# parabolic cylinder functions

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## 11—20 of 59 matching pages

##### 13: 12.12 Integrals
###### Nicholson-type Integral
12.12.4 $(U\left(a,z\right))^{2}+(\overline{U}\left(a,z\right))^{2}=\frac{2^{\frac{3}{2% }}}{\pi}\Gamma\left(\tfrac{1}{2}-a\right)\int_{0}^{\infty}\frac{e^{2at+\frac{1% }{2}z^{2}\tanh t}}{\sqrt{\sinh\left(2t\right)}}\mathrm{d}t,$ $\Re a<\tfrac{1}{2}$ .
##### 14: 12.8 Recurrence Relations and Derivatives
###### §12.8(i) Recurrence Relations
12.8.2 $U'\left(a,z\right)+\tfrac{1}{2}zU\left(a,z\right)+(a+\tfrac{1}{2})U\left(a+1,z% \right)=0,$
12.8.3 $U'\left(a,z\right)-\tfrac{1}{2}zU\left(a,z\right)+U\left(a-1,z\right)=0,$
12.8.6 $V'\left(a,z\right)-\tfrac{1}{2}zV\left(a,z\right)-(a-\tfrac{1}{2})V\left(a-1,z% \right)=0,$
##### 15: 12.13 Sums
###### §12.13 Sums
12.13.1 $U\left(a,x+y\right)=e^{\frac{1}{2}xy+\frac{1}{4}y^{2}}\sum_{m=0}^{\infty}\frac% {(-y)^{m}}{m!}U\left(a-m,x\right),$
12.13.2 $U\left(a,x+y\right)=e^{-\frac{1}{2}xy-\frac{1}{4}y^{2}}\sum_{m=0}^{\infty}% \genfrac{(}{)}{0.0pt}{}{-a-\tfrac{1}{2}}{m}y^{m}U\left(a+m,x\right),$
12.13.3 $V\left(a,x+y\right)=e^{\frac{1}{2}xy+\frac{1}{4}y^{2}}\sum_{m=0}^{\infty}% \genfrac{(}{)}{0.0pt}{}{a-\tfrac{1}{2}}{m}y^{m}V\left(a-m,x\right),$
##### 16: 10.39 Relations to Other Functions
###### ParabolicCylinderFunctions
10.39.3 $K_{\frac{1}{4}}\left(z\right)=\pi^{\frac{1}{2}}z^{-\frac{1}{4}}U\left(0,2z^{% \frac{1}{2}}\right),$
10.39.4 $K_{\frac{3}{4}}\left(z\right)=\tfrac{1}{2}\pi^{\frac{1}{2}}z^{-\frac{3}{4}}% \left(\tfrac{1}{2}U\left(1,2z^{\frac{1}{2}}\right)+U\left(-1,2z^{\frac{1}{2}}% \right)\right).$
##### 17: 12.9 Asymptotic Expansions for Large Variable
###### §12.9 Asymptotic Expansions for Large Variable
12.9.1 $U\left(a,z\right)\sim e^{-\frac{1}{4}z^{2}}z^{-a-\frac{1}{2}}\sum_{s=0}^{% \infty}(-1)^{s}\frac{{\left(\frac{1}{2}+a\right)_{2s}}}{s!(2z^{2})^{s}},$ $|\operatorname{ph}z|\leq\tfrac{3}{4}\pi-\delta(<\tfrac{3}{4}\pi)$ ,
12.9.2 $V\left(a,z\right)\sim\sqrt{\frac{2}{\pi}}e^{\frac{1}{4}z^{2}}z^{a-\frac{1}{2}}% \sum_{s=0}^{\infty}\frac{{\left(\frac{1}{2}-a\right)_{2s}}}{s!(2z^{2})^{s}},$ $|\operatorname{ph}z|\leq\tfrac{1}{4}\pi-\delta(<\tfrac{1}{4}\pi)$ .
12.9.4 $V\left(a,z\right)\sim\sqrt{\frac{2}{\pi}}e^{\frac{1}{4}z^{2}}z^{a-\frac{1}{2}}% \sum_{s=0}^{\infty}\frac{{\left(\tfrac{1}{2}-a\right)_{2s}}}{s!(2z^{2})^{s}}% \pm\frac{i}{\Gamma\left(\tfrac{1}{2}-a\right)}e^{-\frac{1}{4}z^{2}}z^{-a-\frac% {1}{2}}\sum_{s=0}^{\infty}(-1)^{s}\frac{{\left(\tfrac{1}{2}+a\right)_{2s}}}{s!% (2z^{2})^{s}},$ $-\tfrac{1}{4}\pi+\delta\leq\pm\operatorname{ph}z\leq\tfrac{3}{4}\pi-\delta$.
##### 18: 12.5 Integral Representations
###### §12.5(i) Integrals Along the Real Line
12.5.1 $U\left(a,z\right)=\frac{e^{-\frac{1}{4}z^{2}}}{\Gamma\left(\frac{1}{2}+a\right% )}\int_{0}^{\infty}t^{a-\frac{1}{2}}e^{-\frac{1}{2}t^{2}-zt}\mathrm{d}t,$ $\Re a>-\tfrac{1}{2}$ ,
##### 19: 29.16 Asymptotic Expansions
The approximating functions are exponential, trigonometric, and parabolic cylinder functions.