\hSrSSKJr SSKJrJr SSKJr SSKJ r SSK J r J r J r SSKJr SSKJr SS KJrJrJrJr SS KJr SS KJr SS KJr SS KJr SSKJ r SSK!J"r"J#r#J$r$J%r%J&r&J'r'J(r(J)r)J*r*J+r+J,r,J-r-J.r. SSK!J/r/J0r0 SSK1J2r2J3r3J4r4 SSK5J6r6 SSK7J8r8 SSK9J:r: SSK;JJ?r? "SS\5r@SrA"SS\ 5rB"SS\B5rC"SS\B5rD"S S!\ 5rE"S"S#\E5rF"S$S%\E5rG"S&S'\E5rH"S(S)\E5rI"S*S+\E5rJ"S,S-\E5rK"S.S/\E5rL"S0S1\E5rM"S2S3\E5rN"S4S5\E5rO"S6S7\E5rP"S8S9\ 5rQg:);z%Hypergeometric and Meijer G-functions)Counter)SMod)Add)Expr)DefinedFunction DerivativeArgumentIndexError)Tuple)Mul)Ipioozoo)global_parameters)Ne)default_sort_key)Dummy)lcm) sqrtexplogsincosasinatansinhcoshasinhacoshatanhacoth) factorialRisingFactorial)Absre unpolarify) exp_polar)ceiling) Piecewise)AndOr)orderedc.\rSrSrSSS.SjrSSjrSrg) TupleArgNrlogxcdirc p[URVs/sHoDR"X1US.6PM sn6$s snf)Nr1)r/argsas_leading_term)selfr2r3xfs U/var/www/auris/envauris/lib/python3.13/site-packages/sympy/functions/special/hyper.pyr6TupleArg.as_leading_term%s.tyyYy!++QEyYZZYs3c jSSKJn [URVs/sH oT"XQX#5PM sn6$s snf)zCompute limit x->xlim. r)limit)sympy.series.limitsr=r/r5)r7r8xlimdirr=r9s r:r=TupleArg.limit(s/ .$))D)Q%d0)DEEDs0)+)__name__ __module__ __qualname____firstlineno__r6r=__static_attributes__rBr:r/r/s(,![FrIr/cP[UVs/sHn[U5PM sn6$s snf)aI Turn an iterable argument *v* into a tuple and unpolarify, since both hypergeometric and meijer g-functions are unbranched in their parameters. Examples ======== >>> from sympy.functions.special.hyper import _prep_tuple >>> _prep_tuple([1, 2, 3]) (1, 2, 3) >>> _prep_tuple((4, 5)) (4, 5) >>> _prep_tuple((7, 8, 9)) (7, 8, 9) )r/r')vr8s r: _prep_tuplerL3s%" Q/QjmQ/ 00/s#c"\rSrSrSrSrSrSrg)TupleParametersBaseGz_Base class that takes care of differentiation, when some of the arguments are actually tuples. TcSnURSRU5(d#URSRU5(a\[UR5HCup4URUR U5nUS:wdM+X R SU45U-- nME X R S5URSR U5--$![ [4a [X5s$f=f)Nr) r5has enumerate _diffargsdifffdiffr NotImplementedErrorr )r7sresipms r:_eval_derivative$TupleParametersBase._eval_derivativeMs 'Cyy|""diil&6&6q&9&9%dnn5DAq)..q1AAvzz1a&1!336Atyy|'8'8';;; ;"$78 'd& & 'sBC A CC65C6rBN)rDrErFrG__doc__is_commutativer_rHrBrIr:rNrNGs/N 'rIrNc^\rSrSrSrU4Sjr\S5rSSjrSr Sr U4Sjr SU4S jjr \ S 5r\ S 5r\ S 5r\ S 5r\ S5r\ S5r\ S5rSrSrU=r$)hyperZa1 The generalized hypergeometric function is defined by a series where the ratios of successive terms are a rational function of the summation index. When convergent, it is continued analytically to the largest possible domain. Explanation =========== The hypergeometric function depends on two vectors of parameters, called the numerator parameters $a_p$, and the denominator parameters $b_q$. It also has an argument $z$. The series definition is .. math :: {}_pF_q\left(\begin{matrix} a_1, \cdots, a_p \\ b_1, \cdots, b_q \end{matrix} \middle| z \right) = \sum_{n=0}^\infty \frac{(a_1)_n \cdots (a_p)_n}{(b_1)_n \cdots (b_q)_n} \frac{z^n}{n!