є <Zc@s√ddlmZmZdДZedДГZedДГZedДГZedДГZedДГZedДГZ ed ДГZ ed ДГZedДГZedd ДГZ eddДГZedДГZedДГZedДГZdS(i(tdefunt defun_wrappedcCs |j|Г\}}|j|Г}|j}|sСd|jg|dgg||dgggdf}|rК|ddc||7>s(t hypercomb(RRRtkwargs((RRRs:lib/python2.7/site-packages/mpmath/functions/orthogonal.pythermite<scs"ИjЗЗЗfdЖg|НS(s8 Gives the parabolic cylinder function in Whittaker's notation `D_n(z) = U(-n-1/2, z)` (see :func:`~mpmath.pcfu`). It solves the differential equation .. math :: y'' + \left(n + \frac{1}{2} - \frac{1}{4} z^2\right) y = 0. and can be represented in terms of Hermite polynomials (see :func:`~mpmath.hermite`) as .. math :: D_n(z) = 2^{-n/2} e^{-z^2/4} H_n\left(\frac{z}{\sqrt{2}}\right). **Plots** .. literalinclude :: /plots/pcfd.py .. image :: /plots/pcfd.png **Examples** >>> from mpmath import * >>> mp.dps = 25; mp.pretty = True >>> pcfd(0,0); pcfd(1,0); pcfd(2,0); pcfd(3,0) 1.0 0.0 -1.0 0.0 >>> pcfd(4,0); pcfd(-3,0) 3.0 0.6266570686577501256039413 >>> pcfd('1/2', 2+3j) (-5.363331161232920734849056 - 3.858877821790010714163487j) >>> pcfd(2, -10) 1.374906442631438038871515e-9 Verifying the differential equation:: >>> n = mpf(2.5) >>> y = lambda z: pcfd(n,z) >>> z = 1.75 >>> chop(diff(y,z,2) + (n+0.5-0.25*z**2)*y(z)) 0.0 Rational Taylor series expansion when `n` is an integer:: >>> taylor(lambda z: pcfd(5,z), 0, 7) [0.0, 15.0, 0.0, -13.75, 0.0, 3.96875, 0.0, -0.6015625] cstИИИdГS(Ni(R(((RRR(s:lib/python2.7/site-packages/mpmath/functions/orthogonal.pyR)vs(R*(RRRR+((RRRs:lib/python2.7/site-packages/mpmath/functions/orthogonal.pytpcfd@s6cKs-|j|Г\}}|j||j|ГS(sх Gives the parabolic cylinder function `U(a,z)`, which may be defined for `\Re(z) > 0` in terms of the confluent U-function (see :func:`~mpmath.hyperu`) by .. math :: U(a,z) = 2^{-\frac{1}{4}-\frac{a}{2}} e^{-\frac{1}{4} z^2} U\left(\frac{a}{2}+\frac{1}{4}, \frac{1}{2}, \frac{1}{2}z^2\right) or, for arbitrary `z`, .. math :: e^{-\frac{1}{4}z^2} U(a,z) = U(a,0) \,_1F_1\left(-\tfrac{a}{2}+\tfrac{1}{4}; \tfrac{1}{2}; -\tfrac{1}{2}z^2\right) + U'(a,0) z \,_1F_1\left(-\tfrac{a}{2}+\tfrac{3}{4}; \tfrac{3}{2}; -\tfrac{1}{2}z^2\right). **Examples** Connection to other functions:: >>> from mpmath import * >>> mp.dps = 25; mp.pretty = True >>> z = mpf(3) >>> pcfu(0.5,z) 0.03210358129311151450551963 >>> sqrt(pi/2)*exp(z**2/4)*erfc(z/sqrt(2)) 0.03210358129311151450551963 >>> pcfu(0.5,-z) 23.75012332835297233711255 >>> sqrt(pi/2)*exp(z**2/4)*erfc(-z/sqrt(2)) 23.75012332835297233711255 >>> pcfu(0.5,-z) 23.75012332835297233711255 >>> sqrt(pi/2)*exp(z**2/4)*erfc(-z/sqrt(2)) 23.75012332835297233711255 (RR-R(RtaRR+Rt_((s:lib/python2.7/site-packages/mpmath/functions/orthogonal.pytpcfuxs,csыИj|Г\Й}ИjИГЙИjЙИjЙ|dkr╣ИjИdГr╣ЗЗЗЗЗfdЖ}Иj|g|Н}ИjИГr╡ИjИГr╡Иj|Г}n|SЗЗЗЗfdЖ}Иj|Иg|НSdS(s▐ Gives the parabolic cylinder function `V(a,z)`, which can be represented in terms of :func:`~mpmath.pcfu` as .. math :: V(a,z) = \frac{\Gamma(a+\tfrac{1}{2}) (U(a,-z)-\sin(\pi a) U(a,z)}{\pi}. **Examples** Wronskian relation between `U` and `V`:: >>> from mpmath import * >>> mp.dps = 25; mp.pretty = True >>> a, z = 2, 3 >>> pcfu(a,z)*diff(pcfv,(a,z),(0,1))-diff(pcfu,(a,z),(0,1))*pcfv(a,z) 0.7978845608028653558798921 >>> sqrt(2/pi) 0.7978845608028653558798921 >>> a, z = 2.5, 3 >>> pcfu(a,z)*diff(pcfv,(a,z),(0,1))-diff(pcfu,(a,z),(0,1))*pcfv(a,z) 0.7978845608028653558798921 >>> a, z = 0.25, -1 >>> pcfu(a,z)*diff(pcfv,(a,z),(0,1))-diff(pcfu,(a,z),(0,1))*pcfv(a,z) 0.7978845608028653558798921 >>> a, z = 2+1j, 2+3j >>> chop(pcfu(a,z)*diff(pcfv,(a,z),(0,1))-diff(pcfu,(a,z),(0,1))*pcfv(a,z)) 0.7978845608028653558798921 tQics∙ИjИddtГ}tИИИИdГ}tИИИ|dГ}xE|D]=}|djdГ|djdГ|djИИГqRWИjИИИГИjdИjГ}x0|D](}|dj|Г|djdГq┼W||S(NyЁ┐RiiyЁ?ii(RRR(RtexpjpiR R(tjztT1termstT2termstTR(RRRtrR(s:lib/python2.7/site-packages/mpmath/functions/orthogonal.pyth═s + c svИjИdГ}ИjИdГ}Иj|Г}ИjИИj|Гg}|И|ИИdgИИ|ggИ|ИgИg|f}|ИgИ|ИИddgdИИ|ggИ|dИgdИg|f}ИjИИ|Г\}}|dj|Г|dj|Гx:||fD],} | djdГ| djИ|Гq>> from mpmath import * >>> mp.dps = 25; mp.pretty = True >>> a = mpf(0.25) >>> pcfw(a,0) 0.9722833245718180765617104 >>> power(2,-0.75)*sqrt(abs(gamma(0.25+0.5j*a)/gamma(0.75+0.5j*a))) 0.9722833245718180765617104 >>> diff(pcfw,(a,0),(0,1)) -0.5142533944210078966003624 >>> -power(2,-0.25)*sqrt(abs(gamma(0.75+0.5j*a)/gamma(0.25+0.5j*a))) -0.5142533944210078966003624 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