Weber function

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1: 11.10 Anger–Weber Functions

§11.10 Anger–Weber Functions

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Figure 11.10.1: Anger function ${𝐉}_{\nu }\left(x\right)$ for $-8\le x\le 8$ and $\nu =0,\frac{1}{2},1,\frac{3}{2}$. Magnify

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Figure 11.10.2: Weber function ${𝐄}_{\nu }\left(x\right)$ for $-8\le x\le 8$ and $\nu =0,\frac{1}{2},1,\frac{3}{2}$. Magnify

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Figure 11.10.3: Anger function ${𝐉}_{\nu }\left(x\right)$ for $-10\le x\le 10$ and $0\le \nu \le 5$. Magnify 3D Help

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Figure 11.10.4: Weber function ${𝐄}_{\nu }\left(x\right)$ for $-10\le x\le 10$ and $0\le \nu \le 5$. Magnify 3D Help

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2: 11.1 Special Notation

§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: 11.14 Tables

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§11.14(iv) Anger–Weber Functions

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§11.14(v) Incomplete Functions

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Agrest and Maksimov (1971, Chapter 11) defines incomplete Struve, Anger, and Weber functions and includes tables of an incomplete Struve function ${𝐇}_n\left(x,\alpha \right)$ for $n=0,1$, $x=0(.2)10$, and $\alpha =0(.2)1.4,\frac{1}{2}\pi$, together with surface plots.

4: 11.13 Methods of Computation

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§11.13(i) Introduction

… ►The treatment of Lommel and Anger–Weber functions is similar. … ►See §3.6 for implementation of these methods, and with the Weber function ${𝐄}_n\left(x\right)$ as an example.

5: 11.16 Software

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§11.16(v) Anger and Weber Functions

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§11.16(vi) Integrals of Anger and Weber Functions

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6: 11.11 Asymptotic Expansions of Anger–Weber Functions

§11.11 Asymptotic Expansions of Anger–Weber Functions

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§11.11(i) Large $|z|$, Fixed $\nu$

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§11.11(ii) Large $|\nu |$, Fixed $z$

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11.11.17 $${𝐀}_{-\nu }\left(\nu +a{\nu }^{1/3}\right)=2^{1/3}{\nu }^{-1/3}\mathrm{Hi}\left(-2^{1/3}a\right)+O\left({\nu }^{-1}\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{Hi}\left(z\right)$: Scorer function (inhomogeneous Airy function), ${𝐀}_{\nu }\left(z\right)$: Anger–Weber function, $\nu$: real or complex order and $a_k(\lambda )$: expansion function

Sources:

Olver (1997b, pp. 103 and 352); Nemes (2020)

Referenced by:

§11.11(iii), §11.11(iii), §11.11(iii)

Permalink:

http://dlmf.nist.gov/11.11.E17

Encodings:

pMML, png, TeX

See also:

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

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7: 12.1 Special Notation

… ►Unless otherwise noted, primes indicate derivatives with respect to the variable, and fractional powers take their principal values. ►The main functions treated in this chapter are the parabolic cylinder functions (PCFs), also known as Weber parabolic cylinder functions: $U(a,z)$, $V(a,z)$, $\overline{U}(a,z)$, and $W(a,z)$. …

8: 3.6 Linear Difference Equations

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Example 2. Weber Function

►The Weber function ${𝐄}_n\left(1\right)$ satisfies … ►

3.6.15 $${𝐄}_{2n}\left(1\right)\sim \frac{2}{(4n^2-1)\pi },$$

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Symbols:

${𝐄}_{\nu }\left(z\right)$: Weber function, $\sim$: asymptotic equality and $\pi$: the ratio of the circumference of a circle to its diameter

Permalink:

http://dlmf.nist.gov/3.6.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §3.6(vi), §3.6(vi), §3.6 and Ch.3

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3.6.16 $${𝐄}_{2n+1}\left(1\right)\sim \frac{2}{(2n+1)\pi };$$

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Symbols:

${𝐄}_{\nu }\left(z\right)$: Weber function, $\sim$: asymptotic equality and $\pi$: the ratio of the circumference of a circle to its diameter

Permalink:

http://dlmf.nist.gov/3.6.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §3.6(vi), §3.6(vi), §3.6 and Ch.3

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Table 3.6.1: Weber function $w_n={𝐄}_n\left(1\right)$ computed by Olver’s algorithm.

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$n$	$p_n$		$e_n$		$e_n/(p_n{p}_{n+1})$		$w_n$
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9: 10.58 Zeros

§10.58 Zeros

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$b_{n,m}=y_{n+\frac{1}{2},m},$

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${𝗒}_n^{\prime }\left(b_{n,m}\right)=\sqrt{{\displaystyle \frac{\pi }{2y_{n+\frac{1}{2},m}}}}{Y}_{n+\frac{1}{2}}^{\prime }\left(y_{n+\frac{1}{2},m}\right).$

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10: Bibliography N

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The resurgence properties of the large order asymptotics of the Anger-Weber function I. J. Class. Anal. 4 (1), pp. 1–39.

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External Links:

ISSN 1848-5979, Document, Link, MathReview (José Luis López), ZentralBlatt

Referenced by:

(11.11.10), (11.11.8), §11.11(iii).

Encodings:

BibTeX

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The resurgence properties of the large order asymptotics of the Anger-Weber function II. J. Class. Anal. 4 (2), pp. 121–147.

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External Links:

ISSN 1848-5979, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

(11.11.10), (11.11.8), §11.11(iii).

Encodings:

BibTeX

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