}, where $(a)_n = (a)(a+1)\cdots(a+n-1)$ denotes the rising factorial. If one of the $b_q$ is a non-positive integer then the series is undefined unless one of the $a_p$ is a larger (i.e., smaller in magnitude) non-positive integer. If none of the $b_q$ is a non-positive integer and one of the $a_p$ is a non-positive integer, then the series reduces to a polynomial. To simplify the following discussion, we assume that none of the $a_p$ or $b_q$ is a non-positive integer. For more details, see the references. The series converges for all $z$ if $p \le q$, and thus defines an entire single-valued function in this case. If $p = q+1$ the series converges for $|z| < 1$, and can be continued analytically into a half-plane. If $p > q+1$ the series is divergent for all $z$. Please note the hypergeometric function constructor currently does *not* check if the parameters actually yield a well-defined function. Examples ======== The parameters $a_p$ and $b_q$ can be passed as arbitrary iterables, for example: >>> from sympy import hyper >>> from sympy.abc import x, n, a >>> h = hyper((1, 2, 3), [3, 4], x); h hyper((1, 2), (4,), x) >>> hyper((3, 1, 2), [3, 4], x, evaluate=False) # don't remove duplicates hyper((1, 2, 3), (3, 4), x) There is also pretty printing (it looks better using Unicode): >>> from sympy import pprint >>> pprint(h, use_unicode=False) _ |_ /1, 2 | \ | | | x| 2 1 \ 4 | / The parameters must always be iterables, even if they are vectors of length one or zero: >>> hyper((1, ), [], x) hyper((1,), (), x) But of course they may be variables (but if they depend on $x$ then you should not expect much implemented functionality): >>> hyper((n, a), (n**2,), x) hyper((a, n), (n**2,), x) The hypergeometric function generalizes many named special functions. The function ``hyperexpand()`` tries to express a hypergeometric function using named special functions. For example: >>> from sympy import hyperexpand >>> hyperexpand(hyper([], [], x)) exp(x) You can also use ``expand_func()``: >>> from sympy import expand_func >>> expand_func(x*hyper([1, 1], [2], -x)) log(x + 1) More examples: >>> from sympy import S >>> hyperexpand(hyper([], [S(1)/2], -x**2/4)) cos(x) >>> hyperexpand(x*hyper([S(1)/2, S(1)/2], [S(3)/2], x**2)) asin(x) We can also sometimes ``hyperexpand()`` parametric functions: >>> from sympy.abc import a >>> hyperexpand(hyper([-a], [], x)) (1 - x)**a See Also ======== sympy.simplify.hyperexpand gamma meijerg References ========== .. [1] Luke, Y. L. (1969), The Special Functions and Their Approximations, Volume 1 .. [2] https://en.wikipedia.org/wiki/Generalized_hypergeometric_function c >URS[R5(as[[ U65n[[ U65nXV-n//4=nup[ XV45H4upX-n [ U 5Hn XRU /X-5 M M6 O([[ U55n[[ U55n[T U]("U[U5[U5U40UD6$)Nevaluate) poprrgrr rUr-extendlistsuper__new__rL) clsapbqzkwargscacbcommonargr\ck __class__s r:rl hyper.__new__s ::j"3"<"< = =$B$BWFr6 !C&"!2(+  AFMM1#ad(+$, gbk"Bgbk"BwsKO[_aR6RRrIc[U5[U5::d-[U5[U5S-:Xa0[U5S:*S:Xa[U5nX4:wa [XU5$ggg)NrQT)lenr%r'rd)rmrnrorpnzs r:eval hyper.evals] r7c"g #b'SWq["8c!fkd=RABwRR((>S"8rIcHUS:wa [X5e[URVs/sHo"S-PM sn6n[URVs/sHoDS-PM sn6n[ UR6[ UR6- nU[ X5UR 5-$s snfs snf)NrRrQ)r r rnror rdargument)r7argindexanapbnbqfacs r:rX hyper.fdiffs q=$T4 4TWW-W!eW-.TWW-W!eW-.477mCM)54==111.