Lommel functions

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1: 11.9 Lommel Functions

§11.9 Lommel Functions

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Reflection Formulas

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§11.9(ii) Expansions in Series of Bessel Functions

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2: 11.1 Special Notation

… ►For the functions $J_{\nu }\left(z\right)$, $Y_{\nu }\left(z\right)$, $H_{\nu }^{(1)}\left(z\right)$, $H_{\nu }^{(2)}\left(z\right)$, $I_{\nu }\left(z\right)$, and $K_{\nu }\left(z\right)$ see §§10.2(ii), 10.25(ii). ►The functions treated in this chapter are the Struve functions ${𝐇}_{\nu }\left(z\right)$ and ${𝐊}_{\nu }\left(z\right)$, the modified Struve functions ${𝐋}_{\nu }\left(z\right)$ and ${𝐌}_{\nu }\left(z\right)$, the Lommel functions $s_{\mu ,\nu }\left(z\right)$ and $S_{\mu ,\nu }\left(z\right)$, the Anger function ${𝐉}_{\nu }\left(z\right)$, the Weber function ${𝐄}_{\nu }\left(z\right)$, and the associated Anger–Weber function ${𝐀}_{\nu }\left(z\right)$.

3: 11.13 Methods of Computation

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§11.13(i) Introduction

… ►The treatment of Lommel and Anger–Weber functions is similar. …

4: 11.10 Anger–Weber Functions

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§11.10(vi) Relations to Other Functions

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11.10.17 $${𝐉}_{\nu }\left(z\right)=\frac{\sin \left(\pi \nu \right)}{\pi }(s_{0,\nu }\left(z\right)-\nu s_{-1,\nu }\left(z\right)),$$

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Symbols:

${𝐉}_{\nu }\left(z\right)$: Anger function, $s_{\mu ,\nu }\left(z\right)$: Lommel function, $\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: sine function, $z$: complex variable and $\nu$: real or complex order

Referenced by:

§11.10(vi)

Permalink:

http://dlmf.nist.gov/11.10.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §11.10(vi), §11.10 and Ch.11

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11.10.18 $${𝐄}_{\nu }\left(z\right)=-\frac{1}{\pi }(1+\cos \left(\pi \nu \right))s_{0,\nu }\left(z\right)-\frac{\nu }{\pi }(1-\cos \left(\pi \nu \right))s_{-1,\nu }\left(z\right).$$

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Symbols:

$s_{\mu ,\nu }\left(z\right)$: Lommel function, ${𝐄}_{\nu }\left(z\right)$: Weber function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $z$: complex variable and $\nu$: real or complex order

Referenced by:

§11.10(vi)

Permalink:

http://dlmf.nist.gov/11.10.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §11.10(vi), §11.10 and Ch.11

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5: 11.16 Software

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§11.16(iv) Lommel Functions

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6: Bibliography N

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On the large argument asymptotics of the Lommel function via Stieltjes transforms. Asymptot. Anal. 91 (3-4), pp. 265–281.

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External Links:

ISSN 0921-7134, Document, MathReview Entry, ZentralBlatt

Referenced by:

§11.6(iii), §11.6(iii), §11.9(iii), §11.9(iii).

Encodings:

BibTeX

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Error bounds for the large-argument asymptotic expansions of the Lommel and allied functions. Stud. Appl. Math. 140 (4), pp. 508–541.

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External Links:

ISSN 0022-2526, Document, Link, MathReview (Pedro J. Pagola), ZentralBlatt

Referenced by:

§11.11(i), §11.11(i).

Encodings:

BibTeX

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7: Bibliography S

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The sign of Lommel’s function. Trans. Amer. Math. Soc. 163, pp. 123–129.

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External Links:

ISSN 0002-9947, FullText, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§11.9(iv).

Encodings:

BibTeX

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8: Bibliography B

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Tables of the Anger and Lommel-Weber Functions. Technical report Technical Report 53 and AFCRL 796, University Washington Press, Seattle.

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Notes:

Reviewed in Math. Comp., v. 17 (1963), pp.315–317.

Referenced by:

1st item.

Encodings:

BibTeX

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9: Software Index

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	Open Source													With Book						Commercial
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11.16(iv) $s_{\mu ,\nu }\left(z\right)$, $S_{\mu ,\nu }\left(z\right)$												a	✓				✓				✓		a
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►‘✓’ indicates that a software package implements the functions in a section; ‘a’ indicates available functionality through optional or add-on packages; an empty space indicates no known support. … ►In the list below we identify four main sources of software for computing special functions. … ►

Such software ranges from a collection of reusable software parts (e.g., a library) to fully functional interactive computing environments with an associated computing language. Such software is usually professionally developed, tested, and maintained to high standards. It is available for purchase, often with accompanying updates and consulting support.

… ►The following are web-based software repositories with significant holdings in the area of special functions. …