Struve functions and modified Struve functions

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11: 11.6 Asymptotic Expansions

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§11.6(i) Large $|z|$, Fixed $\nu$

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§11.6(ii) Large $|\nu |$, Fixed $z$

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11.6.5 $${𝐇}_{\nu }\left(z\right),{𝐋}_{\nu }\left(z\right)\sim \frac{z}{\pi \nu \sqrt{2}}{\left(\frac{\mathrm{e}z}{2\nu }\right)}^{\nu },$$ $|\mathrm{ph}\nu |\le \pi -\delta$.

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Symbols:

${𝐇}_{\nu }\left(z\right)$: Struve function, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, ${𝐋}_{\nu }\left(z\right)$: modified Struve function, $\mathrm{ph}$: phase, $z$: complex variable, $\nu$: real or complex order and $\delta$: arbitrary small positive constant

Referenced by:

(11.11.7)

Permalink:

http://dlmf.nist.gov/11.6.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §11.6(ii), §11.6 and Ch.11

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11.6.9 $${𝐋}_{\nu }\left(\lambda \nu \right)\sim I_{\nu }\left(\lambda \nu \right),$$ $|\mathrm{ph}\nu |\le \frac{1}{2}\pi -\delta$,

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Symbols:

$\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, ${𝐋}_{\nu }\left(z\right)$: modified Struve function, $\mathrm{ph}$: phase, $\nu$: real or complex order, $\delta$: arbitrary small positive constant and $\lambda$: parameter

Referenced by:

§11.6(iii)

Permalink:

http://dlmf.nist.gov/11.6.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §11.6(iii), §11.6 and Ch.11

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12: 11.13 Methods of Computation

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§11.13(i) Introduction

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13: 11.10 Anger–Weber Functions

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$m_2=\lceil \frac{1}{2}n-\frac{3}{2}\rceil .$

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14: 10.43 Integrals

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10.43.2 $$\int z^{\nu }{𝒵}_{\nu }\left(z\right)dz={\pi }^{\frac{1}{2}}{2}^{\nu -1}\mathrm{\Gamma }\left(\nu +\frac{1}{2}\right)z\left({𝒵}_{\nu }\left(z\right){𝐋}_{\nu -1}\left(z\right)-{𝒵}_{\nu -1}\left(z\right){𝐋}_{\nu }\left(z\right)\right),$$ $\nu \ne -\frac{1}{2}$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, ${𝐋}_{\nu }\left(z\right)$: modified Struve function, ${𝒵}_{\nu }\left(z\right)$: modified cylinder function, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

11.1.8 (Case $\nu =0$ only)

Referenced by:

§10.43(i)

Permalink:

http://dlmf.nist.gov/10.43.E2

Encodings:

pMML, png, TeX

See also:

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

►For the modified Struve function ${𝐋}_{\nu }\left(z\right)$ see §11.2(i). …

15: Bibliography B

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Integral representations and summations of the modified Struve function. Acta Math. Hungar. 141 (3), pp. 254–281.

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External Links:

ISSN 0236-5294, Document, FullText, MathReview (Eugenia N. Petropoulou), ZentralBlatt

Referenced by:

§11.7(v), §11.7(v).

Encodings:

BibTeX

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Tables of modified Struve functions of orders zero and unity.

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Notes:

Part of the Unpublished Mathematical Tables. Reviewed in Math. Comp., v. 18 (1964), no. 86, p. 332.

Referenced by:

3rd item.

Encodings:

BibTeX

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16: Bibliography Z

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A modified Bernoulli number. Nieuw Arch. Wisk. (4) 16 (1-2), pp. 63–72.

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External Links:

ISSN 0028-9825, MathReview (L. Skula), ZentralBlatt

Referenced by:

§24.16(iii).

Encodings:

BibTeX

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Sul calcolo numerico della funzione di Struve ${𝐇}_{\nu }(z)$ . Rend. Sem. Mat. Univ. e Politec. Torino 32, pp. 251–269 (Italian. English summary).

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External Links:

MathReview (Walter Gautschi), ZentralBlatt

Referenced by:

§11.13(i), 4th item.

Encodings:

BibTeX

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Su un integrale definito del prodotto di due funzioni di Struve. Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 112 (1-2), pp. 63–81 (Italian).

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External Links:

ISSN 0001-4419, MathReview, ZentralBlatt

Referenced by:

§11.7(ii).

Encodings:

BibTeX

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Numerical analysis of Struve functions with applications to other special functions. Ann. Numer. Math. 2 (1-4), pp. 199–208.

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External Links:

ISSN 1021-2655, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§11.7(ii).

Encodings:

BibTeX

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Tablitsy modifitsirovannykh funktsii Besselya s mnimym indeksom $K_{i\tau }(x)$ . Vyčisl. Centr Akad. Nauk SSSR, Moscow.

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Notes:

Translated title: Tables of the Modified Bessel Functions with Imaginary Index $K_{i\tau }(x)$

External Links:

MathReview (John Todd), ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

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17: 10.22 Integrals

… ►For the Struve function ${𝐇}_{\nu }\left(z\right)$ see §11.2(i). … ►For $I_{\nu }$ see §10.25(ii). … ►For $I$ and $K$ see §10.25(ii). … ►For $I$ and $K$ see §10.25(ii). … ►Additional infinite integrals over the product of three Bessel functions (including modified Bessel functions) are given in Gervois and Navelet (1984, 1985a, 1985b, 1986a, 1986b). …

18: Bibliography M

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Chebyshev expansions for modified Struve and related functions. Math. Comp. 60 (202), pp. 735–747.

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External Links:

ISSN 0025-5718, FullText, MathReview, ZentralBlatt

Referenced by:

2nd item.

Encodings:

BibTeX

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