-s B Bc @SSKJn SSKJn [ UR 5S:Xar[ UR 5S:XaYURS:XaIUR upEUR SnU"U5U"Xd- U- 5-U"Xd- 5- U"Xe- 5- $U"U5$)Nrgamma hyperexpandrSrQ)'sympy.functions.special.gamma_functionsrsympy.simplify.hyperexpandrr{rnror)r7hintsrrrrrvs r:_eval_expand_funchyper._eval_expand_funcsA: tww<1 TWW!2t}}7I77DA A8E!%!),,U15\9%,F F4  rIc 6SSKJn [SSS9nUVs/sHn[Xv5PM nnUV s/sHn [X5PM n n [ U6[ U 6- n [ U"XU--[ U5- US[45UR4US45$s snfs sn f)Nr)SumnT)integer) sympy.concrete.summationsrrr$r r*r#rconvergence_statement) r7rnrorprqrrrrfaprrfbqcoeffs r:_eval_rewrite_as_Sumhyper._eval_rewrite_as_Sums1 #t $/12r!%r2/12r!%r2T S$Z'#edlYq\9Aq":F3357;TlD D32s BBc(>URSnURUS5nU[RLa-UR US[ U5R (aSOSS9nU[RLa[R$[TU])XUS9$)NrSr-rC)r@r1) r5subsrNaNr=r& is_negativeZeroOnerk_eval_as_leading_term)r7r8r2r3rux0rxs r:rhyper._eval_as_leading_termsuiil XXa^ ;1aBtH,@,@ScJB <55Lw,Q,EErIc >SSKJn URSnURUS5nURSnURSn Xa:XaUS:XdSSKJn U "[ TU]XU55$/n [U5Hln [UV s/sHn [X5PM sn 6n[U Vs/sHn[X5PM sn6nU RUU- Xl--[U 5- 5 Mn [U 6U"X-U5-$s sn fs snf)Nr)OrderrSrQr)sympy.series.orderrr5r=rrrk _eval_nseriesranger r$appendr#r)r7r8rr2r3rrurrnrortermsr\rnumrdenrxs r:rhyper._eval_nseries s,iil YYq!_ YYq\ YYq\R1W ?uw4Q4@A AqAr:r!-r:;Cr:r!-r:;C LL3s7sv.)A,> ? U eADm+, ;:s C? )D c URS$)z)Argument of the hypergeometric function. rSr5r7s r:rhyper.argument%yy|rIc,[URS6$)z5Numerator parameters of the hypergeometric function. rr r5rs r:rnhyper.ap*diil##rIc,[URS6$)z7Denominator parameters of the hypergeometric function. rQrrs r:rohyper.bq/rrIc4URUR-$Nrnrors r:rVhyper._diffargs4ww  rIcX[UR5[UR5- $)z5A quantity related to the convergence of the series. )sumrnrors r:eta hyper.eta8s477|c$''l**rIcp[SURUR-55(aURVs/sH"oR(dMUS:*S:XdM UPM$ nnURVs/sH"oR(dMUS:*S:XdM UPM$ nn[ U5[ U5:a[ R $SnUHHnSnU(a#UR5nX:aSnO SnU(aM#U(aM8[ R s $ U(dU(a[$[ UR5[ UR5S-:Xa[ R$[ UR5[ UR5::a[$[ R $s snfs snf)a Compute the radius of convergence of the defining series. Explanation =========== Note that even if this is not ``oo``, the function may still be evaluated outside of the radius of convergence by analytic continuation. But if this is zero, then the function is not actually defined anywhere else. Examples ======== >>> from sympy import hyper >>> from sympy.abc import z >>> hyper((1, 2), [3], z).radius_of_convergence 1 >>> hyper((1, 2, 3), [4], z).radius_of_convergence 0 >>> hyper((1, 2), (3, 4), z).radius_of_convergence oo c3V# UHoR=(a US:*S:Hv M! g7f)rTN) is_integer).0rs r: .hyper.radius_of_convergence..Ws%L:KQ||0aD 00:Ks')rTFrQ) anyrnro is_Integerr{rrrhrr)r7raintsbintspoppedr cancelleds r:radius_of_convergencehyper.radius_of_convergence=s;4 L$''DGG:KL L L $M1< A!SVaZ (""~rIc SSKJn U"U5$Nrrrr)r7rqrs r:_eval_simplifyhyper._eval_simplify:4  rIrBrR)r)rDrErFrGrarl classmethodr}rXrrrrpropertyrrnrorVrrrrrH __classcell__rxs@r:rdrdZsslS )) 2!DF-2$$$$!!