CylinderD - Maple Help
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CylinderU, CylinderV

Parabolic Cylinder Functions

CylinderD

Whittaker's Parabolic Function

 Calling Sequence CylinderU(a, x) CylinderV(a, x) CylinderD(a, x)

Parameters

 a - algebraic expression (the degree) x - algebraic expression (the argument)

Description

 • CylinderU and CylinderV are the parabolic cylinder functions. They satisfy the first real standard distinct form of the Parabolic Cylinder equation:

$\mathrm{y\text{'}\text{'}}-\left(\frac{{x}^{2}}{4}+a\right)y=0$

 • CylinderD and CylinderU are related in the following way:

$\mathrm{CylinderD}\left(-a-\frac{1}{2},x\right)=\mathrm{CylinderU}\left(a,x\right).$

Examples

 > $\mathrm{aa}≔\mathrm{CylinderU}\left(3,0\right)$
 ${\mathrm{aa}}{≔}\frac{{2}{}{{2}}^{{3}}{{4}}}{}{\mathrm{\Gamma }}{}\left(\frac{{3}}{{4}}\right)}{{5}{}\sqrt{{\mathrm{\pi }}}}$ (1)
 > $\mathrm{evalf}\left(\mathrm{aa}\right)$
 ${0.4650946536}$ (2)
 > $\mathrm{CylinderU}\left(-\frac{5}{2},x\right)$
 $\frac{{{ⅇ}}^{{-}\frac{{{x}}^{{2}}}{{4}}}{}{\mathrm{HermiteH}}{}\left({2}{,}\frac{{x}{}\sqrt{{2}}}{{2}}\right)}{{2}}$ (3)
 > $\mathrm{CylinderD}\left(3.2,1\right)$
 ${-1.819497238}$ (4)
 > $\frac{\partial }{\partial x}\mathrm{CylinderU}\left(a,x\right)$
 ${-}\frac{{x}{}{\mathrm{CylinderU}}{}\left({a}{,}{x}\right)}{{2}}{-}\left({a}{+}\frac{{1}}{{2}}\right){}{\mathrm{CylinderU}}{}\left({a}{+}{1}{,}{x}\right)$ (5)
 > $\mathrm{convert}\left(\mathrm{CylinderD}\left(\frac{3}{2},x\right),\mathrm{CylinderU}\right)$
 ${\mathrm{CylinderU}}{}\left({-2}{,}{x}\right)$ (6)
 > $\mathrm{convert}\left(\mathrm{CylinderU}\left(a,x\right)+\mathrm{CylinderD}\left(b,x\right),\mathrm{CylinderV}\right)$
 $\frac{{\mathrm{\pi }}{}\left({\mathrm{CylinderV}}{}\left({a}{,}{-}{x}\right){-}{\mathrm{sin}}{}\left({\mathrm{\pi }}{}{a}\right){}{\mathrm{CylinderV}}{}\left({a}{,}{x}\right)\right)}{{{\mathrm{cos}}{}\left({\mathrm{\pi }}{}{a}\right)}^{{2}}{}{\mathrm{\Gamma }}{}\left({a}{+}\frac{{1}}{{2}}\right)}{+}\frac{{\mathrm{\pi }}{}\left({\mathrm{CylinderV}}{}\left({-}{b}{-}\frac{{1}}{{2}}{,}{-}{x}\right){-}{\mathrm{sin}}{}\left({\mathrm{\pi }}{}\left({-}{b}{-}\frac{{1}}{{2}}\right)\right){}{\mathrm{CylinderV}}{}\left({-}{b}{-}\frac{{1}}{{2}}{,}{x}\right)\right)}{{{\mathrm{cos}}{}\left({\mathrm{\pi }}{}\left({-}{b}{-}\frac{{1}}{{2}}\right)\right)}^{{2}}{}{\mathrm{\Gamma }}{}\left({-}{b}\right)}$ (7)
 > $\mathrm{series}\left(\mathrm{CylinderV}\left(0,x\right),x\right)$
 $\frac{{{2}}^{{3}}{{4}}}}{{2}{}{\mathrm{\Gamma }}{}\left(\frac{{3}}{{4}}\right)}{+}\frac{{1}}{{2}}{}\frac{{{2}}^{{3}}{{4}}}{}{\mathrm{\Gamma }}{}\left(\frac{{3}}{{4}}\right)}{{\mathrm{\pi }}}{}{x}{+}\frac{{1}}{{96}}{}\frac{{{2}}^{{3}}{{4}}}}{{\mathrm{\Gamma }}{}\left(\frac{{3}}{{4}}\right)}{}{{x}}^{{4}}{+}\frac{{1}}{{160}}{}\frac{{{2}}^{{3}}{{4}}}{}{\mathrm{\Gamma }}{}\left(\frac{{3}}{{4}}\right)}{{\mathrm{\pi }}}{}{{x}}^{{5}}{+}{O}{}\left({{x}}^{{6}}\right)$ (8)