a Ыму@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}|Рr4|а| б}t t|ГГD]F}||dd||7<||dа|б||dаdбРqьt|ГS)zФ Combined calculation of the Hermite polynomial H_n(z) (and its generalization to complex n) and the parabolic cylinder function D. щчр?rщщщ)┌precч╨┐Tй┌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┌wZw2Zrw2Znrw2┌nw┌terms┌T2Zexpu┌iйr%·k/Users/vegardjervell/Documents/master/model/venv/lib/python3.9/site-packages/mpmath/functions/orthogonal.py┌_hermite_paramsB & **: r'cs ИjЗЗЗfddДgfi|дОS)NcstИИИdГS)Nrйr'r%йrrrr%r&┌>єzhermite..й┌ hypercombйrrr┌kwargsr%r)r&┌hermite<sr0cs ИjЗЗЗfddДgfi|дОS)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] cstИИИdГSйNrr(r%r)r%r&r*vr+zpcfd..r,r.r%r)r&┌pcfd@s6r2cKs"|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 r2r)r┌arr/r┌_r%r%r&┌pcfuxs,r5cs░Иа|б\Й}ИаИбЙИjЙИjЙ|dkrДИаИdбrДЗЗЗЗЗfddД}Иj|gfi|дО}ИаИбrАИаИбrАИа|б}|SЗЗЗЗfddД}Иj|Иgfi|дОSdS)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}|D]2}|dаdб|dаdб|dаИИбq:ИаИИИбИаdИjб}|D] }|dа|б|dаdбqФ||S) NyАЁ┐Trrr∙Ё?щr)rr'r┌expjpirr)ZjzZT1termsZT2terms┌Trйrrr┌rrr%r&┌h═s"zpcfv..hc s ИаИdб}ИаИdб}Иа|б}ИjИИа|бg}|И|ИИdgИИ|ggИ|ИgИg|f}|ИgИ|ИИddgdИИ|ggИ|dИgdИg|f}ИаИИ|б\}}|dа|б|dа|б||fD]$} | dаdб| dаИ|бqЄ||fS)Nr rrrr8)Zsquare_exp_argrr┌cospi_sinpir) rr r┌e┌lZY1ZY2┌c┌s┌Y)rrr<rr%r&r=▀s 8LN)r rrZmpq_1_4┌isintr-┌ _is_real_type┌_re)rr3rr/Zntyper=┌vr%r;r&┌pcfvзs rHcsTИа|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; 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 c3sИаИаdИjИбб}ИаdИjИбИаdИjИбd}Иjdd|}ИаdИаdИjИббИаИjИб}Иа|dбИаdИjИб}|Иа|бИаИjИИИа dббV|Иа|бИаИjИИИа dббVdS)Nry@щrrg╨?r ) ┌arg┌gamma┌jZloggammarrr┌expjr5r9)Zphi2┌rho┌k┌Cr)r%r&r"s,.",zpcfw..terms)r rZsum_accuratelyrErF)rr3rr/r4r"rGr%r)r&┌pcfwЁs rQcsВ|аИбrdИИS|аИdбr^|аИdбr:tdГВЗЗfddД}|j|Иgfi|дОSЗЗfddД}|j|Иgfi|дОS)Nrrrz#Gegenbauer function with two limitsc sFd|}ggИ|gИd|gИИ|g|dgddИf}|gSйNrrrr%)r3┌a2r:йrrr%r&r==s8zgegenbauer..hc sFdИ}gg||g|d|g|||gИdgddИf}|gSrRr%)rrSr:)r3rr%r&r=Bs8)r┌NotImplementedErrorr-йrrr3rr/r=r%)r3rrr&┌ gegenbauer3s