# Weber function

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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 Anger–Weber 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: 11.13 Methods of Computation
###### §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 $\mathbf{E}_{n}\left(x\right)$ as an example.
##### 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: $U\left(a,z\right)$, $V\left(a,z\right)$, $\overline{U}\left(a,z\right)$, and $W\left(a,z\right)$. …
##### 8: 3.6 Linear Difference Equations
###### Example 2. WeberFunction
The Weber function $\mathbf{E}_{n}\left(1\right)$ satisfies …
##### 9: 10.58 Zeros
###### §10.58 Zeros
$b_{n,m}=y_{n+\frac{1}{2},m},$
$\mathsf{y}_{n}'\left(b_{n,m}\right)=\sqrt{\frac{\pi}{2y_{n+\frac{1}{2},m}}}Y_{% n+\frac{1}{2}}'\left(y_{n+\frac{1}{2},m}\right).$
##### 10: Bibliography N
• G. Nemes (2014b) The resurgence properties of the large order asymptotics of the Anger-Weber function I. J. Class. Anal. 4 (1), pp. 1–39.
• G. Nemes (2014c) The resurgence properties of the large order asymptotics of the Anger-Weber function II. J. Class. Anal. 4 (2), pp. 121–147.