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Lommel functions

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1: 11.9 Lommel Functions
§11.9 Lommel Functions
Reflection Formulas
§11.9(ii) Expansions in Series of Bessel Functions
2: 11.1 Special Notation
For the functions J ν ( z ) , Y ν ( z ) , H ν ( 1 ) ( z ) , H ν ( 2 ) ( z ) , I ν ( z ) , and K ν ( z ) see §§10.2(ii), 10.25(ii). The functions treated in this chapter are the Struve functions H ν ( z ) and K ν ( z ) , the modified Struve functions L ν ( z ) and M ν ( z ) , the Lommel functions s μ , ν ( z ) and S μ , ν ( z ) , the Anger function J ν ( z ) , the Weber function E ν ( z ) , and the associated Anger–Weber function A ν ( z ) .
3: 11.13 Methods of Computation
§11.13(i) Introduction
The treatment of Lommel and Anger–Weber functions is similar. …
4: 11.16 Software
§11.16(iv) Lommel Functions
5: 11.10 Anger–Weber Functions
§11.10(vi) Relations to Other Functions
11.10.17 J ν ( z ) = sin ( π ν ) π ( s 0 , ν ( z ) - ν s - 1 , ν ( z ) ) ,
11.10.18 E ν ( z ) = - 1 π ( 1 + cos ( π ν ) ) s 0 , ν ( z ) - ν π ( 1 - cos ( π ν ) ) s - 1 , ν ( z ) .
6: Bibliography N
  • G. Nemes (2015b) On the large argument asymptotics of the Lommel function via Stieltjes transforms. Asymptot. Anal. 91 (3-4), pp. 265–281.
  • G. Nemes (2018) Error bounds for the large-argument asymptotic expansions of the Lommel and allied functions. Stud. Appl. Math. 140 (4), pp. 508–541.
  • 7: Bibliography S
  • J. Steinig (1972) The sign of Lommel’s function. Trans. Amer. Math. Soc. 163, pp. 123–129.
  • 8: Bibliography B
  • G. D. Bernard and A. Ishimaru (1962) Tables of the Anger and Lommel-Weber Functions. Technical report Technical Report 53 and AFCRL 796, University Washington Press, Seattle.
  • 9: Software Index
    Open Source With Book Commercial
    11.16(iv) s μ , ν ( z ) , S μ , ν ( z ) a a
    ‘✓’ 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. …
  • Commercial Software.

    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. …