a RG5dB@sddlmZddlmZddlmZddlmZddlm Z ddl m Z m Z m Z ddlmZmZddlmZmZmZdd lmZdd lmZmZdd lmZdd lmZdd lmZm Z m!Z!m"Z"ddl#m$Z$ddl%m&Z&m'Z'ddl(m)Z)m*Z*m+Z+ddl,m-Z-m.Z.m/Z/m0Z0m1Z1ddl2m3Z3m4Z4m5Z5ddl6m7Z7ddl8m9Z9ddl:m;Z;mGddde=Z?Gddde=Z@Gddde=ZAGd d!d!e=ZBGd"d#d#e=ZCd$d%ZDGd&d'd'e=ZEd(d)ZFd*d+ZGGd,d-d-eEZHGd.d/d/eEZIGd0d1d1eEZJGd2d3d3eJZKGd4d5d5eJZLdGd8d9ZMGd:d;d;e ZNGdd?d?eNZPGd@dAdAeNZQGdBdCdCeNZRGdDdEdEe ZSdFS)Hwraps)S)Add)cacheit)Expr)FunctionArgumentIndexError_mexpand)fuzzy_or fuzzy_not)RationalpiI)Pow)DummyWild)sympify) factorial)sincoscsccot)ceiling)explog)cbrtsqrtroot)Absreim polar_lift unpolarify)gammadigamma uppergamma)hyper)spherical_bessel_fn)mpworkprecc@s^eZdZdZeddZeddZeddZdd d Z d d Z d dZ ddZ ddZ dS) BesselBasea Abstract base class for Bessel-type functions. This class is meant to reduce code duplication. All Bessel-type functions can 1) be differentiated, with the derivatives expressed in terms of similar functions, and 2) be rewritten in terms of other Bessel-type functions. Here, Bessel-type functions are assumed to have one complex parameter. To use this base class, define class attributes ``_a`` and ``_b`` such that ``2*F_n' = -_a*F_{n+1} + b*F_{n-1}``. cCs |jdS)z( The order of the Bessel-type function. rargsselfr0Z/var/www/html/django/DPS/env/lib/python3.9/site-packages/sympy/functions/special/bessel.pyorder4szBesselBase.ordercCs |jdS)z+ The argument of the Bessel-type function. r,r.r0r0r1argument9szBesselBase.argumentcCsdSNr0clsnuzr0r0r1eval>szBesselBase.evalcCsN|dkrt|||jd||jd|j|jd||jd|jSNr;r3)r _b __class__r2r4_ar/argindexr0r0r1fdiffBs  zBesselBase.fdiffcCs*|j}|jdur&||j|SdSNF)r4is_extended_negativer>r2 conjugater/r9r0r0r1_eval_conjugateHs zBesselBase._eval_conjugatecCsx|j|j}}||rdS|||s,dS|||}|jrdt|ttt t t t fsZ|j sdt|jStt|j |jgSrC)r2r4has_eval_is_meromorphicsubs is_integer isinstancebesseljbesselihn1hn2jnynis_zeror is_infiniter )r/xar8r9z0r0r0r1rIMs    zBesselBase._eval_is_meromorphiccKs|j|j|j}}}|jr|djrn|j |j||d|d|j|d||d||S|djrd|j|d||d|||j|j||d|S|SNr3r;) r2r4r>is_real is_positiver?r=_eval_expand_func is_negative)r/hintsr8r9fr0r0r1r[Zs & &zBesselBase._eval_expand_funccKsddlm}||S)Nr) besselsimp)sympy.simplify.simplifyr_)r/kwargsr_r0r0r1_eval_simplifyes zBesselBase._eval_simplifyN)r;)__name__ __module__ __qualname____doc__propertyr2r4 classmethodr:rBrGrIr[rbr0r0r0r1r+$s      r+csdeZdZdZejZejZeddZ ddZ ddZ dd Z dd d Z ddZdfdd ZZS)rMa3 Bessel function of the first kind. Explanation =========== The Bessel $J$ function of order $\nu$ is defined to be the function satisfying Bessel's differential equation .. math :: z^2 \frac{\mathrm{d}^2 w}{\mathrm{d}z^2} + z \frac{\mathrm{d}w}{\mathrm{d}z} + (z^2 - \nu^2) w = 0, with Laurent expansion .. math :: J_\nu(z) = z^\nu \left(\frac{1}{\Gamma(\nu + 1) 2^\nu} + O(z^2) \right), if $\nu$ is not a negative integer. If $\nu=-n \in \mathbb{Z}_{<0}$ *is* a negative integer, then the definition is .. math :: J_{-n}(z) = (-1)^n J_n(z). Examples ======== Create a Bessel function object: >>> from sympy import besselj, jn >>> from sympy.abc import z, n >>> b = besselj(n, z) Differentiate it: >>> b.diff(z) besselj(n - 1, z)/2 - besselj(n + 1, z)/2 Rewrite in terms of spherical Bessel functions: >>> b.rewrite(jn) sqrt(2)*sqrt(z)*jn(n - 1/2, z)/sqrt(pi) Access the parameter and argument: >>> b.order n >>> b.argument z See Also ======== bessely, besseli, besselk References ========== .. [1] Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 9", Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables .. [2] Luke, Y. L. (1969), The Special Functions and Their Approximations, Volume 1 .. [3] https://en.wikipedia.org/wiki/Bessel_function .. [4] http://functions.wolfram.com/Bessel-TypeFunctions/BesselJ/ cCsZ|jrX|jrtjS|jr"|jdus,t|jr2tjSt|jrL|jdurLtjS|j rXtj S|tj tj fvrntjS| r||| | t|| S|jr| rtj| t| |S|t}|rt|t||S|jrt|}||kr:t||Sn8|\}}|dkr:td|t|tt||St|}||krVt||SdSNFTrr;)rSrOnerKr rZZeror\ComplexInfinity is_imaginaryNaNInfinityNegativeInfinitycould_extract_minus_signrM NegativeOneextract_multiplicativelyrrNr#extract_branch_factorrrr7r8r9ZnewznZnnur0r0r1r:s:      " z besselj.evalcKs(ttt|dt|tt |SNr;)rrrrNr"r/r8r9rar0r0r1_eval_rewrite_as_besselisz besselj._eval_rewrite_as_besselicKs<|jdur8tt|t| |tt|t||SdSrC)rKrrbesselyrrxr0r0r1_eval_rewrite_as_besselys