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21: 11.11 Asymptotic Expansions of Anger–Weber Functions

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11.11.2 $${𝐉}_{\nu }\left(z\right)\sim J_{\nu }\left(z\right)+\frac{\sin \left(\pi \nu \right)}{\pi z}\left(\sum _{k=0}^{\mathrm{\infty }}\frac{F_k(\nu )}{z^{2k}}-\frac{\nu }{z}\sum _{k=0}^{\mathrm{\infty }}\frac{G_k(\nu )}{z^{2k}}\right),$$

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Symbols:

${𝐉}_{\nu }\left(z\right)$: Anger function, $J_{\nu }\left(z\right)$: Bessel function of the first kind, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: sine function, $z$: complex variable, $\nu$: real or complex order, $k$: nonnegative integer, $F_k(\nu )$: expansion functions and $G_k(\nu )$: expansion functions

Proof sketch:

Derivable from (11.10.15).

Referenced by:

§11.11(i)

Permalink:

http://dlmf.nist.gov/11.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §11.11(i), §11.11 and Ch.11

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11.11.3 $${𝐄}_{\nu }\left(z\right)\sim -Y_{\nu }\left(z\right)-\frac{1+\cos \left(\pi \nu \right)}{\pi z}\sum _{k=0}^{\mathrm{\infty }}\frac{F_k(\nu )}{z^{2k}}-\frac{\nu (1-\cos \left(\pi \nu \right))}{\pi z^2}\sum _{k=0}^{\mathrm{\infty }}\frac{G_k(\nu )}{z^{2k}},$$

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Symbols:

$Y_{\nu }\left(z\right)$: Bessel function of the second kind, ${𝐄}_{\nu }\left(z\right)$: Weber function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $z$: complex variable, $\nu$: real or complex order, $k$: nonnegative integer, $F_k(\nu )$: expansion functions and $G_k(\nu )$: expansion functions

Proof sketch:

Derivable from (11.10.16).

Permalink:

http://dlmf.nist.gov/11.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §11.11(i), §11.11 and Ch.11

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11.11.4 $${𝐀}_{\nu }\left(z\right)\sim \frac{1}{\pi z}\sum _{k=0}^{\mathrm{\infty }}\frac{F_k(\nu )}{z^{2k}}-\frac{\nu }{\pi z^2}\sum _{k=0}^{\mathrm{\infty }}\frac{G_k(\nu )}{z^{2k}}.$$

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Symbols:

${𝐀}_{\nu }\left(z\right)$: Anger–Weber function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $z$: complex variable, $\nu$: real or complex order, $k$: nonnegative integer, $F_k(\nu )$: expansion functions and $G_k(\nu )$: expansion functions

Permalink:

http://dlmf.nist.gov/11.11.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §11.11(i), §11.11 and Ch.11

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11.11.8 $${𝐀}_{\nu }\left(\lambda \nu \right)\sim \frac{1}{\pi }\sum _{k=0}^{\mathrm{\infty }}\frac{(2k)!a_k(\lambda )}{{\nu }^{2k+1}},$$ $\nu \to \mathrm{\infty }$, $|\mathrm{ph}\nu |\le \pi -\delta$,

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Symbols:

${𝐀}_{\nu }\left(z\right)$: Anger–Weber 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 $a_k(\lambda )$: expansion function

Sources:

Meijer (1932); Watson (1944, §10.15); Olver (1997b, pp. 103 and 352); Nemes (2014b); Nemes (2014c)

Referenced by:

(11.11.18), (11.11.19), §11.11(iii)

Permalink:

http://dlmf.nist.gov/11.11.E8

Encodings:

pMML, png, TeX

Clarification (effective with 1.1.2):

The constraint which originally was $\nu \to \mathrm{\infty }$, , has been replaced to be $\nu \to \mathrm{\infty }$, $|\mathrm{ph}\nu |\le \pi -\delta$.

Suggested 2021-04-05 by Gergő Nemes

See also:

Annotations for §11.11(iii), §11.11 and Ch.11

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11.11.10 $${𝐀}_{-\nu }\left(\lambda \nu \right)\sim -\frac{1}{\pi }\sum _{k=0}^{\mathrm{\infty }}\frac{(2k)!a_k(-\lambda )}{{\nu }^{2k+1}},$$ $\nu \to \mathrm{\infty }$, $|\mathrm{ph}\nu |\le \pi -\delta$.