++22h  !!rIrdc ^\rSrSrSrU4SjrSSjrSrSrSr Sr S r S r \ S 5r\ S 5r\ S 5r\ S5r\ S5r\ S5r\ S5r\ S5r\ S5r\ S5r\ S5rSrU=r$)meijergia The Meijer G-function is defined by a Mellin-Barnes type integral that resembles an inverse Mellin transform. It generalizes the hypergeometric functions. Explanation =========== The Meijer G-function depends on four sets of parameters. There are "*numerator parameters*" $a_1, \ldots, a_n$ and $a_{n+1}, \ldots, a_p$, and there are "*denominator parameters*" $b_1, \ldots, b_m$ and $b_{m+1}, \ldots, b_q$. Confusingly, it is traditionally denoted as follows (note the position of $m$, $n$, $p$, $q$, and how they relate to the lengths of the four parameter vectors): .. math :: G_{p,q}^{m,n} \left(\begin{matrix}a_1, \cdots, a_n & a_{n+1}, \cdots, a_p \\ b_1, \cdots, b_m & b_{m+1}, \cdots, b_q \end{matrix} \middle| z \right). However, in SymPy the four parameter vectors are always available separately (see examples), so that there is no need to keep track of the decorating sub- and super-scripts on the G symbol. The G function is defined as the following integral: .. math :: \frac{1}{2 \pi i} \int_L \frac{\prod_{j=1}^m \Gamma(b_j - s) \prod_{j=1}^n \Gamma(1 - a_j + s)}{\prod_{j=m+1}^q \Gamma(1- b_j +s) \prod_{j=n+1}^p \Gamma(a_j - s)} z^s \mathrm{d}s, where $\Gamma(z)$ is the gamma function. There are three possible contours which we will not describe in detail here (see the references). If the integral converges along more than one of them, the definitions agree. The contours all separate the poles of $\Gamma(1-a_j+s)$ from the poles of $\Gamma(b_k-s)$, so in particular the G function is undefined if $a_j - b_k \in \mathbb{Z}_{>0}$ for some $j \le n$ and $k \le m$. The conditions under which one of the contours yields a convergent integral are complicated and we do not state them here, see the references. Please note currently the Meijer G-function constructor does *not* check any convergence conditions. Examples ======== You can pass the parameters either as four separate vectors: >>> from sympy import meijerg, Tuple, pprint >>> from sympy.abc import x, a >>> pprint(meijerg((1, 2), (a, 4), (5,), [], x), use_unicode=False) __1, 2 /1, 2 4, a | \ /__ | | x| \_|4, 1 \ 5 | / Or as two nested vectors: >>> pprint(meijerg([(1, 2), (3, 4)], ([5], Tuple()), x), use_unicode=False) __1, 2 /1, 2 3, 4 | \ /__ | | x| \_|4, 1 \ 5 | / As with the hypergeometric function, the parameters may be passed as arbitrary iterables. Vectors of length zero and one also have to be passed as iterables. The parameters need not be constants, but if they depend on the argument then not much implemented functionality should be expected. All the subvectors of parameters are available: >>> from sympy import pprint >>> g = meijerg([1], [2], [3], [4], x) >>> pprint(g, use_unicode=False) __1, 1 /1 2 | \ /__ | | x| \_|2, 2 \3 4 | / >>> g.an (1,) >>> g.ap (1, 2) >>> g.aother (2,) >>> g.bm (3,) >>> g.bq (3, 4) >>> g.bother (4,) The Meijer G-function generalizes the hypergeometric functions. In some cases it can be expressed in terms of hypergeometric functions, using Slater's theorem. For example: >>> from sympy import hyperexpand >>> from sympy.abc import a, b, c >>> hyperexpand(meijerg([a], [], [c], [b], x), allow_hyper=True) x**c*gamma(-a + c + 1)*hyper((-a + c + 1,), (-b + c + 1,), -x)/gamma(-b + c + 1) Thus