z besselj._eval_rewrite_as_besselycKs"td|tt|tj|jSrw)rrrQrHalfr4rxr0r0r1_eval_rewrite_as_jnszbesselj._eval_rewrite_as_jnNrcCsR|j\}}||}||jvr:||d|t|dS||||dSdSNr;r3r)r-as_leading_term free_symbolsr$funcrJ)r/rUlogxcdirr8r9argr0r0r1_eval_as_leading_terms    zbesselj._eval_as_leading_termcCs|j\}}|jr|jrdSdSNTr-rKis_extended_realr/r8r9r0r0r1_eval_is_extended_reals  zbesselj._eval_is_extended_realc s$ddlm}|j\}}z||\}} WnttfyB|YS0| jrt|| } ||||} |d|||| } | t j ur| St | d|  } | |t |d}|g}td| ddD]4}|| |||9}t ||  }||qt|| Stt|||||SNr)Orderr;r3)sympy.series.orderrr-leadterm ValueErrorNotImplementedErrorrZr _eval_nseriesremoveOrrkr r$rangeappendrsuperrM)r/rUrvrrrr8r9_rnewnorttermskr>r0r1rs*       zbesselj._eval_nseries)Nr)r)rcrdrerfrrjr?r=rhr:ryr{r}rrr __classcell__r0r0rr1rMjsD # rMcsdeZdZdZejZejZeddZ ddZ ddZ dd Z dd d Z ddZdfdd ZZS)rza_ Bessel function of the second kind. Explanation =========== The Bessel $Y$ function of order $\nu$ is defined as .. math :: Y_\nu(z) = \lim_{\mu \to \nu} \frac{J_\mu(z) \cos(\pi \mu) - J_{-\mu}(z)}{\sin(\pi \mu)}, where $J_\mu(z)$ is the Bessel function of the first kind. It is a solution to Bessel's equation, and linearly independent from $J_\nu$. Examples ======== >>> from sympy import bessely, yn >>> from sympy.abc import z, n >>> b = bessely(n, z) >>> b.diff(z) bessely(n - 1, z)/2 - bessely(n + 1, z)/2 >>> b.rewrite(yn) sqrt(2)*sqrt(z)*yn(n - 1/2, z)/sqrt(pi) See Also ======== besselj, besseli, besselk References ========== .. [1] http://functions.wolfram.com/Bessel-TypeFunctions/BesselY/ cCsv|jr6|jrtjSt|jdur&tjSt|jr6tjS|tjtjfvrLtjS|jrr| rrtj | t | |SdSrC) rSrrpr rlrnrorkrKrqrrrzr6r0r0r1r:7s z bessely.evalcKs<|jdur8tt|tt|t||t| |SdSrC)rKrrrrMrxr0r0r1_eval_rewrite_as_besseljGs z bessely._eval_rewrite_as_besseljcKs|j|j}|r|tSdSr5)rr-rewriterNr/r8r9raajr0r0r1ryKs z bessely._eval_rewrite_as_besselicKs"td|tt|tj|jSrw)rrrRrr|r4rxr0r0r1_eval_rewrite_as_ynPszbessely._eval_rewrite_as_ynNrc Cs|j\}}dtt|dt||}|jrN|d|  t|dtntj}|d| tt|t|dtj }t |||gj ||d} || j vs|| j vr| S| |||dSdS)Nr;r3)rr)r-rrrMrZrrrkr% EulerGammarrrrrJcancel) r/rUrrr8r9Zterm_oneZterm_twoZ term_threerr0r0r1rSs ,,zbessely._eval_as_leading_termcCs|j\}}|jr|jrdSdSrr-rKrZrr0r0r1r^s  zbessely._eval_is_extended_realc s ddlm}|j\}}z||\}} WnttfyB|YS0| jr |jr t|| } t ||} dt t |d|  ||||} gg} }||||}|d |||| }|tjur|St|d| }|tjkrX|| t|dt }| |td|dD]8}||||d|9}t|| }| |q||t t|}|t|dtj}||td| ddD]V}|| |||9}t|| }|t||dt|d}||q| t| t|Stt| ||||Sr)rrr-rrrrZrKrrMrrrrrrkr rjrrrr%rrrrz)r/rUrvrrrr8r9rrrbnrVbcrrrrrprr0r1rcsB     $      zbessely._eval_nseries)Nr)r)rcrdrerfrrjr?r=rhr:rryrrrrrr0r0rr1rz s(  rzc@sJeZdZdZej ZejZeddZ ddZ ddZ dd Z d d Z d S) rNa Modified Bessel function of the first kind. Explanation =========== The Bessel $I$ function is a solution to the modified Bessel equation .. math :: z^2 \frac{\mathrm{d}^2 w}{\mathrm{d}z^2} + z \frac{\mathrm{d}w}{\mathrm{d}z} + (z^2 + \nu^2)^2 w = 0. It can be defined as .. math :: I_\nu(z) = i^{-\nu} J_\nu(iz), where $J_\nu(z)$ is the Bessel function of the first kind. Examples ======== >>> from sympy import besseli >>> from sympy.abc import z, n >>> besseli(n, z).diff(z) besseli(n - 1, z)/2 + besseli(n + 1, z)/2 See Also ======== besselj, bessely, besselk References ========== .. [1] http://functions.wolfram.com/Bessel-TypeFunctions/BesselI/ cCsV|jrX|jrtjS|jr"|jdus,t|jr2tjSt|jrL|jdurLtjS|j rXtj St |tj tj fvrrtjS|r||| | t|| S|jr|rt| |S|t}|rt| t|| S|jrt|}||kr6t||Sn8|\}}|dkr6td|t|tt||St|}||krRt||SdSri)rSrrjrKr rZrkr\rlrmrnr!rorprqrNrsrrMr#rtrrrur0r0r1r:s:       " z besseli.evalcKs(tt t|dt|tt|Srw)rrrrMr"rxr0r0r1rsz besseli._eval_rewrite_as_besseljcKs|j|j}|r|tSdSr5rr-rrzrr0r0r1r{s z besseli._eval_rewrite_as_besselycKs|j|jtSr5)rr-rrQrxr0r0r1r}szbesseli._eval_rewrite_as_jncCs|j\}}|jr|jrdSdSrrrr0r0r1rs  zbesseli._eval_is_extended_realN)rcrdrerfrrjr?r=rhr:rr{r}rr0r0r0r1rNs' #rNc@sReZdZdZejZej ZeddZ ddZ ddZ dd Z d d Z d d ZdS)besselka Modified Bessel function of the second kind. Explanation =========== The Bessel $K$ function of order $\nu$ is defined as .. math :: K_\nu(z) = \lim_{\mu \to \nu} \frac{\pi}{2} \frac{I_{-\mu}(z) -I_\mu(z)}{\sin(\pi \mu)}, where $I_\mu(z)$ is the modified Bessel function of the