\hkSrSSKJr SSKJr SSKJr SSKJr SSK J r J r J r J r SSKJrJr SSKJr SS KJrJrJrJrJrJrJr S rS rSS jrSrSSjrSSjr Sr!Sr"Sr#Sr$Sr%Sr&SSjr'Sr(g )a Algorithms for solving the Risch differential equation. Given a differential field K of characteristic 0 that is a simple monomial extension of a base field k and f, g in K, the Risch Differential Equation problem is to decide if there exist y in K such that Dy + f*y == g and to find one if there are some. If t is a monomial over k and the coefficients of f and g are in k(t), then y is in k(t), and the outline of the algorithm here is given as: 1. Compute the normal part n of the denominator of y. The problem is then reduced to finding y' in k, where y == y'/n. 2. Compute the special part s of the denominator of y. The problem is then reduced to finding y'' in k[t], where y == y''/(n*s) 3. Bound the degree of y''. 4. Reduce the equation Dy + f*y == g to a similar equation with f, g in k[t]. 5. Find the solutions in k[t] of bounded degree of the reduced equation. See Chapter 6 of "Symbolic Integration I: Transcendental Functions" by Manuel Bronstein. See also the docstring of risch.py. )mul)reduce)oo)Dummy)PolygcdZZcancel)imre)sqrt)gcdex_diophantinefrac_in derivation splitfactorNonElementaryIntegralExceptionDecrementLevelrecognize_log_derivativecJUR(a[$U[X"5:Xa%URU5R 5SS$/nUnUR U5nSnUR(a?UR XF45 XD-nUS-nUR U5nUR(aM?Sn[SU5n[U5S:waSUR5n XS-n UR U 5nUR(a XyS- nU n[U5S:waMSU$)a9 Computes the order of a at p, with respect to t. Explanation =========== For a, p in k[t], the order of a at p is defined as nu_p(a) = max({n in Z+ such that p**n|a}), where a != 0. If a == 0, nu_p(a) = +oo. To compute the order at a rational function, a/b, use the fact that nu_p(a/b) == nu_p(a) - nu_p(b). r) is_zerorras_polyETremappendlenpop) apt power_listp1r tracks_powernproductfinalproductfs K/var/www/auris/envauris/lib/python3.13/site-packages/sympy/integrals/rde.pyorder_atr+)s yy DJyy| #A&& J B b AL ))2+, U EE"I ))) A1ajG j/Q  8# EE(O 99 qMAG j/Q  HcvUR(a[$URU5URU5- $)z Computes the order of a/d at oo (infinity), with respect to t. For f in k(t), the order or f at oo is defined as deg(d) - deg(a), where f == a/d. )rrdegree)rdr!s r* order_at_oor0Ts+ yy 88A;! $$r,NcU=(d [S5n[X5upE[XDRUR55nUR U5nUR [Xv55n[ UR U5RUR5URUR5URUR55upU[X2R5[X5-- RUR5RURUR55n [X5n U RRU5(d[SUR5X44$U R5V s/sHo[;dMU S:dMU PM n n [[ U Vs/sH1n[U[XR5[X5-- U5PM3 sn[SUR55n[X5nX-UU-- nX-nUR#USS9unnUUU44$s sn fs snf)aP Weak normalization. Explanation =========== Given a derivation D on k[t] and f == a/d in k(t), return q in k[t] such that f - Dq/q is weakly normalized with respect to t. f in k(t) is said to be "weakly normalized" with respect to t if residue_p(f) is not a positive integer for any normal irreducible p