rWcsв|аИбs0ЗЗЗfddД}|j||gfi|дОS|аИбs`ЗЗfddД}|j||Иgfi|дОS|а|И|б|j|d|ИИИddИdfi|дОS)NcsJggИ|dg|dИdg|ИИ|dgИdgdИdffSйNrrr%)rйr3┌b┌xr%r&r=Kszjacobi..hcsFggИg|dИ|g||И|dgИdgИddffSrXr%)rr3)rZr[r%r&r=Osrr)rr-rDZbinomial┌hyp2f1)rrr3rZr[r/r=r%rYr&┌jacobiHs r]cs$ЗЗfddД}|j||gfi|дОS)Ncs4gg|Иdg|dИdgИg|dgИffSr1r%)r3rTr%r&r=Zszlaguerre..hr,rVr%rTr&┌laguerreUsr^cKsИ|а|бrbt|Г}||dkd@rb|s*|S|а|б}|d|jdkrJ|S|dkrb|j|7_|j||ddd|dfi|дОS)Nrrщ■ щ щ√ r)rD┌int┌magr r\)rrr[r/rcr%r%r&┌legendre^s rdrcsР|а|б}|а|б}|s,|j|Иfi|дОS|dkrXЗfddД}|j|||gfi|дОS|dkrДЗfddД}|j|||gfi|дОStdГВdS)Nrc sP|d}dИdИg||ggd|g||dgd|gddИf}|fSйNrrr%йr┌m┌gr:йrr%r&r=wsBzlegenp..hr8c sP|d}ИdИdg||ggd|g||dgd|gddИf}|fSrer%rfrir%r&r=}sB·requires type=2 or type=3)rrdr-┌ ValueErrorйrrrgr┌typer/r=r%rir&┌legenpms rncs╞Иа|б}Иа|б}ИаИбЙИdvr,ИjS|dkrZЗЗfddД}Иj|||gfi|дОS|dkr║tИГdkrФЗЗfddД}Иj|||gfi|дОSЗЗfddД}Иj|||gfi|дОStd ГВdS) N)rщ rcs╥Иа|б\}}d|Иj}|}dИ}dИ}|d}dИd} ||||gdd||ggd|g||dgd|g| f} |||gd||g||dg||d|dg||dg|dg| f}| |fSйNrrro)r>r)rrg┌cos┌sinrBrAr3rZrr rr#йrrr%r&r=Рs 2 zlegenq..hr8rcsОИа|бdИjИИdИdgd|dd||dd|d|g||dg|dgdd||dd||g|dgИdf}|gS)Nrrrg°?r_)r9r)rrgrrsr%r&r=вs&,¤c s╘dИа|бИj}Иа|б}dИ}Иd}|d}dИd}||||gdd||ggd|g||dgd|g|f}||||gdd||g||dg||d|dg||dg|dg|f} || fSrp)Zsinpirr9) rrgrBrAr3rZrr rr#rsr%r&r=лs 6 rj)r┌nanr-┌absrkrlr%rsr&┌legenqДs rvcKsN|s,|а|бr,t|а|бГddkr,|dS|j||dd|dfi|дОS)Nrrr)rrйrDrbrFr\йrrr[r/r%r%r&┌chebyt║s$rycKsZ|s,|а|бr,t|а|бГddkr,|dS|d|j||ddd|dfi|дОS)Nrrr)r8rrwrxr%r%r&┌chebyu└s$rzc sьИа|б}Иа|б}ИаИбЙИаИбЙИа|б}|o<|dk}Иа|б}|rv|dkrv|rvИj|d|ИИfi|дОSИdkrФ|rФ|dkrФИjdS|r─|r─t|Г|kr▓ИjdSЗЗЗfddД} nЗЗЗfddД} Иj| ||gfi|дОS)Nrrr7c s└t|Г}dИа|Ибd|dИа||бИjИа||бИаИбdИа|бdg}d|Иа|бdddd|d|dg}||gg||||dg|dgИаdИбdffS)Nrorrr)rurMZfacrrr┌sign)r@rgZabsmrP┌Pйr┌phi┌thetar%r&r=╪s, ¤," zspherharm..hcsЄИа||dбs2Иа||dбs2Иаd|бrJdgdgggggdffSИаdИб\}}dИа|Ибd|dИjИа||dбИа||dб|d|dg}ddddd|d|g}||gd|g||dgd|g|dffS)Nrrrorrgр┐)rZcos_sinrMrrK)r@rgrqrrrPr|r}r%r&r=фs2 ■)rrD┌ spherharmZzerorur-) rr@rgrr~r/Zl_isintZ l_naturalZm_isintr=r%r}r&rА╞s" rАN)r)r)Z functionsrrr'r0r2r5rHrQrWr]r^rdrnrvryrzrАr%r%r%r&┌s:9 7 . 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