the Meijer G-function also subsumes many named functions as special cases. You can use ``expand_func()`` or ``hyperexpand()`` to (try to) rewrite a Meijer G-function in terms of named special functions. For example: >>> from sympy import expand_func, S >>> expand_func(meijerg([[],[]], [[0],[]], -x)) exp(x) >>> hyperexpand(meijerg([[],[]], [[S(1)/2],[0]], (x/2)**2)) sin(x)/sqrt(pi) See Also ======== hyper sympy.simplify.hyperexpand References ========== .. [1] Luke, Y. L. (1969), The Special Functions and Their Approximations, Volume 1 .. [2] https://en.wikipedia.org/wiki/Meijer_G-function c>^[U5S:XaUSUS4USUS4US/n[U5S:wa [S5eSnU"US5U"US5snm[UT5R[[ [*5(a [ S 5e[U4S jUS55(a [ S 5e[TU]$"XTUS40UD6$) NrrQrSrRz4args must be either as, as', bs, bs', z or as, bs, zc[U5S:wa [S5eUVs/sHn[[U55PM nn[ [ US5[ US55$s snf)NrSzwrong argumentrrQ)r{ TypeErrorrjr-r/rL)r]r\s r:trmeijerg.__new__..trsZ1v{ 011+,-1agaj!1A-K!-{1Q4/@A A.sA#z$G-function parameters must be finitec3v># UH.nTSH!nX- R=(a X- S:v M# M0 g7f)rN)r)rrrarg1s r:r"meijerg.__new__..s>21aA!!/aeai/)00s69zNno parameter a1, ..., an may differ from any b1, ..., bm by a positive integer) r{rr rTrr ValueErrorrrkrl)rmr5rqrarg0rrxs @r:rlmeijerg.__new__ s t9>!Wd1g&a$q'(:DGDD t9>() ) B Q["T!W+ d t  S2# . .CD D 2Q2 2 2AB Bws$QB6BBrIcUS:waURUS5$[UR5S:a[UR5nUS==S-ss'[ X R UR URUR5nSUR- URSS- U-U--$[UR 5S:a}[UR 5nUS==S- ss'[ URUR X@RUR5nSUR- UR SU-U- -$[R$)NrRrQr) _diff_wrt_parameterr{anrjraotherbmbotherrrr)r7rrGrs r:rX meijerg.fdiff"s q=++HQK8 8 tww<1 TWW A aDAID;;dmmLAT]]?twwqzA~t&;a&?@ @ \Q TWW A aDAIDadmmLAT]]?dggajo&9: :66MrIc [UR5n[UR5n[UR5n[UR5nU[ U5:aUR U5 O{U[ U5-nU[ U5:aUR U5 OLU[ U5-nU[ U5:aUR U5 OUR U[ U5- 5 /n/nX%U4X4U44Hupn U(dMUR 5n Sn [U 5H*up[X- R5S5(aM(U n O U c [S5eU W nU R U 5 U RX45 U(aMM [UR5U-nUHunnSnUU- nUnUS:a SnUU- nUn[U5H^nUU[URUU-S-4-URURURUU-S-4-UR5--nM` M UHunnSnUU- nUnUS:a SnUU- nUn[U5H^nUU[URURUU-S-4-URUU-S-4-URUR5--nM` M U$)NrQz)Derivative not expressible as G-function?r)rjrrrrr{rhrUrsimplifyrYrrrrr)r7idxrrnrropairs1pairs2l1l2pairsr8foundr\yr[rrsignrbaserws r:rmeijerg._diff_wrt_parameter2s,$''] $++  $''] $++  R= FF3K 3r7NCSW}s s2wR=FF3KFF3R=)!v.0@AMBE"FFH%bMDA//1155 !*=-/?@@qEq  aV$"B $-- %DAqDAAD1uE1XtGDGGtax!|o$=t{{$(GGT[[D1HqL?-J$(MM333DAqDAAD1uE1XtGDGGT[[D1HqL?-J$(GGtax!|o$=t{{$(MM333 rIc2SnU"UR5nU"UR5n[UR5[UR5pTXE:Xa&[ X24;a[ $S[ -[X25-$XE:a S[ -U-$S[ -U-$)am Return a number $P$ such that $G(x*exp(I*P)) == G(x)$. Examples ======== >>> from sympy import meijerg, pi, S >>> from sympy.abc import z >>> meijerg([1], [], [], [], z).get_period() 2*pi >>> meijerg([pi], [], [], [], z).get_period() oo >>> meijerg([1, 2], [], [], [], z).get_period() oo >>> meijerg([1,1], [2], [1, S(1)/2, S(1)/3], [1], z).get_period() 12*pi c[U5HkupUR(d[s $[US-[ U55H1n[ X U- R 5S5(aM)[s s $ Mm [SU56$)NrQc38# UHoRv M g7fr)q)rr8s r:r6meijerg.get_period..compute..s(aas)rU is_Rationalrrr{rrr)lr\rjs r:compute#meijerg.get_period..computesk!! }}Iq1uc!f-AaD224a88! .