in k[t] such that f is in R_p (Definition 6.1.1). If f has an elementary integral, this is equivalent to no logarithm of integral(f) whose argument depends on t has a positive integer coefficient, where the arguments of the logarithms not in k(t) are in k[t]. Returns (q, f - Dq/q) zrrTinclude)rrrdiffr!quorrrr resultantexprhas real_rootsr rrr )rr/DEr2dndsg d_sqf_partd1a1br$iNr&qdqsnsds r*weak_normalizerrI`s( U3ZA  FB B AJ J* +B aeeBi//5rzz"$$7G "$$ EB T!TT]:b- - -66rtt<FF 244 A Q A 66::a==Q v&&LLN8Nq2g!a%NA8sANAqST!TT]:b+===rBAN Q  A A B qtB B YYr4Y (FB Bx= 9Ns I &I .I 8I c[X5upV[X45upxURU5n URURUR55R U RU RUR555n XZ-n X-n U R U5S(a[ eX-n U RUSS9upX-U[X5-U-- nURUSS9unnXU4X4U 4$)a Normal part of the denominator. Explanation =========== Given a derivation D on k[t] and f, g in k(t) with f weakly normalized with respect to t, either raise NonElementaryIntegralException, in which case the equation Dy + f*y == g has no solution in k(t), or the quadruplet (a, b, c, h) such that a, h in k[t], b, c in k, and for any solution y in k(t) of Dy + f*y == g, q = y*h in k satisfies a*Dq + b*q == c. This constitutes step 1 in the outline given in the rde.py docstring. rTr3) rrr5r!r6divrr r)fafdgagdr;r<r=enesr hrccacdbabds r* normal_denomrXs  FB  FB r A rwwrtt}!!!%%rtt "56A A AuuRy|,, B YYr4Y (FB :a$$R' 'B YYr4Y (FB Bx"1 %%r,c \ US:Xa URnUS:Xa![URUR5nOUS:Xa'[URS-S-UR5nOkUS;aWUR5R U5nUR5R U5n XU [SUR54$[ SU-5e[ XUR5[ X'UR5- n [ X7UR5[ XGUR5- n [SU [SU 5- 5n U (GdSS KJ n US:XaURR [URUR55n[U5 [URS5*URS5- URS5- UR5unn[XR5unnU "UUUUU5nUbUunnnUS:Xa [U U5n S S S 5 GOUS:XGaURR [URS-S-UR55n[U5 [[UR[S 55*UR[S 55- UR[S 55- 5UR5unn[[!UR[S 55*UR[S 55- UR[S 55- 5UR5unn[XR5unn[#[SUR5U-UU5(aSU "U[[S 5UR5-U-UU--UU-UUU5nUbUunnnUS:Xa [U U5n S S S 5 [%SU *X- 5nUU-nX|*-nUU-nUUR U5-[XR5U-['Xu5R U5-U--nUU-U-R U5n UnUXU4$!,(df  N=f!,(df  N=f) a Special part of the denominator. Explanation =========== case is one of {'exp', 'tan', 'primitive'} for the hyperexponential, hypertangent, and primitive cases, respectively. For the hyperexponential (resp. hypertangent) case, given a derivation D on k[t] and a in k[t], b, c, in k with Dt/t in k (resp. Dt/(t**2 + 1) in k, sqrt(-1) not in k), a != 0, and gcd(a, t) == 1 (resp. gcd(a, t**2 + 1) == 1), return the quadruplet (A, B, C, 1/h) such that A, B, C, h in k[t] and for any solution q in k of a*Dq + b*q == c, r = qh in k[t] satisfies A*Dr + B*r == C. For ``case == 'primitive'``, k == k[t], so it returns (a, b, c, 1) in this case. This constitutes step 2 of the outline given in the rde.py docstring. autoexptanrr) primitivebasez@case must be one of {'exp', 'tan', 'primitive', 'base'}, not %s.rparametric_log_derivN)caserr!to_fieldr6 ValueErrorr+minprder`r/rrevalr r r rmaxr)rrVrWrTrUr;rbr BCnbncr&r`dcoeffalphaaalphadetaaetadAQmr2betaabetadrDpNpnrRs r* special_denomrys. v~ww u} rtt   q1bdd # & & KKM  b ! KKM  b !aa''!