first kind. It is a solution of the modified Bessel equation, and linearly independent from $Y_\nu$. Examples ======== >>> from sympy import besselk >>> from sympy.abc import z, n >>> besselk(n, z).diff(z) -besselk(n - 1, z)/2 - besselk(n + 1, z)/2 See Also ======== besselj, besseli, bessely References ========== .. [1] http://functions.wolfram.com/Bessel-TypeFunctions/BesselK/ cCsv|jr6|jrtjSt|jdur&tjSt|jr6tjS|tjttjttjfvrXtjS|j rr| rrt | |SdSrC) rSrror rlrnrrprkrKrqrr6r0r0r1r:s z besselk.evalcKs8|jdur4ttt|t| |t||dSdS)NFr;)rKrrrNrxr0r0r1ry(s z besselk._eval_rewrite_as_besselicKs|j|j}|r|tSdSr5)ryr-rrM)r/r8r9raair0r0r1r,s z besselk._eval_rewrite_as_besseljcKs|j|j}|r|tSdSr5rrr0r0r1r{1s z besselk._eval_rewrite_as_besselycKs|j|j}|r|tSdSr5)r{r-rrR)r/r8r9raayr0r0r1r6s zbesselk._eval_rewrite_as_yncCs|j\}}|jr|jrdSdSrrrr0r0r1r;s  zbesselk._eval_is_extended_realN)rcrdrerfrrjr?r=rhr:ryrr{rrr0r0r0r1rs% rc@s$eZdZdZejZejZddZdS)hankel1a Hankel function of the first kind. Explanation =========== This function is defined as .. math :: H_\nu^{(1)} = J_\nu(z) + iY_\nu(z), where $J_\nu(z)$ is the Bessel function of the first kind, and $Y_\nu(z)$ is the Bessel function of the second kind. It is a solution to Bessel's equation. Examples ======== >>> from sympy import hankel1 >>> from sympy.abc import z, n >>> hankel1(n, z).diff(z) hankel1(n - 1, z)/2 - hankel1(n + 1, z)/2 See Also ======== hankel2, besselj, bessely References ========== .. [1] http://functions.wolfram.com/Bessel-TypeFunctions/HankelH1/ cCs(|j}|jdur$t|j|SdSrC)r4rDhankel2r2rErFr0r0r1rGis zhankel1._eval_conjugateN rcrdrerfrrjr?r=rGr0r0r0r1rAs$rc@s$eZdZdZejZejZddZdS)ra Hankel function of the second kind. Explanation =========== This function is defined as .. math :: H_\nu^{(2)} = J_\nu(z) - iY_\nu(z), where $J_\nu(z)$ is the Bessel function of the first kind, and $Y_\nu(z)$ is the Bessel function of the second kind. It is a solution to Bessel's equation, and linearly independent from $H_\nu^{(1)}$. Examples ======== >>> from sympy import hankel2 >>> from sympy.abc import z, n >>> hankel2(n, z).diff(z) hankel2(n - 1, z)/2 - hankel2(n + 1, z)/2 See Also ======== hankel1, besselj, bessely References ========== .. [1] http://functions.wolfram.com/Bessel-TypeFunctions/HankelH2/ cCs(|j}|jdur$t|j|SdSrC)r4rDrr2rErFr0r0r1rGs zhankel2._eval_conjugateNrr0r0r0r1ros%rcstfdd}|S)Ncs|jr|||SdSr5)rKrfnr0r1gszassume_integer_order..gr)rrr0rr1assume_integer_ordersrc@s*eZdZdZddZddZd ddZd S) SphericalBesselBasea- Base class for spherical Bessel functions. These are thin wrappers around ordinary Bessel functions, since spherical Bessel functions differ from the ordinary ones just by a slight change in order. To use this class, define the ``_eval_evalf()`` and ``_expand()`` methods. cKs tddS)z@ Expand self into a polynomial. Nu is guaranteed to be Integer. Z expansionNrr/r]r0r0r1_expandszSphericalBesselBase._expandcKs|jjr|jfi|S|Sr5)r2 is_Integerrrr0r0r1r[sz%SphericalBesselBase._eval_expand_funcr;cCs:|dkrt||||jd|j||jd|jSr<)r r>r2r4r@r0r0r1rBs  zSphericalBesselBase.fdiffN)r;)rcrdrerfrr[rBr0r0r0r1rs rcCs8t||t|tj|dt| d|t|SNr3)r(rrrrrrvr9r0r0r1_jns$rcCs8tj|dt| d|t|t||t|Sr)rrrr(rrrr0r0r1_yns$rc@sDeZdZdZeddZddZddZdd Zd d Z d d Z dS)rQa Spherical Bessel function of the first kind. Explanation =========== This function is a solution to the spherical Bessel equation .. math :: z^2 \frac{\mathrm{d}^2 w}{\mathrm{d}z^2} + 2z \frac{\mathrm{d}w}{\mathrm{d}z} + (z^2 - \nu(\nu + 1)) w = 0. It can be defined as .. math :: j_\nu(z) = \sqrt{\frac{\pi}{2z}} J_{\nu + \frac{1}{2}}(z), where $J_\nu(z)$ is the Bessel function of the first kind. The spherical Bessel functions of integral order are calculated using the formula: .. math:: j_n(z) = f_n(z) \sin{z} + (-1)^{n+1} f_{-n-1}(z) \cos{z}, where the coefficients $f_n(z)$ are available as :func:`sympy.polys.orthopolys.spherical_bessel_fn`. Examples ======== >>> from sympy import Symbol, jn, sin, cos, expand_func, besselj, bessely >>> z = Symbol("z") >>> nu = Symbol("nu", integer=True) >>> print(expand_func(jn(0, z))) sin(z)/z >>> expand_func(jn(1, z)) == sin(z)/z**2 - cos(z)/z True >>> expand_func(jn(3, z)) (-6/z**2 + 15/z**4)*sin(z) + (1/z - 15/z**3)*cos(z) >>> jn(nu, z).rewrite(besselj) sqrt(2)*sqrt(pi)*sqrt(1/z)*besselj(nu + 1/2, z)/2 >>> jn(nu, z).rewrite(bessely) (-1)**nu*sqrt(2)*sqrt(pi)*sqrt(1/z)*bessely(-nu - 1/2, z)/2 >>> jn(2, 5.2+0.3j).evalf(20) 0.099419756723640344491 - 0.054525080242173562897*I See Also ======== besselj, bessely, besselk, yn References ========== .. [1] http://dlmf.nist.gov/10.47 cCsD|jr*|jrtjS|jr*|jr$tjStjS|tjtjfvr@tjSdSr5) rSrrjrKrZrkrlrpror6r0r0r1r:szjn.evalcKs ttd|t|tj|Srw)rrrMrr|rxr0r0r1rszjn._eval_rewrite_as_besseljcKs,tj|ttd|t| tj|Srw)rrrrrrzr|rxr0r0r1r{szjn._eval_rewrite_as_besselycKstj|t| d|Sr)rrrrRrxr0r0r1rszjn._eval_rewrite_as_yncKst|j|jSr5)rr2r4rr0r0r1rsz jn._expandcCs|jjr|t|SdSr5r2rrrM _eval_evalfr/precr0r0r1r szjn._eval_evalfN) rcrdrerfrhr:rr{rrrr0r0r0r1rQs9 rQc@s@eZdZdZeddZeddZddZdd Zd d Z d S) rRa Spherical Bessel function of the second kind. Explanation =========== This function is another solution to the spherical Bessel equation, and linearly independent from $j_n$. It can be defined as .. math :: y_\nu(z) = \sqrt{\frac{\pi}{2z}} Y_{\nu + \frac{1}{2}}(z), where $Y_\nu(z)$ is the Bessel function of the second kind. For integral orders $n$, $y_n$ is calculated using the formula: .. math:: y_n(z) = (-1)^{n+1} j_{-n-1}(z) Examples ======== >>> from sympy import Symbol, yn, sin, cos, expand_func, besselj, bessely >>> z = Symbol("z") >>> nu = Symbol("nu", integer=True) >>> print(expand_func(yn(0, z))) -cos(z)/z >>> expand_func(yn(1, z)) == -cos(z)/z**2-sin(z)/z True >>> yn(nu, z).rewrite(besselj) (-1)**(nu + 1)*sqrt(2)*sqrt(pi)*sqrt(1/z)*besselj(-nu - 1/2, z)/2 >>> yn(nu, z).rewrite(bessely) sqrt(2)*sqrt(pi)*sqrt(1/z)*bessely(nu + 1/2, z)/2 >>> yn(2, 5.2+0.3j).evalf(20) 0.18525034196069722536 + 0.014895573969924817587*I See Also ======== besselj, bessely, besselk, jn References ========== .. [1] http://dlmf.nist.gov/10.47 cKs0tj|dttd|t| tj|SrX)rrrrrrMr|rxr0r0r1rTszyn._eval_rewrite_as_besseljcKs ttd|t|tj|Srw)rrrzrr|rxr0r0r1r{Xszyn._eval_rewrite_as_besselycKstj|dt| d|Sr)rrrrQrxr0r0r1r}\szyn._eval_rewrite_as_jncKst|j|jSr5)rr2r4rr0r0r1r_sz yn._expandcCs|jjr|t|SdSr5)r2rrrzrrr0r0r1rbszyn._eval_evalfN) rcrdrerfrrr{r}rrr0r0r0r1rR%s.  rRc@sLeZdZeddZeddZddZddZd d Zd d Z d dZ dS)SphericalHankelBasecKsN|j}ttd|t|tj||ttj|dt| tj|Sr<)_hankel_kind_signrrrMrr|rrrr/r8r9rahksr0r0r1ris&z,SphericalHankelBase._eval_rewrite_as_besseljcKsJ|j}ttd|tj|t| tj||tt|tj|Srw)rrrrrrrzr|rrr0r0r1r{rs(z,SphericalHankelBase._eval_rewrite_as_besselycKs(|j}t||t|tt||Sr5)rrQrrRrrr0r0r1r{sz'SphericalHankelBase._eval_rewrite_as_yncKs(|j}t|||tt||tSr5)rrQrrRrrr0r0r1r}sz'SphericalHankelBase._eval_rewrite_as_jncKsJ|jjr|jfi|S|j}|j}|j}t|||tt||SdSr5)r2rrr4rrQrrR)r/r]r8r9rr0r0r1r[s z%SphericalHankelBase._eval_expand_funccKs2|j}|j}|j}t|||tt||Sr5)r2r4rrrrexpand)r/r]rvr9rr0r0r1rs zSphericalHankelBase._expandcCs|jjr|t|SdSr5rrr0r0r1rszSphericalHankelBase._eval_evalfN) rcrdrerrr{rr}r[rrr0r0r0r1rgs   rc@s"eZdZdZejZeddZdS)rOa Spherical Hankel function of the first kind. Explanation =========== This function is defined as .. math:: h_\nu^(1)(z) = j_\nu(z) + i y_\nu(z), where $j_\nu(z)$ and $y_\nu(z)$ are the spherical Bessel function of the first and second kinds. For integral orders $n$, $h_n^(1)$ is calculated using the formula: .. math:: h_n^(1)(z) = j_{n}(z) + i (-1)^{n+1} j_{-n-1}(z) Examples ======== >>> from sympy import Symbol, hn1, hankel1, expand_func, yn, jn >>> z = Symbol("z") >>> nu = Symbol("nu", integer=True) >>> print(expand_func(hn1(nu, z))) jn(nu, z) + I*yn(nu, z) >>> print(expand_func(hn1(0, z))) sin(z)/z - I*cos(z)/z >>> print(expand_func(hn1(1, z))) -I*sin(z)/z - cos(z)/z + sin(z)/z**2 - I*cos(z)/z**2 >>> hn1(nu, z).rewrite(jn) (-1)**(nu + 1)*I*jn(-nu - 1, z) + jn(nu, z) >>> hn1(nu, z).rewrite(yn) (-1)**nu*yn(-nu - 1, z) + I*yn(nu, z) >>> hn1(nu, z).rewrite(hankel1) sqrt(2)*sqrt(pi)*sqrt(1/z)*hankel1(nu, z)/2 See Also ======== hn2, jn, yn, hankel1, hankel2 References ========== .. [1] http://dlmf.nist.gov/10.47 cKsttd|t||Srw)rrrrxr0r0r1_eval_rewrite_as_hankel1szhn1._eval_rewrite_as_hankel1N) rcrdrerfrrjrrrr0r0r0r1rOs0rOc@s$eZdZdZej ZeddZdS)rPa Spherical Hankel function of the second kind. Explanation =========== This function is defined as .. math:: h_\nu^(2)(z) = j_\nu(z) - i y_\nu(z), where $j_\nu(z)$ and $y_\nu(z)$ are the spherical Bessel function of the first and second kinds. For integral orders $n$, $h_n^(2)$ is calculated using the formula: .. math:: h_n^(2)(z) = j_{n} - i (-1)^{n+1} j_{-n-1}(z) Examples ======== >>> from sympy import Symbol, hn2, hankel2, expand_func, jn, yn >>> z = Symbol("z") >>> nu = Symbol("nu", integer=True) >>> print(expand_func(hn2(nu, z))) jn(nu, z) - I*yn(nu, z) >>> print(expand_func(hn2(0, z))) sin(z)/z + I*cos(z)/z >>> print(expand_func(hn2(1, z))) I*sin(z)/z - cos(z)/z + sin(z)/z**2 + I*cos(z)/z**2 >>> hn2(nu, z).rewrite(hankel2) sqrt(2)*sqrt(pi)*sqrt(1/z)*hankel2(nu, z)/2 >>> hn2(nu, z).rewrite(jn) -(-1)**(nu + 1)*I*jn(-nu - 1, z) + jn(nu, z) >>> hn2(nu, z).rewrite(yn) (-1)**nu*yn(-nu - 1, z) - I*yn(nu, z) See Also ======== hn1, jn, yn, hankel1, hankel2 References ========== .. [1] http://dlmf.nist.gov/10.47 cKsttd|t||Srw)rrrrxr0r0r1_eval_rewrite_as_hankel2 szhn2._eval_rewrite_as_hankel2N) rcrdrerfrrjrrrr0r0r0r1rPs0rPsympyc sddlm}dkrTddlmddlm}||fddtd|dDSd krdd lmzdd l m fd d }Wqt yddl m fdd }Yq0nt dfdd}|}|||}|g} t|dD]} ||||}| |q| S)a Zeros of the spherical Bessel function of the first kind. Explanation =========== This returns an array of zeros of $jn$ up to the $k$-th zero. * method = "sympy": uses `mpmath.besseljzero `_ * method = "scipy": uses the `SciPy's sph_jn `_ and `newton `_ to find all roots, which is faster than computing the zeros using a general numerical solver, but it requires SciPy and only works with low precision floating point numbers. (The function used with method="sympy" is a recent addition to mpmath; before that a general solver was used.) Examples ======== >>> from sympy import jn_zeros >>> jn_zeros(2, 4, dps=5) [5.7635, 9.095, 12.323, 15.515] See Also ======== jn, yn, besselj, besselk, bessely Parameters ========== n : integer order of Bessel function k : integer number of zeros to return r)rr) besseljzero) dps_to_preccs0g|](}ttdt|qS)g?)r _from_mpmathr _to_mpmathint).0l)rrvrr0r1 Cs zjn_zeros..r3scipy)newton) spherical_jncs |Sr5r0rU)rvrr0r1Jzjn_zeros..)sph_jncs|ddS)Nrr0r)rvrr0r1rMrUnknown method.cs dkr||}ntd|S)Nrrr)r^rUr)methodrr0r1solverQs zjn_zeros..solver)mathrmpmathrmpmath.libmp.libmpfrrZscipy.optimizerZ scipy.specialr ImportErrorrrr) rvrrdpsZmath_pirr^rrrootsir0)rrrvrrrrr1jn_zeross2-          rc@s4eZdZdZddZddZd ddZd d d Zd S)AiryBasezg Abstract base class for Airy functions. This class is meant to reduce code duplication. cCs||jdSNr)rr-rEr.r0r0r1rGlszAiryBase._eval_conjugatecCs |jdjSr)r-rr.r0r0r1roszAiryBase._eval_is_extended_realTcKsL|jd}|}|j}||||d}t||||d}||fS)Nrr;)r-rErr)r/deepr]r9Zzcr^uvr0r0r1 as_real_imagrs  zAiryBase.as_real_imagcKs&|jfd|i|\}}||tjS)Nr)rr ImaginaryUnit)r/rr]re_partim_partr0r0r1_eval_expand_complexzszAiryBase._eval_expand_complexN)T)T)rcrdrerfrGrrrr0r0r0r1rds  rc@s^eZdZdZdZdZeddZdddZe e dd Z d d Z d d Z ddZddZdS)airyaia The Airy function $\operatorname{Ai}$ of the first kind. Explanation =========== The Airy function $\operatorname{Ai}(z)$ is defined to be the function satisfying Airy's differential equation .. math:: \frac{\mathrm{d}^2 w(z)}{\mathrm{d}z^2} - z w(z) = 0. Equivalently, for real $z$ .. math:: \operatorname{Ai}(z) := \frac{1}{\pi} \int_0^\infty \cos\left(\frac{t^3}{3} + z t\right) \mathrm{d}t. Examples ======== Create an Airy function object: >>> from sympy import airyai >>> from sympy.abc import z >>> airyai(z) airyai(z) Several special values are known: >>> airyai(0) 3**(1/3)/(3*gamma(2/3)) >>> from sympy import oo >>> airyai(oo) 0 >>> airyai(-oo) 0 The Airy function obeys the mirror symmetry: >>> from sympy import conjugate >>> conjugate(airyai(z)) airyai(conjugate(z)) Differentiation with respect to $z$ is supported: >>> from sympy import diff >>> diff(airyai(z), z) airyaiprime(z) >>> diff(airyai(z), z, 2) z*airyai(z) Series expansion is also supported: >>> from sympy import series >>> series(airyai(z), z, 0, 3) 3**(5/6)*gamma(1/3)/(6*pi) - 3**(1/6)*z*gamma(2/3)/(2*pi) + O(z**3) We can numerically evaluate the Airy function to arbitrary precision on the whole complex plane: >>> airyai(-2).evalf(50) 0.22740742820168557599192443603787379946077222541710 Rewrite $\operatorname{Ai}(z)$ in terms of hypergeometric functions: >>> from sympy