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Symbols:

${𝐀}_{\nu }\left(z\right)$: Anger–Weber 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 $a_k(\lambda )$: expansion function

Sources:

Meijer (1932); Watson (1944, §10.15); Olver (1997b, pp. 103 and 352); Nemes (2014b); Nemes (2014c)

Proof sketch:

Apply Laplace’s method to the integral (11.10.4).

Referenced by:

(11.11.18), (11.11.19), §11.11(iii), §11.11(iii)

Permalink:

http://dlmf.nist.gov/11.11.E10

Encodings:

pMML, png, TeX

Clarification (effective with 1.1.2):

The constraint which was originally $\nu \to +\mathrm{\infty }$, has been extended to be $\nu \to \mathrm{\infty }$, $|\mathrm{ph}\nu |\le \pi -\delta$.

Suggested 2021-04-05 by Gergő Nemes

See also:

Annotations for §11.11(iii), §11.11 and Ch.11

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22: 10.60 Sums

… ►For collections of sums of series relevant to spherical Bessel functions or Bessel functions of half odd integer order see Erdélyi et al. (1953b, pp. 43–45 and 98–105), Gradshteyn and Ryzhik (2000, §§8.51, 8.53), Hansen (1975), Magnus et al. (1966, pp. 106–108 and 123–138), and Prudnikov et al. (1986b, pp. 635–637 and 651–700). …

23: Bibliography D

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Bessel functions $J_n(z)$ and $Y_n(z)$ of integer order and complex argument. Comput. Phys. Comm. 78 (1-2), pp. 181–189.

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Notes:

Double-precision Pascal, maximum accuracy 7D.

External Links:

ISSN 0010-4655, Document, Code, MathReview (J. P. Singhal), ZentralBlatt

Referenced by:

2nd item.

Encodings:

BibTeX

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24: 28.24 Expansions in Series of Cross-Products of Bessel Functions or Modified Bessel Functions

… ►For further power series of Mathieu radial functions of integer order for small parameters and improved convergence rate see Larsen et al. (2009).

25: 28.12 Definitions and Basic Properties

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26: 11.6 Asymptotic Expansions

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11.6.1 $${𝐊}_{\nu }\left(z\right)\sim \frac{1}{\pi }\sum _{k=0}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(k+\frac{1}{2}\right){(\frac{1}{2}z)}^{\nu -2k-1}}{\mathrm{\Gamma }\left(\nu +\frac{1}{2}-k\right)},$$ $|\mathrm{ph}z|\le \pi -\delta$,

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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, $\mathrm{ph}$: phase, $z$: complex variable, $\nu$: real or complex order, $k$: nonnegative integer and $\delta$: arbitrary small positive constant

A&S Ref:

12.1.29

Referenced by:

§11.6(i), §11.6(i), §11.6(i), §11.6(i)

Permalink:

http://dlmf.nist.gov/11.6.E1

Encodings:

pMML, png, TeX

See also:

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

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11.6.2 $${𝐌}_{\nu }\left(z\right)\sim \frac{1}{\pi }\sum _{k=0}^{\mathrm{\infty }}{(-1)}^{k+1}\frac{\mathrm{\Gamma }\left(k+\frac{1}{2}\right){(\frac{1}{2}z)}^{\nu -2k-1}}{\mathrm{\Gamma }\left(\nu +\frac{1}{2}-k\right)},$$ $|\mathrm{ph}z|\le \frac{1}{2}\pi -\delta$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝐌}_{\nu }\left(z\right)$: modified Struve function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $z$: complex variable, $\nu$: real or complex order, $k$: nonnegative integer and $\delta$: arbitrary small positive constant

A&S Ref:

12.2.6

Referenced by:

§11.6(i), §11.6(i), §11.6(i)

Permalink:

http://dlmf.nist.gov/11.6.E2

Encodings:

pMML, png, TeX

See also:

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

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

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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

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11.6.7 $${𝐌}_{\nu }\left(\lambda \nu \right)\sim -\frac{{(\frac{1}{2}\lambda \nu )}^{\nu -1}}{\sqrt{\pi }\mathrm{\Gamma }\left(\nu +\frac{1}{2}\right)}\sum _{k=0}^{\mathrm{\infty }}\frac{k!c_k(\mathrm{i}\lambda )}{{\nu }^k},$$ $|\mathrm{ph}\nu |\le \frac{1}{2}\pi -\delta$.