% (a() )rIrS)rrr{rnrorrr)r7rbetaalphar]rs r: get_periodmeijerg.get_periods* *tww 477|S\1 6e]" R4E(( ( UR49 R4: rIc SSKJn U"U5$rr)r7rrs r:rmeijerg._eval_expand_funcrrIc SSKnURRU5nUR[5(aAUR [5up4[ U5S:wagUSRS[- nO[Rn[[U[- 55S-nU[RU- -[[U-U- 5-nUSU- URSURS4Vs/sHnUR!U5PM snupxpUR%U5 UR'XXx5n SSS5 [(R*"W U5$s snf!["a gf=f!,(df  N:=f)NrrQ)mpmathr _eval_evalfrTr( as_coeff_mulr{r5r rrr)rrrr _to_mpmathrworkprecrr _from_mpmath) r7precrznumbranchrrurprrnrorKs r:rmeijerg._eval_evalfsL }}((. 88I  ,,Y7LD6{aAY^^A&q(FVVF Cr N #a 'aeeAgs1V8a<00  $ac499Q<1FHF"nnT2FHNQ2 __T "rq,A#  D))H   # "s0%E$;EE$-E4E$$ E10E14 Fc:SSKJn U"U5RXUS9$)Nrrr1)rrr6)r7r8r2r3rs r:rmeijerg._eval_as_leading_terms :4 00D0IIrIc ^^SSKJm URT-[UU4SjUR56-[UU4SjUR 56-[UU4SjUR 56- [UU4SjUR56- $)z!Get the defining integrand D(s). rrc3:># UHnT"UT- 5v M g7frrBrrrrZs r:r$meijerg.integrand..s2'QE!a%LL'c3@># UHnT"SU- T-5v M g7frQNrBrrrrZs r:rr,s!6gE!a%!)$$gc3@># UHnT"SU- T-5v M g7fr/rBr+s r:rr,s!:kE!a%!)$$kr1c3:># UHnT"UT- 5v M g7frrBr0s r:rr,s6+QE!a%LL+r-)rrrr rrrr)r7rZrs `@r: integrandmeijerg.integrandswA}}a2$''2346dgg678:dkk:;<6$++67 8 8rIc URS$)z#Argument of the Meijer G-function. rSrrs r:rmeijerg.argumentrrIc2[URSS6$)z#First set of numerator parameters. rrrs r:r meijerg.andiil1o&&rIcX[URSSURSS-6$)zCombined numerator parameters. rrQrrs r:rn meijerg.ap,tyy|A1a8::rIc2[URSS6$)z$Second set of numerator parameters. rrQrrs r:rmeijerg.aotherr:rIc2[URSS6$)z%First set of denominator parameters. rQrrrs r:r meijerg.bmr:rIcX[URSSURSS-6$)z!Combined denominator parameters. rQrrrs r:ro meijerg.bqr=rIc2[URSS6$)z&Second set of denominator parameters. rQrrs r:rmeijerg.botherr:rIc4URUR-$rrrs r:rVmeijerg._diffargsrrIcX[UR5[UR5- $)OA quantity related to the convergence region of the integral, c.f. references. )rrornrs r:nu meijerg.nu s477|c$''l**rIc[UR5[UR5-[[UR5[UR 5-5S- - $)rIrS)r{rrrrnrors r:delta meijerg.deltasC477|c$''l*Qs477|c$''l/J-KA-MMMrIc$UR(+$)z2Returns true if expression has numeric data only. ) free_symbolsrs r: is_numbermeijerg.is_numbers$$$$rIrBr)rDrErFrGrarlrXrrrrrr4rrrrnrrrorrVrJrMrQrHrrs@r:rrsDC0 Tl(T!*@J8'';;'''';;''!!++ NN %%rIrct\rSrSrSr\S5r\S5r\S5r\S5r \S5r Sr S r S r g ) HyperRepia A base class for "hyper representation functions". This is used exclusively in ``hyperexpand()``, but fits more logically here. pFq is branched at 1 if p == q+1. For use with slater-expansion, we want define an "analytic continuation" to all polar numbers, which is continuous on circles and on the ray t*exp_polar(I*pi). Moreover, we want a "nice" expression for the various cases. This base class contains the core logic, concrete derived classes only supply the actual functions. c\[[[USS55USS-nX:waU"U6$g)Nr)tuplemapr')rmr5newargss r:r} HyperRep.eval-s8JSb 23d23i? ?