Struve functions

SearchAdvancedHelp

(0.003 seconds)

11—20 of 25 matching pages

11: 11.6 Asymptotic Expansions

… ►

§11.6(i) Large $|z|$, Fixed $\nu$

… ►

§11.6(ii) Large $|\nu |$, Fixed $z$

►

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

ⓘ

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

… ► … ►

11.6.6 $${𝐊}_{\nu }\left(\lambda \nu \right)\sim \frac{{(\frac{1}{2}\lambda \nu )}^{\nu -1}}{\sqrt{\pi }\mathrm{\Gamma }\left(\nu +\frac{1}{2}\right)}\sum _{k=0}^{\mathrm{\infty }}\frac{k!c_k(\lambda )}{{\nu }^k},$$ $|\mathrm{ph}\nu |\le \frac{1}{2}\pi -\delta$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝐊}_{\nu }\left(z\right)$: Struve function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\mathrm{ph}$: phase, $\nu$: real or complex order, $k$: nonnegative integer, $\delta$: arbitrary small positive constant, $\lambda$: parameter and $c_k(\lambda )$: expansion function

A&S Ref:

12.1.34

Referenced by:

§11.6(iii)

Permalink:

http://dlmf.nist.gov/11.6.E6

Encodings:

pMML, png, TeX

See also:

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

…

12: 11.13 Methods of Computation

… ►

§11.13(i) Introduction

►Subsequent subsections treat the computation of Struve functions. …

13: 11.16 Software

… ►

§11.16(ii) Struve Functions

… ►

§11.16(iii) Integrals of Struve Functions

…

14: 11 Struve and Related Functions

Chapter 11 Struve and Related Functions

…

15: Richard B. Paris

… ►

11 Struve and Related Functions

…

16: 11.10 Anger–Weber Functions

… ►

11.10.22 $${𝐄}_n\left(z\right)=-{𝐇}_n\left(z\right)+\frac{1}{\pi }\sum _{k=0}^{m_1}\frac{\mathrm{\Gamma }\left(k+\frac{1}{2}\right)}{\mathrm{\Gamma }\left(n+\frac{1}{2}-k\right)}{(\frac{1}{2}z)}^{n-2k-1},$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝐇}_{\nu }\left(z\right)$: Struve function, ${𝐄}_{\nu }\left(z\right)$: Weber function, $\pi$: the ratio of the circumference of a circle to its diameter, $z$: complex variable, $n$: integer order and $k$: nonnegative integer

A&S Ref:

12.3.6 (with upper limit corrected in later printings.)

Referenced by:

§11.10(vi), §11.10(vii)

Permalink:

http://dlmf.nist.gov/11.10.E22

Encodings:

pMML, png, TeX

See also:

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

… ►

11.10.23 $${𝐄}_{-n}\left(z\right)=-{𝐇}_{-n}\left(z\right)+\frac{{(-1)}^{n+1}}{\pi }\sum _{k=0}^{m_2}\frac{\mathrm{\Gamma }\left(n-k-\frac{1}{2}\right)}{\mathrm{\Gamma }\left(k+\frac{3}{2}\right)}{(\frac{1}{2}z)}^{-n+2k+1},$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝐇}_{\nu }\left(z\right)$: Struve function, ${𝐄}_{\nu }\left(z\right)$: Weber function, $\pi$: the ratio of the circumference of a circle to its diameter, $z$: complex variable, $n$: integer order and $k$: nonnegative integer

A&S Ref:

12.3.7 (with upper limit corrected.)

Referenced by:

§11.10(vi), Ch.11

Permalink:

http://dlmf.nist.gov/11.10.E23

Encodings:

pMML, png, TeX

See also:

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

… ►

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

… ►

11.10.26 ${𝐄}_0\left(z\right)=-{𝐇}_0\left(z\right),$ ${𝐄}_1\left(z\right)={\displaystyle \frac{2}{\pi }}-{𝐇}_1\left(z\right).$

ⓘ

Symbols:

${𝐇}_{\nu }\left(z\right)$: Struve function, ${𝐄}_{\nu }\left(z\right)$: Weber function, $\pi$: the ratio of the circumference of a circle to its diameter and $z$: complex variable

Referenced by:

§11.10(vii)

Permalink:

http://dlmf.nist.gov/11.10.E26

Encodings:

pMML, pMML, png, png, TeX, TeX

See also:

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

►

11.10.27 $${\frac{\partial }{\partial \nu }{𝐉}_{\nu }\left(z\right)|}_{\nu =0}=\frac{1}{2}\pi {𝐇}_0\left(z\right),$$

ⓘ

Symbols:

${𝐉}_{\nu }\left(z\right)$: Anger function, ${𝐇}_{\nu }\left(z\right)$: Struve function, $\pi$: the ratio of the circumference of a circle to its diameter, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $z$: complex variable and $\nu$: real or complex order

Referenced by:

§11.10(vii)

Permalink:

http://dlmf.nist.gov/11.10.E27

Encodings:

pMML, png, TeX

See also:

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

…

17: Bibliography Z

… ►

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

ⓘ

External Links:

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

Referenced by:

§11.7(ii).

Encodings:

BibTeX

…

18: Bibliography N

… ►

Approximations for the Bessel and Struve functions. Math. Comp. 43 (168), pp. 551–556.

ⓘ

External Links:

ISSN 0025-5718, FullText, MathReview (S. Conde), ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

…

19: Bibliography

… ►

Efficient approximation of the Struve functions ${𝐇}_n$ occurring in the calculation of sound radiation quantities. The Journal of the Acoustical Society of America 140 (6), pp. 4154–4160.

ⓘ

External Links:

Document

Referenced by:

§11.13(i), §11.13(i).

Encodings:

BibTeX

… ►

Tablitsy funktsii Struve i integralov ot nikh. “Nauka”, Moscow (Russian).

ⓘ

Notes:

Translated title: Tables of Struve Functions and their Integrals

External Links:

MathReview (Walter Gautschi), ZentralBlatt

Referenced by:

2nd item, 2nd item.

Encodings:

BibTeX

… ►

Derivatives and integrals with respect to the order of the Struve functions ${𝐇}_{\nu }(x)$ and ${𝐋}_{\nu }(x)$ . J. Math. Anal. Appl. 137 (1), pp. 17–36.

ⓘ

External Links:

ISSN 0022-247X, Document, MathReview (A. McD. Mercer), ZentralBlatt

Referenced by:

§11.4(vi), §11.7(iv).

Encodings:

BibTeX

…

20: Bibliography B

… ►

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

ⓘ

External Links:

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

Referenced by:

§11.7(v), §11.7(v).

Encodings:

BibTeX

… ►

Tables of modified Struve functions of orders zero and unity.

ⓘ

Notes:

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

Referenced by:

3rd item.

Encodings:

BibTeX

… ►

On the derivatives of the Bessel and Struve functions with respect to the order. Integral Transforms Spec. Funct. 16 (3), pp. 187–198.

ⓘ

External Links:

ISSN 1065-2469, Document, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§10.15, §10.38, §11.4(vi).

Encodings:

BibTeX

…