import hyper >>> airyai(z).rewrite(hyper) -3**(2/3)*z*hyper((), (4/3,), z**3/9)/(3*gamma(1/3)) + 3**(1/3)*hyper((), (2/3,), z**3/9)/(3*gamma(2/3)) See Also ======== airybi: Airy function of the second kind. airyaiprime: Derivative of the Airy function of the first kind. airybiprime: Derivative of the Airy function of the second kind. References ========== .. [1] https://en.wikipedia.org/wiki/Airy_function .. [2] http://dlmf.nist.gov/9 .. [3] http://www.encyclopediaofmath.org/index.php/Airy_functions .. [4] http://mathworld.wolfram.com/AiryFunctions.html r3TcCs|jr^|tjurtjS|tjur&tjS|tjur6tjS|jr^tjdtddt tddS|jrtjdtddt tddSdS)Nr;) is_NumberrrnrorkrprSrjr r$r7rr0r0r1r:s   "z airyai.evalcCs$|dkrt|jdSt||dSNr3r) airyaiprimer-r r@r0r0r1rBsz airyai.fdiffcGs:|dkrtjSt|}t|dkr|d}td|| td||dtt|tddtddt|t |dtddtt|tddtddt|dt |dtdd|Stj dtddtt |tj tdttddt|tj t|td||SdS)Nrr3rrr;) rrkrlenrrrr rr$rjrvrUprevious_termsrr0r0r1 taylor_terms" L@Hzairyai.taylor_termcKs`tdd}tdd}t| tdd}t|jr\|t| t| ||t|||SdSNr3rr;r rr r\rrMr/r9raotttrVr0r0r1rs    zairyai._eval_rewrite_as_besseljcKstdd}tdd}t|tdd}t|jrX|t|t| ||t|||S|t||t| |||t|| t|||SdSrr rr rZrrNrr0r0r1rys    *zairyai._eval_rewrite_as_besselicKs~tjdtddttdd}|tddttdd}|tgtddg|dd|tgtddg|ddS)Nrr;r3 r)rrjr r$rr'r/r9raZpf1Zpf2r0r0r1_eval_rewrite_as_hypers"zairyai._eval_rewrite_as_hyperc Ks |jd}|j}t|dkr|}td|gd}td|gd}td|gd}td|gd}||||||} | dur| |}d|jr| |}| |}| |}|||||||||} ||||||} tj| tj t | | tj t dt | SdS Nrr3r)excludedmrvr) r-rrpoprmatchrKrr|rjrrairybi r/r]rZsymbsr9rrr rvMpfnewargr0r0r1r[s$   $zairyai._eval_expand_funcN)r3rcrdrerfnargs unbranchedrhr:rB staticmethodrrrryrr[r0r0r0r1rsX   rc@s^eZdZdZdZdZeddZdddZe e dd Z d d Z d d Z ddZddZdS)r a The Airy function $\operatorname{Bi}$ of the second kind. Explanation =========== The Airy function $\operatorname{Bi}(z)$ is defined to be the function satisfying Airy's differential equation .. math:: \frac{\mathrm{d}^2 w(z)}{\mathrm{d}z^2} - z w(z) = 0. Equivalently, for real $z$ .. math:: \operatorname{Bi}(z) := \frac{1}{\pi} \int_0^\infty \exp\left(-\frac{t^3}{3} + z t\right) + \sin\left(\frac{t^3}{3} + z t\right) \mathrm{d}t. Examples ======== Create an Airy function object: >>> from sympy import airybi >>> from sympy.abc import z >>> airybi(z) airybi(z) Several special values are known: >>> airybi(0) 3**(5/6)/(3*gamma(2/3)) >>> from sympy import oo >>> airybi(oo) oo >>> airybi(-oo) 0 The Airy function obeys the mirror symmetry: >>> from sympy import conjugate >>> conjugate(airybi(z)) airybi(conjugate(z)) Differentiation with respect to $z$ is supported: >>> from sympy import diff >>> diff(airybi(z), z) airybiprime(z) >>> diff(airybi(z), z, 2) z*airybi(z) Series expansion is also supported: >>> from sympy import series >>> series(airybi(z), z, 0, 3) 3**(1/3)*gamma(1/3)/(2*pi) + 3**(2/3)*z*gamma(2/3)/(2*pi) + O(z**3) We can numerically evaluate the Airy function to arbitrary precision on the whole complex plane: >>> airybi(-2).evalf(50) -0.41230258795639848808323405461146104203453483447240 Rewrite $\operatorname{Bi}(z)$ in terms of hypergeometric functions: >>> from sympy import hyper >>> airybi(z).rewrite(hyper) 3**(1/6)*z*hyper((), (4/3,), z**3/9)/gamma(1/3) + 3**(5/6)*hyper((), (2/3,), z**3/9)/(3*gamma(2/3)) See Also ======== airyai: Airy function of the first kind. airyaiprime: Derivative of the Airy function of the first kind. airybiprime: Derivative of the Airy function of the second kind. References ========== .. [1] https://en.wikipedia.org/wiki/Airy_function .. [2] http://dlmf.nist.gov/9 .. [3] http://www.encyclopediaofmath.org/index.php/Airy_functions .. [4] http://mathworld.wolfram.com/AiryFunctions.html r3TcCs|jr^|tjurtjS|tjur&tjS|tjur6tjS|jr^tjdtddt tddS|jrtjdtddt tddSdS)Nrr3r;) rrrnrorprkrSrjr r$rr0r0r1r:s   "z airybi.evalcCs$|dkrt|jdSt||dSr) airybiprimer-r r@r0r0r1rBsz airybi.fdiffcGs|dkrtjSt|}t|dkr|d}td|tttddt|tj t |tj td|tj tt tddt|tj t |dtd|Stj t ddtt|tj tdtttddt|tj t |td||SdS)Nrr3rrr;r)rrkrrrrrr rrjrrr|rr$rr0r0r1rs @<Hzairybi.taylor_termcKs`tdd}tdd}t| tdd}t|jr\t| dt| ||t|||SdSrrrr0r0r1rs    zairybi._eval_rewrite_as_besseljcKstdd}tdd}t|tdd}t|jr\t|tdt| ||t|||St||}t|| }t||t| ||||t|||SdSrrr/r9rarrrVrrr0r0r1rys   .  