43(iii) Fractional Integrals

►The Bickley function ${\mathrm{Ki}}_{\alpha }\left(x\right)$ is defined by … ►

10.43.14 $${\mathrm{Ki}}_0\left(x\right)=K_0\left(x\right),$$

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Symbols:

${\mathrm{Ki}}_{\alpha }\left(x\right)$: Bickley function, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind and $x$: real variable

A&S Ref:

11.2.8

Permalink:

http://dlmf.nist.gov/10.43.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §10.43(iii), §10.43(iii), §10.43 and Ch.10

… ►For further properties of the Bickley function, including asymptotic expansions and generalizations, see Amos (1983c, 1989) and Luke (1962, Chapter 8). ►

§10.43(iv) Integrals over the Interval ($0,\mathrm{\infty }$)

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4: Bibliography

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Integrals involving Bickley and Bessel functions in radiative transfer, and generalized exponential integral functions. J. Heat Transfer 118 (3), pp. 789–792.

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External Links:

Document

Referenced by:

§10.73(iv), §8.24(iii).

Encodings:

BibTeX

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Algorithm 609. A portable FORTRAN subroutine for the Bickley functions ${\mathrm{Ki}}_n(x)$ . ACM Trans. Math. Software 9 (4), pp. 480–493.

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Notes:

Single-precision Fortran, maximum accuracy 18S.

External Links:

ISSN 0098-3500, Document, GAMS (module 609; package TOMS), MathReview (Marietta J. Tretter), ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

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Uniform asymptotic expansions for exponential integrals $E_n(x)$ and Bickley functions ${\mathrm{Ki}}_n(x)$ . ACM Trans. Math. Software 9 (4), pp. 467–479.

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External Links:

ISSN 0098-3500, Document, MathReview (Marietta J. Tretter), ZentralBlatt

Referenced by:

§10.43(iii).

Encodings:

BibTeX

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5: Bibliography B

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Bessel Functions. Part II: Functions of Positive Integer Order. British Association for the Advancement of Science, Mathematical Tables, Volume 10, Cambridge University Press, Cambridge.

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Notes:

Reprinted, with corrections, in 1960.

External Links:

ISBN 0-521-04321-2, MathReview (R. C. Archibald), ZentralBlatt

Referenced by:

§10.18(iii), §10.19(ii), §10.21(vi), §10.41(ii), 2nd item, 2nd item, §3.6(iii).

Encodings:

BibTeX

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A short table of the functions ${\mathrm{Ki}}_n(x)$, from $n=1$ to $n=16$ . Phil. Mag. Series 7 20, pp. 343–347.

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External Links:

Document, ZentralBlatt

Referenced by:

§10.43(iii), 2nd item.

Encodings:

BibTeX

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Rational Chebyshev approximations for the Bickley functions $K{i}_n(x)$ . Math. Comp. 32 (143), pp. 876–886.

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External Links:

ISSN 0025-5718, FullText, MathReview (H. Hochstadt), ZentralBlatt

Referenced by:

§10.76(iii).

Encodings:

BibTeX

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6: 10.18 Modulus and Phase Functions

§10.18 Modulus and Phase Functions

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§10.18(i) Definitions

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§10.18(ii) Basic Properties

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§10.18(iii) Asymptotic Expansions for Large Argument

… ►In (10.18.17) and (10.18.18) the remainder after $n$ terms does not exceed the $(n+1)$th term in absolute value and is of the same sign, provided that $n>\nu -\frac{1}{2}$ for (10.18.17) and $-\frac{3}{2}\le \nu \le \frac{3}{2}$ for (10.18.18).

7: 10.75 Tables

§10.75 Tables

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Bickley et al. (1952) tabulates $J_n\left(x\right)$, $Y_n\left(x\right)$ or $x^n{Y}_n\left(x\right)$, $n=2(1)20$, $x=0$($.01$ or $.1$) $10(.1)25$, 8D (for $J_n\left(x\right)$), 8S (for $Y_n\left(x\right)$ or $x^n{Y}_n\left(x\right)$); $J_n\left(x\right)$, $Y_n\left(x\right)$, $n=0(1)20$, $x=0$ or $0.1(.1)25$, 10D (for $J_n\left(x\right)$), 10S (for $Y_n\left(x\right)$).

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§10.75(iv) Integrals of Bessel Functions

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Bickley et al. (1952) tabulates $x^{-n}{I}_n\left(x\right)$ or ${\mathrm{e}}^{-x}{I}_n\left(x\right)$, $x^n{K}_n\left(x\right)$ or ${\mathrm{e}}^x{K}_n\left(x\right)$, $n=2(1)20$, $x=0$(.01 or .1) 10(.1) 20, 8S; $I_n\left(x\right)$, $K_n\left(x\right)$, $n=0(1)20$, $x=0$ or $0.1(.1)20$, 10S.

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Bickley and Nayler (1935) tabulates ${\mathrm{Ki}}_n\left(x\right)$ (§10.43(iii)) for $n=1(1)16$, $x=0(.05)0.2(.1)$ 2, 3, 9D.

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8: 10.19 Asymptotic Expansions for Large Order

§10.19 Asymptotic Expansions for Large Order

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§10.19(i) Asymptotic Forms

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§10.19(ii) Debye’s Expansions

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§10.19(iii) Transition Region

… ►See also §10.20(i).

9: 10.21 Zeros

… ► … ►For the next three terms in (10.21.19) and the next two terms in (10.21.20) see Bickley et al. (1952, p. xxxvii) or Olver (1960, pp. xvii–xviii). … ► … ►The zeros of the functions … ►

§10.21(xiii) Rayleigh Function

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10: 10.41 Asymptotic Expansions for Large Order

… ►For $U_4(p)$, $U_5(p)$, $U_6(p)$, see Bickley et al. (1952, p. xxxv). … ► … ►

§10.41(iv) Double Asymptotic Properties

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§10.41(v) Double Asymptotic Properties (Continued)

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