=  rIc[e)z0An expression for F(x) which holds for |x| < 1. rYrmr8s r: _expr_smallHyperRep._expr_small3 "!rIc[e)z1An expression for F(-x) which holds for |x| < 1. r[r\s r:_expr_small_minusHyperRep._expr_small_minus8r_rIc[e)z5An expression for F(exp_polar(2*I*pi*n)*x), |x| > 1. r[rmr8rs r: _expr_bigHyperRep._expr_big=r_rIc[e)z 1. r[rds r:_expr_big_minusHyperRep._expr_big_minusBr_rIcURSRSS9up4SnURSSU4-nUR(dSnU[R-nXd4-nU(aUR "U6nUR "U6n OUR"U6nUR"U6n X:XaU$[U [U5S:4US45$)NrT allow_halfFrQ) r5extract_branch_factorrrHalfrarhr]rer*r) r7r5rqr8rminusrX newerargssmallbigs r:_eval_rewrite_as_nonrep HyperRep._eval_rewrite_as_nonrepGsyy}22d2C))CR.A4'||E KAdN **G4E&& 2C$$g.E..),C <L#s1vz*UDM::rIcURSRSS9up4URSSU4-nUR(dUR"U6$UR"U6$)NrTrk)r5rmrrar])r7r5rqr8rs r:_eval_rewrite_as_nonrepsmall%HyperRep._eval_rewrite_as_nonrepsmallZs^yy}22d2Cyy"~$||))40 0&&rIrBN)rDrErFrGrarr}r]rarerhrsrvrHrBrIr:rTrTsu  !! """""""";&'rIrTcX\rSrSrSr\S5r\S5r\S5r\S5r Sr g) HyperRep_power1ibz>Return a representative for hyper([-a], [], z) == (1 - z)**a. cSU- U-$NrQrBrmrr8s r:r]HyperRep_power1._expr_smalleAzrIcSU-U-$r{rBr|s r:ra!HyperRep_power1._expr_small_minusir~rIcUR(aURX5$US- U-[SU-S- [-[-U-5-$NrQrS)rr]rrr rmrr8rs r:reHyperRep_power1._expr_bigmsE <<??1( (Az#qsQwl1nQ.///rIcUR(aURX5$SU-U-[SU-[-[-U-5-$r)rrarrr rs r:rhHyperRep_power1._expr_big_minusssA <<((. .Az#ac"fQhqj/))rIrBN rDrErFrGrarr]rarerhrHrBrIr:ryrybsSI00 **rIrycX\rSrSrSr\S5r\S5r\S5r\S5r Sr g) HyperRep_power2izz;Return a representative for hyper([a, a - 1/2], [2*a], z). cNSSU-S- -S[SU- 5-SSU-- --$NrSrQrr|s r:r]HyperRep_power2._expr_small}21Q37|Qa!e_AaC888rIcNSSU-S- -S[SU-5-SSU-- --$rrr|s r:ra!HyperRep_power2._expr_small_minusrrIcSnUR(aSnUS-nSSU-S- -SU[-[US- 5--SSU-- --[SU-[-[-U-5-$)NrrQrS)is_oddr rrrrmrr8rsgns r:reHyperRep_power2._expr_bigsu 88C FA1Q37|QQtAE{!22a!A#g>> Ab1  rIcSnUR(aSnUSSU-S- --[SU-5U-SSU-- --[S[-[-U-U-5-$)NrQrrSr)rrrrr rs r:rhHyperRep_power2._expr_big_minussc 88C1qsQw<a!es!2a!A#g >>s2b5719Q;?OOOrIrBNrrBrIr:rrzsUF9999PPrIrcX\rSrSrSr\S5r\S5r\S5r\S5r Sr g) HyperRep_log1iz2Represent -z*hyper([1, 1], [2], z) == log(1 - z). c[SU- 5$r{rr\s r:r]HyperRep_log1._expr_small1q5zrIc[SU-5$r{rr\s r:raHyperRep_log1._expr_small_minusrrIcL[US- 5SU-S- [-[--$rrrr rds r:reHyperRep_log1._expr_bigs%1q5zQqS1WbLN**rIcF[SU-5SU-[-[--$rrrds r:rhHyperRep_log1._expr_big_minuss!1q5zAaCF1H$$rIrBNrrBrIr:rrsS=++%%rIrc:\rSrSrSr\S5rSrSrSr Sr g) HyperRep_atanhiz?Represent hyper([1/2, 1], [3/2], z) == atanh(sqrt(z))/sqrt(z). cB[[U55[U5- $r)r!rr\s r:r]HyperRep_atanh._expr_smallT!W~d1g%%rIcB[[U55[U5- $r)rrr\s r:ra HyperRep_atanh._expr_small_minussDG}T!W$$rIcUR(a1[[U55[[-S- -[U5- $[[U55[[-S- - [U5- $NrS)is_evenr"rr rrds r:reHyperRep_atanh._expr_bigsQ 99$q'NQrT!V+T!W4 4$q'NQrT!V+T!W4 4rIcUR(a [[U55[U5- $[[U55[- [U5- $r)rrrrrds r:rhHyperRep_atanh._expr_big_minuss< 99Q=a( (aMB&Q/ /rIrBNrrBrIr:rrs&J&&%5 0rIrcX\rSrSrSr\S5r\S5r\S5r\S5r Sr g) HyperRep_asin1iz@Represent hyper([1/2, 1/2], [3/2], z) == asin(sqrt(z))/sqrt(z). cB[[U55[U5- $r)rrrmrps r:r]HyperRep_asin1._expr_smallsDG}T!W$$rIcB[[U55[U5- $r)rrrs r:ra HyperRep_asin1._expr_small_minusrrIc[RU-[RU- [-[ U5- [ [ [ U55-[ U5- --$r)r NegativeOnernrrr r rmrprs r:reHyperRep_asin1._expr_bigsG}}a!&&1*ba!81U47^;KDQRG;S!STTrIc[RU-[[U55[U5- U[-[ -[U5- --$r)rrrrrr rs r:rhHyperRep_asin1._expr_big_minuss<}}atAwQ!7!B$q&a.!HIIrIrBNrrBrIr:rrsWK%%&&UUJJrIrcX\rSrSrSr\S5r\S5r\S5r\S5r Sr g) HyperRep_asin2izFRepresent hyper([1, 1], [3/2], z) == asin(sqrt(z))/sqrt(z)/sqrt(1-z). cv[RU5[R[RU5- $r)rr]ryrrnrs r:r]HyperRep_asin2._expr_smalls.))