# cylinder functions

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##### 1: 12.14 The Function $W\left(a,x\right)$
###### §12.14 The Function$W\left(a,x\right)$
This equation is important when $a$ and $z$ $(=x)$ are real, and we shall assume this to be the case. …
##### 2: 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 $W\left(a,z\right)$. …An older notation, due to Whittaker (1902), for $U\left(a,z\right)$ is $D_{\nu}\left(z\right)$. …
##### 3: 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 $\nu$. …
##### 4: 12.2 Differential Equations
###### §12.2(i) Introduction
Standard solutions are $U\left(a,\pm z\right)$, $V\left(a,\pm z\right)$, $\overline{U}\left(a,\pm x\right)$ (not complex conjugate), $U\left(-a,\pm iz\right)$ for (12.2.2); $W\left(a,\pm x\right)$ for (12.2.3); $D_{\nu}\left(\pm z\right)$ for (12.2.4), where …
###### §12.2(iii) Wronskians
When $z$ $(=x)$ is real the solution $\overline{U}\left(a,x\right)$ is defined by …
##### 5: 10.2 Definitions
###### CylinderFunctions
The notation $\mathscr{C}_{\nu}\left(z\right)$ denotes $J_{\nu}\left(z\right)$, $Y_{\nu}\left(z\right)$, ${H^{(1)}_{\nu}}\left(z\right)$, ${H^{(2)}_{\nu}}\left(z\right)$, or any nontrivial linear combination of these functions, the coefficients in which are independent of $z$ and $\nu$. …
##### 6: 14.15 Uniform Asymptotic Approximations
Here we introduce the envelopes of the parabolic cylinder functions $U\left(-c,x\right)$, $\overline{U}\left(-c,x\right)$, which are defined in §12.2. For $U\left(-c,x\right)$ or $\overline{U}\left(-c,x\right)$, with $c$ and $x$ nonnegative, …
14.15.24 $\mathsf{P}^{-\mu}_{\nu}\left(x\right)=\frac{1}{\left(\nu+\frac{1}{2}\right)^{1% /4}2^{(\nu+\mu)/2}\Gamma\left(\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{3}{4}\right)% }\left(\frac{\zeta^{2}-\alpha^{2}}{x^{2}-a^{2}}\right)^{1/4}\*\left(U\left(\mu% -\nu-\tfrac{1}{2},\left(2\nu+1\right)^{1/2}\zeta\right)+O\left(\nu^{-2/3}% \right)\mathrm{env}\mskip-1.0mu U\left(\mu-\nu-\tfrac{1}{2},\left(2\nu+1\right% )^{1/2}\zeta\right)\right),$
14.15.25 $\mathsf{Q}^{-\mu}_{\nu}\left(x\right)=\frac{\pi}{\left(\nu+\frac{1}{2}\right)^% {1/4}2^{(\nu+\mu+2)/2}\Gamma\left(\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{3}{4}% \right)}\*\left(\frac{\zeta^{2}-\alpha^{2}}{x^{2}-a^{2}}\right)^{1/4}\*\left(% \overline{U}\left(\mu-\nu-\tfrac{1}{2},\left(2\nu+1\right)^{1/2}\zeta\right)+O% \left(\nu^{-2/3}\right)\mathrm{env}\mskip-1.0mu \overline{U}\left(\mu-\nu-% \tfrac{1}{2},\left(2\nu+1\right)^{1/2}\zeta\right)\right),$
14.15.30 $\mathsf{P}^{-\mu}_{\nu}\left(x\right)=\frac{1}{\left(\nu+\frac{1}{2}\right)^{1% /4}2^{(\nu+\mu)/2}\Gamma\left(\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{3}{4}\right)% }\left(\frac{\zeta^{2}+\alpha^{2}}{x^{2}+a^{2}}\right)^{1/4}\*U\left(\mu-\nu-% \tfrac{1}{2},\left(2\nu+1\right)^{1/2}\zeta\right)\left(1+O\left(\nu^{-1}\ln% \nu\right)\right),$
##### 7: 10.29 Recurrence Relations and Derivatives
With $\mathscr{Z}_{\nu}\left(z\right)$ defined as in §10.25(ii),
$\mathscr{Z}_{\nu-1}\left(z\right)-\mathscr{Z}_{\nu+1}\left(z\right)=(2\nu/z)% \mathscr{Z}_{\nu}\left(z\right),$
$\mathscr{Z}_{\nu-1}\left(z\right)+\mathscr{Z}_{\nu+1}\left(z\right)=2\mathscr{% Z}_{\nu}'\left(z\right).$
$\mathscr{Z}_{\nu}'\left(z\right)=\mathscr{Z}_{\nu-1}\left(z\right)-(\nu/z)% \mathscr{Z}_{\nu}\left(z\right),$
$\mathscr{Z}_{\nu}'\left(z\right)=\mathscr{Z}_{\nu+1}\left(z\right)+(\nu/z)% \mathscr{Z}_{\nu}\left(z\right).$
##### 8: 10.36 Other Differential Equations
The quantity $\lambda^{2}$ in (10.13.1)–(10.13.6) and (10.13.8) can be replaced by $-\lambda^{2}$ if at the same time the symbol $\mathscr{C}$ in the given solutions is replaced by $\mathscr{Z}$. …
10.36.1 $z^{2}(z^{2}+\nu^{2})w^{\prime\prime}+z(z^{2}+3\nu^{2})w^{\prime}-\left((z^{2}+% \nu^{2})^{2}+z^{2}-\nu^{2}\right)w=0,$ $w=\mathscr{Z}_{\nu}'\left(z\right)$,
10.36.2 ${z^{2}w^{\prime\prime}+z(1\pm 2z)w^{\prime}+(\pm z-\nu^{2})w=0},$ $w=e^{\mp z}\mathscr{Z}_{\nu}\left(z\right)$.
##### 9: 12.16 Mathematical Applications
###### §12.16 Mathematical Applications
For examples see §§13.20(iii), 13.20(iv), 14.15(v), and 14.26. …
##### 10: 10.13 Other Differential Equations
###### §10.13 Other Differential Equations
10.13.1 $w^{\prime\prime}+\left(\lambda^{2}-\frac{\nu^{2}-\tfrac{1}{4}}{z^{2}}\right)w=0,$ $w=z^{\frac{1}{2}}\mathscr{C}_{\nu}\left(\lambda z\right)$,
In (10.13.9)–(10.13.11) $\mathscr{C}_{\nu}\left(z\right)$, $\mathscr{D}_{\mu}(z)$ are any cylinder functions of orders $\nu,\mu$, respectively, and $\vartheta=z(\!\ifrac{\mathrm{d}}{\mathrm{d}z})$.
10.13.9 ${z^{2}w^{\prime\prime\prime}+3zw^{\prime\prime}+(4z^{2}+1-4\nu^{2})w^{\prime}+% 4zw=0},$ $w=\mathscr{C}_{\nu}\left(z\right)\mathscr{D}_{\nu}(z)$,
10.13.10 ${z^{3}w^{\prime\prime\prime}+z(4z^{2}+1-4\nu^{2})w^{\prime}+(4\nu^{2}-1)w=0},$ $w=z\mathscr{C}_{\nu}\left(z\right)\mathscr{D}_{\nu}(z)$,