# Anger–Weber functions

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##### 1: 11.10 Anger–Weber Functions
###### §11.10 Anger–WeberFunctions Figure 11.10.1: Anger function J ν ⁡ ( x ) for - 8 ≤ x ≤ 8 and ν = 0 , 1 2 , 1 , 3 2 . Magnify Figure 11.10.2: Weber function E ν ⁡ ( x ) for - 8 ≤ x ≤ 8 and ν = 0 , 1 2 , 1 , 3 2 . Magnify Figure 11.10.3: Anger function J ν ⁡ ( x ) for - 10 ≤ x ≤ 10 and 0 ≤ ν ≤ 5 . Magnify 3D Help Figure 11.10.4: Weber function E ν ⁡ ( x ) for - 10 ≤ x ≤ 10 and 0 ≤ ν ≤ 5 . Magnify 3D Help
##### 2: 11.1 Special Notation
###### §11.1 Special Notation
For the functions $J_{\nu}\left(z\right)$, $Y_{\nu}\left(z\right)$, ${H^{(1)}_{\nu}}\left(z\right)$, ${H^{(2)}_{\nu}}\left(z\right)$, $I_{\nu}\left(z\right)$, and $K_{\nu}\left(z\right)$ see §§10.2(ii), 10.25(ii). The functions treated in this chapter are the Struve functions $\mathbf{H}_{\nu}\left(z\right)$ and $\mathbf{K}_{\nu}\left(z\right)$, the modified Struve functions $\mathbf{L}_{\nu}\left(z\right)$ and $\mathbf{M}_{\nu}\left(z\right)$, the Lommel functions $s_{{\mu},{\nu}}\left(z\right)$ and $S_{{\mu},{\nu}}\left(z\right)$, the Anger function $\mathbf{J}_{\nu}\left(z\right)$, the Weber function $\mathbf{E}_{\nu}\left(z\right)$, and the associated AngerWeber function $\mathbf{A}_{\nu}\left(z\right)$.
##### 3: 11.14 Tables
###### §11.14(v) Incomplete Functions
• Agrest and Maksimov (1971, Chapter 11) defines incomplete Struve, Anger, and Weber functions and includes tables of an incomplete Struve function $\mathbf{H}_{n}\left(x,\alpha\right)$ for $n=0,1$, $x=0(.2)10$, and $\alpha=0(.2)1.4,\tfrac{1}{2}\pi$, together with surface plots.

• ##### 4: 11.11 Asymptotic Expansions of Anger–Weber Functions
###### §11.11(ii) Large $|\nu|$, Fixed $z$
11.11.16 $\mathbf{A}_{-\nu}\left(\nu\right)\sim\frac{2^{4/3}}{3^{7/6}\Gamma\left(\tfrac{% 2}{3}\right)\nu^{1/3}},$
11.11.17 $\mathbf{A}_{-\nu}\left(\nu+a\nu^{1/3}\right)=2^{1/3}\nu^{-1/3}\mathrm{Hi}\left% (-2^{1/3}a\right)+O\left(\nu^{-1}\right),$
##### 5: 11.13 Methods of Computation
###### §11.13(i) Introduction
The treatment of Lommel and AngerWeber functions is similar. …
##### 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(v) $\mathbf{J}_{\nu}\left(z\right)$, $\mathbf{E}_{\nu}\left(z\right)$, $\mathbf{A}_{\nu}\left(z\right)$ 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. …
##### 10: Bibliography Z
• M. R. Zaghloul and A. N. Ali (2011) Algorithm 916: computing the Faddeyeva and Voigt functions. ACM Trans. Math. Software 38 (2), pp. Art. 15, 22.
• M. R. Zaghloul (2017) Algorithm 985: Simple, Efficient, and Relatively Accurate Approximation for the Evaluation of the Faddeyeva Function. ACM Trans. Math. Softw. 44 (2), pp. 22:1–22:9.
• R. Zanovello (1977) Integrali di funzioni di Anger, Weber ed Airy-Hardy. Rend. Sem. Mat. Univ. Padova 58, pp. 275–285 (Italian).
• R. Zanovello (1995) Numerical analysis of Struve functions with applications to other special functions. Ann. Numer. Math. 2 (1-4), pp. 199–208.
• I. J. Zucker (1979) The summation of series of hyperbolic functions. SIAM J. Math. Anal. 10 (1), pp. 192–206.