!,  ( ( 34 4rIcv[RU5[R[RU5- $r)rraryrrnrs r:ra HyperRep_asin2._expr_small_minuss.//2  . .qvvq 9: :rIcv[RX5[R[RX5- $r)rreryrrnrs r:reHyperRep_asin2._expr_bigs.''-  & &qvvq 45 5rIcv[RX5[R[RX5- $r)rrhryrrnrs r:rhHyperRep_asin2._expr_big_minuss.--a3  , ,QVVQ :; ;rIrBNrrBrIr:rrsSQ44::55;;rIrcX\rSrSrSr\S5r\S5r\S5r\S5r Sr g) HyperRep_sqrts1iz>rIc ZSU-U-[SU-[[U55-5-$r)rrrrs r:ra!HyperRep_sqrts1._expr_small_minuss+Az#ac$tAw-/000rIcUR(ao[U5S-SU--[S[-[-U-U-5-[U5S- SU--[S[-[-US- -U-5--S- $US-n[U5S- SU--[S[-[-U-US--5-[U5S-SU--[S[-[-U-U-5--S- $rrrrrr rmrrprs r:reHyperRep_sqrts1._expr_bigs 99!Wq[AaC(QrT!VAXaZ8!Wq[AaC(QrT!VQU^A-=)>>?@AB B FA!Wq[AaC(QrT!VAXq1u-=)>>!Wq[AaC(QrT!VAXaZ89:;< Represent hyper([a, -a], [1/2], z) == cos(2*a*asin(sqrt(z))). cH[SU-[[U55-5$r)rrrrs r:r]HyperRep_cosasin._expr_smallDs1Q3tDG}$%%rIcH[SU-[[U55-5$r)rrrrs r:ra"HyperRep_cosasin._expr_small_minusHsAaCd1g&''rIc|[SU-[[U55-U[-[-SU-S- --5$r)rr rrr rs r:reHyperRep_cosasin._expr_bigLs8AaCd1g&2a1q)99::rIcv[SU-[[U55-SU-[-[-U--5$r)rrrrr rs r:rh HyperRep_cosasin._expr_big_minusPs3AaCd1g&1R!344rIrBNrrBrIr:rr?sUI&&((;;55rIrcX\rSrSrSr\S5r\S5r\S5r\S5r Sr g) HyperRep_sinasiniUz\Represent 2*a*z*hyper([1 - a, 1 + a], [3/2], z) == sqrt(z)/sqrt(1-z)*sin(2*a*asin(sqrt(z))) c ~[U5[SU- 5- [SU-[[U55-5-$r)rrrrs r:r]HyperRep_sinasin._expr_smallYs4AwtAE{"3qs4Q='8#999rIc [U5*[SU-5- [SU-[[U55-5-$r)rrrrs r:ra"HyperRep_sinasin._expr_small_minus]s6QxQU #D1U47^);$<<>> from sympy import appellf1, symbols >>> x, y, a, b1, b2, c = symbols('x y a b1 b2 c') >>> appellf1(2., 1., 6., 4., 5., 6.) 0.0063339426292673 >>> appellf1(12., 12., 6., 4., 0.5, 0.12) 172870711.659936 >>> appellf1(40, 2, 6, 4, 15, 60) appellf1(40, 2, 6, 4, 15, 60) >>> appellf1(20., 12., 10., 3., 0.5, 0.12) 15605338197184.4 >>> appellf1(40, 2, 6, 4, x, y) appellf1(40, 2, 6, 4, x, y) >>> appellf1(a, b1, b2, c, x, y) appellf1(a, b1, b2, c, x, y) References ========== .. [1] https://en.wikipedia.org/wiki/Appell_series .. [2] https://functions.wolfram.com/HypergeometricFunctions/AppellF1/ c[U5[U5:aX2p2XepeU"XX4XV5$X#:Xa$[U5[U5:a XepeU"XX4XV5$US:XaUS:Xa[R$gg)Nr)rrr)rmrb1b2rvr8rs r:r} appellf1.evalsw B "22"6 6qqbQ* * X*1-0@0CCqqbQ* * 6a1f55L6rIc URup#pEpgUS:XaX#-U- [US-US-XES-Xg5-$US:XaX$-U- [US-X4S-US-Xg5-$US;a[XRUS- 5$[X5e)NrrQ)rQrSrRr)r5rr r )r7rrrrrvr8rs r:rXappellf1.fdiffs!YYra q=DFHQUBFBAqDD D ]DFHQUBQAqDD D  %dIIhqj$9: :$T4 4rIrBN)r) rDrErFrGrarr}rXrHrBrIr:rris! D   5rIrN)Rra collectionsr sympy.corerrsympy.core.addrsympy.core.exprrsympy.core.functionrr r sympy.core.containersr sympy.core.mulr sympy.core.numbersr rrrsympy.core.parametersrsympy.core.relationalrsympy.core.sortingrsympy.core.symbolrsympy.external.gmpyrsympy.functionsrrrrrrrrrrr r!r"r#r$$sympy.functions.elementary.complexesr%r&r'&sympy.functions.elementary.exponentialr(#sympy.functions.elementary.integersr)$sympy.functions.elementary.piecewiser*sympy.logic.boolalgr+r,sympyr-r/rLrNrdrrTryrrrrrrrrrrrrBrIr:rsG+ OO'--3$/##00006DD<7:)FuF,1('/'&j! j!Z R%!R%j C'C'L*h*0PhP8%H%&0X0,JXJ&;X;0NhN:1h1>8H825x5,FxF(8585rI