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1: 10.43 Integrals

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

10.43.12 $${\mathrm{Ki}}_{\alpha }\left(x\right)={\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-x\cosh t}}{{(\cosh t)}^{\alpha }}dt,$$ $x>0$.

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Symbols:

${\mathrm{Ki}}_{\alpha }\left(x\right)$: Bickley function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\int$: integral and $x$: real variable

A&S Ref:

11.2.10 (with other conditions)

Referenced by:

§10.43(iii)

Permalink:

http://dlmf.nist.gov/10.43.E12

Encodings:

pMML, png, TeX

See also:

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

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10.43.13 $${\mathrm{Ki}}_{\alpha }\left(x\right)={\int }_x^{\mathrm{\infty }}{\mathrm{Ki}}_{\alpha -1}\left(t\right)dt,$$

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Symbols:

${\mathrm{Ki}}_{\alpha }\left(x\right)$: Bickley function, $dx$: differential of $x$ , $\int$: integral and $x$: real variable

A&S Ref:

11.2.8

Referenced by:

§10.43(iii)

Permalink:

http://dlmf.nist.gov/10.43.E13

Encodings:

pMML, png, TeX

See also:

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

►

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

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10.43.17 $$\alpha {\mathrm{Ki}}_{\alpha +1}\left(x\right)+x{\mathrm{Ki}}_{\alpha }\left(x\right)+(1-\alpha ){\mathrm{Ki}}_{\alpha -1}\left(x\right)-x{\mathrm{Ki}}_{\alpha -2}\left(x\right)=0.$$

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Symbols:

${\mathrm{Ki}}_{\alpha }\left(x\right)$: Bickley function and $x$: real variable

A&S Ref:

11.2.14

Referenced by:

§10.43(iii)

Permalink:

http://dlmf.nist.gov/10.43.E17

Encodings:

pMML, png, TeX

See also:

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

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2: Software Index

… ►

These are collections of software (e.g. libraries) or interactive systems of a somewhat broad scope. Contents may be adapted from research software or may be contributed by project participants who donate their services to the project. The software is made freely available to the public, typically in source code form. While formal support of the collection may not be provided by its developers, within active projects there is often a core group who donate time to consider bug reports and make updates to the collection.

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3: Bibliography

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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

… ►

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

…

4: Bibliography B

… ►

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

…

5: 10.75 Tables

… ►

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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