zairybi._eval_rewrite_as_besselicKsztjtddttdd}|tddttdd}|tgtddg|dd|tgtddg|ddS)Nrrr;r3rr)rrjrr$r r'rr0r0r1rszairybi._eval_rewrite_as_hyperc Ks |jd}|j}t|dkr|}td|gd}td|gd}td|gd}td|gd}||||||} | dur| |}d|jr| |}| |}| |}|||||||||} ||||||} tjt dtj | t | tj | t | SdSr) r-rrr rr rKrr|rrjrr r r0r0r1r[s$   $zairybi._eval_expand_funcN)r3rr0r0r0r1r +sZ    r c@sVeZdZdZdZdZeddZdddZdd Z d d Z d d Z ddZ ddZ dS)ra+ The derivative $\operatorname{Ai}^\prime$ of the Airy function of the first kind. Explanation =========== The Airy function $\operatorname{Ai}^\prime(z)$ is defined to be the function .. math:: \operatorname{Ai}^\prime(z) := \frac{\mathrm{d} \operatorname{Ai}(z)}{\mathrm{d} z}. Examples ======== Create an Airy function object: >>> from sympy import airyaiprime >>> from sympy.abc import z >>> airyaiprime(z) airyaiprime(z) Several special values are known: >>> airyaiprime(0) -3**(2/3)/(3*gamma(1/3)) >>> from sympy import oo >>> airyaiprime(oo) 0 The Airy function obeys the mirror symmetry: >>> from sympy import conjugate >>> conjugate(airyaiprime(z)) airyaiprime(conjugate(z)) Differentiation with respect to $z$ is supported: >>> from sympy import diff >>> diff(airyaiprime(z), z) z*airyai(z) >>> diff(airyaiprime(z), z, 2) z*airyaiprime(z) + airyai(z) Series expansion is also supported: >>> from sympy import series >>> series(airyaiprime(z), z, 0, 3) -3**(2/3)/(3*gamma(1/3)) + 3**(1/3)*z**2/(6*gamma(2/3)) + O(z**3) We can numerically evaluate the Airy function to arbitrary precision on the whole complex plane: >>> airyaiprime(-2).evalf(50) 0.61825902074169104140626429133247528291577794512415 Rewrite $\operatorname{Ai}^\prime(z)$ in terms of hypergeometric functions: >>> from sympy import hyper >>> airyaiprime(z).rewrite(hyper) 3**(1/3)*z**2*hyper((), (5/3,), z**3/9)/(6*gamma(2/3)) - 3**(2/3)*hyper((), (1/3,), z**3/9)/(3*gamma(1/3)) See Also ======== airyai: Airy function of the first kind. airybi: Airy function of the second kind. airybiprime: Derivative of the Airy function of the second kind. References ========== .. [1] https://en.wikipedia.org/wiki/Airy_function .. [2] http://dlmf.nist.gov/9 .. [3] http://www.encyclopediaofmath.org/index.php/Airy_functions .. [4] http://mathworld.wolfram.com/AiryFunctions.html r3TcCsR|jr&|tjurtjS|tjur&tjS|jrNtjdtddttddSdS)Nrr3) rrrnrorkrSrrr r$rr0r0r1r:1s  zairyaiprime.evalcCs.|dkr |jdt|jdSt||dSr)r-rr r@r0r0r1rB<szairyaiprime.fdiffcCsR|jd|}t|tj|dd}Wdn1s<0Yt||SNrr3) derivative)r-rr*r)rrrr/rr9resr0r0r1rBs ,zairyaiprime._eval_evalfcKsPtdd}t| tdd}t|jrL|dt| ||t|||SdSNr;r)r rr r\rMr/r9rarrVr0r0r1rHs  z$airyaiprime._eval_rewrite_as_besseljcKstdd}tdd}|t|tdd}t|jrP|dt||t| |St|tdd}t||}t|| }||d|t||||t| ||SdSr)r rr rZrNrr0r0r1ryNs     z$airyaiprime._eval_rewrite_as_besselicKs|dddtddttdd}dtddttdd}|tgtddg|dd|tgtddg|ddS)Nr;rr3r)r r$rr'rr0r0r1rZs(z"airyaiprime._eval_rewrite_as_hyperc Ks |jd}|j}t|dkr|}td|gd}td|gd}td|gd}td|gd}||||||} | dur| |}d|jr| |}| |}| |}|||||||||} ||||||} tj| tj t | | tj t dt | SdSr) r-rrr rr rKrr|rjrrrr r0r0r1r[_s$   $zairyaiprime._eval_expand_funcN)r3rcrdrerfrrrhr:rBrrryrr[r0r0r0r1rsQ  rc@sVeZdZdZdZdZeddZdddZdd Z d d Z d d Z ddZ ddZ dS)ra< The derivative $\operatorname{Bi}^\prime$ of the Airy function of the first kind. Explanation =========== The Airy function $\operatorname{Bi}^\prime(z)$ is defined to be the function .. math:: \operatorname{Bi}^\prime(z) := \frac{\mathrm{d} \operatorname{Bi}(z)}{\mathrm{d} z}. Examples ======== Create an Airy function object: >>> from sympy import airybiprime >>> from sympy.abc import z >>> airybiprime(z) airybiprime(z) Several special values are known: >>> airybiprime(0) 3**(1/6)/gamma(1/3) >>> from sympy import oo >>> airybiprime(oo) oo >>> airybiprime(-oo) 0 The Airy function obeys the mirror symmetry: >>> from sympy import conjugate >>> conjugate(airybiprime(z)) airybiprime(conjugate(z)) Differentiation with respect to $z$ is supported: >>> from sympy import diff >>> diff(airybiprime(z), z) z*airybi(z) >>> diff(airybiprime(z), z, 2) z*airybiprime(z) + airybi(z) Series expansion is also supported: >>> from sympy import series >>> series(airybiprime(z), z, 0, 3) 3**(1/6)/gamma(1/3) + 3**(5/6)*z**2/(6*gamma(2/3)) + O(z**3) We can numerically evaluate the Airy function to arbitrary precision on the whole complex