B <Zс>у@s╚ddlmZmZddДZeddДГZeddДГZedd ДГZed dДГZedd ДГZeddДГZ eddДГZ eddДГZeddДГZed"ddДГZ ed#ddДГZeddДГZeddДГZedd ДГZd!S)$щ)┌defun┌ defun_wrappedcCs@|а|б\}}|а|б}|j}|sld|jg|dgg||dgggdf}|rf|dd||7<|fS|а|бpа|а|бdkpа|а|бdkoа|а|бdk}|jdd}|rь|j|j|||dНdd d Н} |j||j d|dН|dН} n|} |j| | |dН}|j d||dН}|j|d d Н} |j| d d Н}|Рrdd| g||ggg||||dgg| f}|g}nld|g||ggg||||dgg| f}d|j|g|dddgg||g||dgd|g|f}||g}|Рr8|а| б}xVt t|ГГD]F}||dd||7<||dа|б||dаdбРqюWt|ГS)zФ Combined calculation of the Hermite polynomial H_n(z) (and its generalization to complex n) and the parabolic cylinder function D. щgр?rщщщ)┌precg╨┐T)┌exact)┌_convert_param┌convert┌mpq_1_2┌pi┌isnpint┌reZimr┌fmul┌sqrtZfdivZfneg┌exp┌range┌len┌append┌tuple)┌ctx┌n┌zZparabolic_cylinderZntyp┌q┌T1Zcan_use_2f0Zexpprec┌u┌wZw2Zrw2Znrw2Znw┌terms┌T2Zexpu┌iйr!·:lib/python3.7/site-packages/mpmath/functions/orthogonal.py┌_hermite_params@ &**: r#csИjЗЗЗfddДgf|ОS)NcstИИИdГS)Nr)r#r!)rrrr!r"┌>szhermite..)┌ hypercomb)rrr┌kwargsr!)rrrr"┌hermite<sr'csИjЗЗЗfddДgf|ОS)a8 Gives the parabolic cylinder function in Whittaker's notation `D_n(z) = U(-n-1/2, z)` (see :func:`~mpmath.pcfu`). It solves the differential equation .. math :: y'' + \left(n + \frac{1}{2} - \frac{1}{4} z^2\right) y = 0. and can be represented in terms of Hermite polynomials (see :func:`~mpmath.hermite`) as .. math :: D_n(z) = 2^{-n/2} e^{-z^2/4} H_n\left(\frac{z}{\sqrt{2}}\right). **Plots** .. literalinclude :: /plots/pcfd.py .. image :: /plots/pcfd.png **Examples** >>> from mpmath import * >>> mp.dps = 25; mp.pretty = True >>> pcfd(0,0); pcfd(1,0); pcfd(2,0); pcfd(3,0) 1.0 0.0 -1.0 0.0 >>> pcfd(4,0); pcfd(-3,0) 3.0 0.6266570686577501256039413 >>> pcfd('1/2', 2+3j) (-5.363331161232920734849056 - 3.858877821790010714163487j) >>> pcfd(2, -10) 1.374906442631438038871515e-9 Verifying the differential equation:: >>> n = mpf(2.5) >>> y = lambda z: pcfd(n,z) >>> z = 1.75 >>> chop(diff(y,z,2) + (n+0.5-0.25*z**2)*y(z)) 0.0 Rational Taylor series expansion when `n` is an integer:: >>> taylor(lambda z: pcfd(5,z), 0, 7) [0.0, 15.0, 0.0, -13.75, 0.0, 3.96875, 0.0, -0.6015625] cstИИИdГS)Nr)r#r!)rrrr!r"r$vszpcfd..)r%)rrrr&r!)rrrr"┌pcfd@s6r(cKs"|а|б\}}|а||j|бS)aх 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 )r r(r)r┌arr&r┌_r!r!r"┌pcfuxs,r+csиИа|б\Й}ИаИбЙИjЙИjЙ|dkrАИаИdбrАЗЗЗЗЗfddД}Иj|gf|О}ИаИбr|ИаИбr|Иа|б}|SЗЗЗЗfddД}Иj|Иgf|ОSdS)a▐ 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 ┌Qrcs╞ИjИdddН}tИИИИdГ}tИИИ|dГ}x:|D]2}|dаdб|dаdб|dаИИбq.hc s$ИаИdб}ИаИdб}Иа|б}ИjИИа|бg}|И|ИИdgИИ|ggИ|ИgИg|f}|ИgИ|ИИddgdИИ|ggИ|dИgdИg|f}ИаИИ|б\}}|dа|б|dа|бx0||fD]$} | dаdб| dаИ|бqЇW||fS)Ng╨┐gр?rrr-)Zsquare_exp_argrr ┌cospi_sinpir) rrr┌e┌lZY1ZY2┌c┌s┌Y)rrr0rr!r"r1▀s 8LN)r rrZmpq_1_4┌isintr%┌ _is_real_type┌_re)rr)rr&Zntyper1┌vr!)rrrr0rr"┌pcfvзs r<csTИа|б\Й}ИаИбЙЗЗЗfddД}Иа|б}ИаИбrPИаИбrPИа|б}|S)aI Gives the parabolic cylinder function `W(a,z)` defined in (DLMF 12.14). **Examples** Value at the origin:: >>> from mpmath import * >>> mp.dps = 25; 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