# cylinder functions

(0.006 seconds)

## 11—20 of 83 matching pages

##### 13: 10.44 Sums
10.44.1 $\mathscr{Z}_{\nu}\left(\lambda z\right)=\lambda^{\pm\nu}\sum_{k=0}^{\infty}% \frac{(\lambda^{2}-1)^{k}(\frac{1}{2}z)^{k}}{k!}\mathscr{Z}_{\nu\pm k}\left(z% \right),$ $|\lambda^{2}-1|<1$.
If $\mathscr{Z}=I$ and the upper signs are taken, then the restriction on $\lambda$ is unnecessary. …
10.44.3 $\mathscr{Z}_{\nu}\left(u\pm v\right)=\sum_{k=-\infty}^{\infty}(\pm 1)^{k}% \mathscr{Z}_{\nu+k}\left(u\right)I_{k}\left(v\right),$ $|v|<|u|$.
The restriction $|v|<|u|$ is unnecessary when $\mathscr{Z}=I$ and $\nu$ is an integer. …
##### 14: 10.66 Expansions in Series of Bessel Functions
###### §10.66 Expansions in Series of Bessel Functions
10.66.1 $\operatorname{ber}_{\nu}x+i\operatorname{bei}_{\nu}x=\sum_{k=0}^{\infty}\frac{% e^{(3\nu+k)\pi i/4}x^{k}J_{\nu+k}\left(x\right)}{2^{k/2}k!}=\sum_{k=0}^{\infty% }\frac{e^{(3\nu+3k)\pi i/4}x^{k}I_{\nu+k}\left(x\right)}{2^{k/2}k!}.$
##### 15: 10.23 Sums
###### §10.23(i) Multiplication Theorem
If $\mathscr{C}=J$ and the upper signs are taken, then the restriction on $\lambda$ is unnecessary.
###### §10.23(ii) Addition Theorems
The restriction $|v|<|u|$ is unnecessary when $\mathscr{C}=J$ and $\nu$ is an integer. … The restriction $|ve^{\pm i\alpha}|<|u|$ is unnecessary in (10.23.7) when $\mathscr{C}=J$ and $\nu$ is an integer, and in (10.23.8) when $\mathscr{C}=J$. …
##### 16: 12.17 Physical Applications
###### §12.17 Physical Applications
By using instead coordinates of the parabolic cylinder $\xi,\eta,\zeta$, defined by … Dean (1966) describes the role of PCFs in quantum mechanical systems closely related to the one-dimensional harmonic oscillator. … Lastly, parabolic cylinder functions arise in the description of ultra cold atoms in harmonic trapping potentials; see Busch et al. (1998) and Edwards et al. (1999).
##### 17: 12.7 Relations to Other Functions
###### §12.7(i) Hermite Polynomials
12.7.1 $U\left(-\tfrac{1}{2},z\right)=D_{0}\left(z\right)=e^{-\frac{1}{4}z^{2}},$
##### 18: 10.25 Definitions
###### Symbol $\mathscr{Z}_{\nu}\left(z\right)$
Corresponding to the symbol $\mathscr{C}_{\nu}$ introduced in §10.2(ii), we sometimes use $\mathscr{Z}_{\nu}\left(z\right)$ to denote $I_{\nu}\left(z\right)$, $e^{\nu\pi i}K_{\nu}\left(z\right)$, or any nontrivial linear combination of these functions, the coefficients in which are independent of $z$ and $\nu$. …
##### 20: 12.3 Graphics
###### §12.3(i) Real Variables Figure 12.3.8: V ⁡ ( a , x ) , - 2.5 ≤ a ≤ 2.5 , - 2.5 ≤ x ≤ 2.5 . Magnify 3D Help