plane: >>> airybiprime(-2).evalf(50) 0.27879516692116952268509756941098324140300059345163 Rewrite $\operatorname{Bi}^\prime(z)$ in terms of hypergeometric functions: >>> from sympy import hyper >>> airybiprime(z).rewrite(hyper) 3**(5/6)*z**2*hyper((), (5/3,), z**3/9)/(6*gamma(2/3)) + 3**(1/6)*hyper((), (1/3,), z**3/9)/gamma(1/3) See Also ======== airyai: Airy function of the first kind. airybi: Airy function of the second kind. airyaiprime: Derivative of the Airy function of the first kind. References ========== .. [1] https://en.wikipedia.org/wiki/Airy_function .. [2] http://dlmf.nist.gov/9 .. [3] http://www.encyclopediaofmath.org/index.php/Airy_functions .. [4] http://mathworld.wolfram.com/AiryFunctions.html r3TcCs~|jrX|tjurtjS|tjur&tjS|tjur6tjS|jrXdtddttddS|jrzdtddttddSdS)Nrr3r) rrrnrorprkrSr r$rr0r0r1r:s   zairybiprime.evalcCs.|dkr |jdt|jdSt||dSr)r-r r r@r0r0r1rBszairybiprime.fdiffcCsR|jd|}t|tj|dd}Wdn1s<0Yt||Sr)r-rr*r)r rrrr0r0r1rs ,zairybiprime._eval_evalfcKsRtdd}|t| tdd}t|jrN| tdt| |t||SdSrrrr0r0r1rs  z$airybiprime._eval_rewrite_as_besseljcKstdd}tdd}|t|tdd}t|jrT|tdt| |t||St|tdd}t||}t|| }t||t| |||d|t|||SdSrrrr0r0r1rys   "  z$airybiprime._eval_rewrite_as_besselicKs||ddtddttdd}tddttdd}|tgtddg|dd|tgtddg|ddS)Nr;rrr3rr)rr$r r'rr0r0r1rs$z"airybiprime._eval_rewrite_as_hyperc Ks |jd}|j}t|dkr|}td|gd}td|gd}td|gd}td|gd}||||||} | dur| |}d|jr| |}| |}| |}|||||||||} ||||||} tjt d| tj t | | tj t | SdSr) r-rrr rr rKrr|rrjrrr r0r0r1r[s$   $zairybiprime._eval_expand_funcN)r3rr0r0r0r1rysS   rc@sFeZdZdZeddZdddZddZd d Zd d Z d dZ dS)marcumqa The Marcum Q-function. Explanation =========== The Marcum Q-function is defined by the meromorphic continuation of .. math:: Q_m(a, b) = a^{- m + 1} \int_{b}^{\infty} x^{m} e^{- \frac{a^{2}}{2} - \frac{x^{2}}{2}} I_{m - 1}\left(a x\right)\, dx Examples ======== >>> from sympy import marcumq >>> from sympy.abc import m, a, b >>> marcumq(m, a, b) marcumq(m, a, b) Special values: >>> marcumq(m, 0, b) uppergamma(m, b**2/2)/gamma(m) >>> marcumq(0, 0, 0) 0 >>> marcumq(0, a, 0) 1 - exp(-a**2/2) >>> marcumq(1, a, a) 1/2 + exp(-a**2)*besseli(0, a**2)/2 >>> marcumq(2, a, a) 1/2 + exp(-a**2)*besseli(0, a**2)/2 + exp(-a**2)*besseli(1, a**2) Differentiation with respect to $a$ and $b$ is supported: >>> from sympy import diff >>> diff(marcumq(m, a, b), a) a*(-marcumq(m, a, b) + marcumq(m + 1, a, b)) >>> diff(marcumq(m, a, b), b) -a**(1 - m)*b**m*exp(-a**2/2 - b**2/2)*besseli(m - 1, a*b) References ========== .. [1] https://en.wikipedia.org/wiki/Marcum_Q-function .. [2] http://mathworld.wolfram.com/MarcumQ-Function.html cCsZ|tjur@|tjur$|tjur$tjSt||dtjt|S|tjurn|tjurnddt|dtjS||kr|tjurdt|d td|dtjS|dkrtjtjt|d td|dt|d td|dS|jr,|jr|jrtjSt||dtjt|S|jrV|jrVddt|dtjSdSr~) rrkr&r|r$rrjrNrS)r7r rVrr0r0r1r:Ns"  &Dz marcumq.evalr;cCs|j\}}}|dkr6|t||| td|||S|dkr|| ||dt|d|d dt|d||St||dS)Nr;r3r)r-r rrNr )r/rAr rVrr0r0r1rBfs  "Bz marcumq.fdiffcKsjddlm}|dtd}|d||||t|d|d dt|d||||tjgS)Nr)IntegralrUr3r;)sympy.integrals.integralsr!getrrrNrro)r/r rVrrar!rUr0r0r1_eval_rewrite_as_Integralos   @z!marcumq._eval_rewrite_as_IntegralcKsbddlm}|dtd}t|d|d d||||t||||d|tjgS)Nr)Sumrr;r3)sympy.concrete.summationsr%r#rrrNrro)r/r rVrrar%rr0r0r1_eval_rewrite_as_Sumus zmarcumq._eval_rewrite_as_Sumc s|kr|dkr4dtd tdddS|jr|dkrtfddtd|D}tjtd tdddtd |SdS)Nr3r;rcsg|]}t|dqS)r;)rN)rrrVr0r1rrz4marcumq._eval_rewrite_as_besseli..)rrNrsumrrr|)r/r rVrrarr0r(r1ryzs $z marcumq._eval_rewrite_as_besselicCstdd|jDrdSdS)Ncss|] }|jVqdSr5)rS)rrr0r0r1 rz(marcumq._eval_is_zero..T)allr-r.r0r0r1 _eval_is_zeroszmarcumq._eval_is_zeroN)r;) rcrdrerfrhr:rBr$r'ryr,r0r0r0r1r s0  r N)rr)T functoolsr sympy.corerZsympy.core.addrsympy.core.cachersympy.core.exprrsympy.core.functionrr r sympy.core.logicr r sympy.core.numbersr rrsympy.core.powerrsympy.core.symbolrrsympy.core.sympifyr(sympy.functions.combinatorial.factorialsr(sympy.functions.elementary.trigonometricrrrr#sympy.functions.elementary.integersr&sympy.functions.elementary.exponentialrr(sympy.functions.elementary.miscellaneousrrr$sympy.functions.elementary.complexesrr r!r"r#'sympy.functions.special.gamma_functionsr$r%r&sympy.functions.special.hyperr'Zsympy.polys.orthopolysr(rr)r*r+rMrzrNrrrrrrrrQrRrrOrPrrrr rrr r0r0r0r1sb            F"`R./XB988 T-2%