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Symbols:

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

Referenced by:

§11.6(iii)

Permalink:

http://dlmf.nist.gov/11.6.E7

Encodings:

pMML, png, TeX

See also:

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

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27: 11.9 Lommel Functions

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11.9.3 $$s_{\mu ,\nu }\left(z\right)=z^{\mu +1}\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{z^{2k}}{a_{k+1}(\mu ,\nu )},$$

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Defines:

$s_{\mu ,\nu }\left(z\right)$: Lommel function

Symbols:

$z$: complex variable, $\nu$: real or complex order, $k$: nonnegative integer and $a_k(\mu ,\nu )$: expansion function

Permalink:

http://dlmf.nist.gov/11.9.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §11.9(i), §11.9 and Ch.11

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11.9.4 $$a_k(\mu ,\nu )=\prod _{m=1}^k\left({(\mu +2m-1)}^2-{\nu }^2\right)=4^k{\left(\frac{\mu -\nu +1}{2}\right)}_k{\left(\frac{\mu +\nu +1}{2}\right)}_k,$$ $k=0,1,2,\mathrm{\dots }$.

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\nu$: real or complex order, $k$: nonnegative integer and $a_k(\mu ,\nu )$: expansion function

Referenced by:

§11.9(iii), Erratum (V1.1.3) for Additions

Permalink:

http://dlmf.nist.gov/11.9.E4

Encodings:

pMML, png, TeX

Addition (effective with 1.1.3):

An alternative Pochhammer symbol representation was added.

See also:

Annotations for §11.9(i), §11.9 and Ch.11

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11.9.7 $$s_{\mu ,\nu }\left(z\right)=2^{\mu +1}\sum _{k=0}^{\mathrm{\infty }}\frac{(2k+\mu +1)\mathrm{\Gamma }\left(k+\mu +1\right)}{k!(2k+\mu -\nu +1)(2k+\mu +\nu +1)}{J}_{2k+\mu +1}\left(z\right),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $s_{\mu ,\nu }\left(z\right)$: Lommel function, $!$: factorial (as in $n!$), $z$: complex variable, $\nu$: real or complex order and $k$: nonnegative integer

Permalink:

http://dlmf.nist.gov/11.9.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §11.9(ii), §11.9 and Ch.11

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11.9.8 $$s_{\mu ,\nu }\left(z\right)=2^{(\mu +\nu -1)/2}\mathrm{\Gamma }\left(\frac{1}{2}\mu +\frac{1}{2}\nu +\frac{1}{2}\right)z^{(\mu +1-\nu )/2}\sum _{k=0}^{\mathrm{\infty }}\frac{{(\frac{1}{2}z)}^k}{k!(2k+\mu -\nu +1)}{J}_{k+\frac{1}{2}(\mu +\nu +1)}\left(z\right).$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $s_{\mu ,\nu }\left(z\right)$: Lommel function, $!$: factorial (as in $n!$), $z$: complex variable, $\nu$: real or complex order and $k$: nonnegative integer

Permalink:

http://dlmf.nist.gov/11.9.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §11.9(ii), §11.9 and Ch.11

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11.9.9 $$S_{\mu ,\nu }\left(z\right)\sim z^{\mu -1}\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k{a}_k(-\mu ,\nu )z^{-2k},$$ $z\to \mathrm{\infty }$, .

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Symbols:

$S_{\mu ,\nu }\left(z\right)$: Lommel function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $z$: complex variable, $\nu$: real or complex order, $k$: nonnegative integer, $\delta$: arbitrary small positive constant and $a_k(\mu ,\nu )$: expansion function

Referenced by:

§11.9(iii), §11.9(iii)

Permalink:

http://dlmf.nist.gov/11.9.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §11.9(iii), §11.9 and Ch.11

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28: 14.1 Special Notation

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$x$, $y$, $\tau$	real variables.
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$m$, $n$	unless stated otherwise, nonnegative integers, used for order and degree, respectively.
…

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29: 28.28 Integrals, Integral Representations, and Integral Equations

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§28.28(iv) Integrals of Products of Mathieu Functions of Integer Order

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30: 14.16 Zeros

… ►Throughout this section we assume that $\mu$ and $\nu$ are real, and when they are not integers we write …where $m$, $n\in \mathbb{Z}$ and ${\delta }_{\mu }$, ${\delta }_{\nu }\in (0,1)$. … ►

$\mu >0$, $\mu >\nu$, $\mu \notin \mathbb{Z}$, and $\sin \left((\mu -\nu )\pi \right)$ and $\sin \left(\mu \pi \right)$ have opposite signs.

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$\mu \le \nu$, $\mu \notin \mathbb{Z}$, and $\lfloor \mu \rfloor$ is odd.

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