%&' ' " "!6 6B " "!6 6B ArC2JA . 5=TTXXd244./F#!("''!*RWWQZ)?q )I244!P$VTT2 d(tRH=GAq!Av1I$#U]TTXXd244719bdd34F#!(RWWT"X->,>rwwtBx?P,PQRQWQWX\]_X`Qa,a)bdfdhdh!i&r27748+<*# UHoRTR5v M! g7fN)r.r!).0rCr;s r* bound_degree..(s,A"$$s'*r^rrr])limited_integratezLength of m should be 1!is_log_deriv_k_t_radical_in_fieldNr[r_)r\other_nonlinearzScase must be one of {'exp', 'tan', 'primitive', 'other_nonlinear', 'base'}, not %s.)rbr.r!rhr rLCas_expr is_Integerrr/Tlevelrrfrrrdrrrr`r6r)rrBcQr;rb parametricdadbdcalphar&rprqt1rnrorzazdrtrrraar2betarurvr`deltalams ` r* bound_degreer sv( v~ww "$$B "$$B ,, , YYrtt_ AIIbddO&&(0022 "$$$$&' (E v~ 2BQ' ( a! !A F H-$#, Hsa;'S#R$5CSR57S,>S$ R2.S1R22S5 S?SSS S S'c[SUR5n[SUR5n[SUR5nUR(aXUSXW4$US:SLa[eUR U5nUR U5R(d[eUR U5UR U5UR U5p!nURUR5S:XaCUR5R U5nUR5R U5nXX6U4$[XU5upU[X5- nU [X5- nX0RUR5-nXvU -- nX`-nGMD)az Rothstein's Special Polynomial Differential Equation algorithm. Explanation =========== Given a derivation D on k[t], an integer n and ``a``,``b``,``c`` in k[t] with ``a != 0``, either raise NonElementaryIntegralException, in which case the equation a*Dq + b*q == c has no solution of degree at most ``n`` in k[t], or return the tuple (B, C, m, alpha, beta) such that B, C, alpha, beta in k[t], m in ZZ, and any solution q in k[t] of degree at most n of a*Dq + b*q == c must be of the form q == alpha*h + beta, where h in k[t], deg(h) <= m, and Dh + B*h == C. This constitutes step 4 of the outline given in the rde.py docstring. rrT) rr!rrrrr6r.rcrr) rrBrSr&r;zerorrr>r$r2s r*spdersE" 244=D BDDME 244=D  994. . Ed?0 0 EE!HuuQx0 0%%(AEE!HaeeAha 88BDD>Q    #A   #A!D) ) q) Z  1! ! XXbdd^   / r,cB[SUR5nUR(dURUR5URUR5- nSUs=::a U::d[e [e[UR UR5R 5UR UR5R 5- URU--URSS9nXF-nUS- nU[Xc5- X-- nUR(dMU$)a Poly Risch Differential Equation - No cancellation: deg(b) large enough. Explanation =========== Given a derivation D on k[t], ``n`` either an integer or +oo, and ``b``,``c`` in k[t] with ``b != 0`` and either D == d/dt or deg(b) > max(0, deg(D) - 1), either raise NonElementaryIntegralException, in which case the equation ``Dq + b*q == c`` has no solution of degree at most n in k[t], or a solution q in k[t] of this equation with ``deg(q) < n``. rFexpandr)rr!rr.rrrrrBrSr&r;rErtr s r*no_cancel_b_largers Q Aii HHRTTNQXXbdd^ +A{{0 00 0 244##%aiio&8&8&::2447BBDD  E E 1! !AC 'iii Hr,c[SUR5nUR(Gd;US:XaSnODURUR5URRUR5- S-nSUs=::a U::d[ e [ eUS:a[UR UR5R5XSRR UR5R5-- URU--URSS9nGOURUR5URUR5:wa[ eURUR5S:XaVX@R URURS- 5UR URURS- 54$[UR UR5R5UR UR5R5- URSS9nXF-nUS- nU[Xc5- X-- nUR(dGM;U$)ah Poly Risch Differential Equation - No cancellation: deg(b) small enough. Explanation =========== Given a derivation D on k[t], ``n`` either an integer or +oo, and ``b``,``c`` in k[t] with deg(b) < deg(D) - 1 and either D == d/dt or deg(D) >= 2, either raise NonElementaryIntegralException, in which case the equation Dq + b*q == c has no solution of degree at most n in k[t], or a solution q in k[t] of this equation with deg(q) <= n, or the tuple (h, b0, c0) such that h in k[t], b0, c0, in k, and for any solution q in k[t] of degree at most n of Dq + bq == c, y == q - h is a solution in k of Dy + b0*y == c0. rrFr) rr!rr.r/rrrrrrrs r*no_cancel_b_smallrs Q Aiii 6ARTT!22Q6AA{{0 00 0 q5QYYrtt_'')1TT\\"$$-?-B-B-D+DEbddAgMU$Axx~"$$/44xx~"99RTT"((Q,%78IIbdd288a<0133QYYrtt_'')!))BDD/*<*<*>>A E E 1! !AC '/iii2 Hr,c[SUR5n[URUR5R 5*UR RUR5R 5- 5nUR (aUR(aUnOSnUR(Gd [XaRUR5UR RUR5- S-5nSUs=::a U::d[e [e[XsR RUR5R 5-URUR5R 5-5nUR(aXGU4$US:aP[URUR5R 5U- URU--URSS9n OURUR5UR RUR5S- :wa[eURUR5R 5URUR5R 5- n XI-nUS- nU[X5- X -- nUR(dGM U$)ai Poly Risch Differential Equation - No cancellation: deg(b) == deg(D) - 1 Explanation =========== Given a derivation D on k[t] with deg(D) >= 2, n either an integer or +oo, and b, c in k[t] with deg(b) == deg(D) - 1, either raise NonElementaryIntegralException, in which case the equation Dq + b*q == c has no solution of degree at most n in k[t], or a solution q in k[t] of this equation with deg(q) <= n, or the tuple (h, m, C) such that h in k[t], m in ZZ, and C in k[t], and for any solution q in k[t] of degree at most n of Dq + b*q == c, y == q - h is a solution in k[t] of degree at most m of Dy + b*y == C. rrarFr) rr!r rrr/r is_positiverrhr.rr) rBrSr&r;rElcMrtur s r*no_cancel_equalrs Q A 244##%%bddll244&8&;&;&== >B }}  iii 88BDD>BDDKK$559 :A{{0 00 0 1TT\\"$$'**,,qyy/A/A/CC D 99!9  q5QYYrtt_'')!+BDD!G3RTT%HAxx~RTT!2Q!6644IIbddO&&(244););)== E E 1! !AC ''iii* Hr,crSSKJn [U5 [XR5upVU"XVU5nUbUup(US:Xa [ S5eSSS5 UR (aU$X!RUR5:a[e[SUR5n UR (dURUR5n X*:a[e[U5 [UR5UR5up[WWXU5upSSS5 [W R5WR5- URU --URSS9nX- n U S- nXU-[X5--nUR (dMU $!,(df  GNR=f!,(df  N=f)a Poly Risch Differential Equation - Cancellation: Primitive case. Explanation =========== Given a derivation D on k[t], n either an integer or +oo, ``b`` in k, and ``c`` in k[t] with Dt in k and ``b != 0``, either raise NonElementaryIntegralException, in which case the equation Dq + b*q == c has no solution of degree at most n in k[t], or a solution q in k[t] of this equation with deg(q) <= n. rrNz7is_deriv_in_field() is required to solve this problem.rFr)rfrrrr!NotImplementedErrorrr.rrrrischDErr)rBrSr&r;rrVrWrrr2rErta2aa2dsarHstms r*cancel_primitiver.sa8  DD! -bb 9 =DAAv)++,,  yy88BDD>,, Q Aii HHRTTN 50 0 B qttvrtt,HCRSr2FB 2::< ,RTT1W4bdd5I  E sUZ( ((iii HA  0 s9F"7F( F%( F6cSSKJn URR[ UR UR 55R 5n[U5 [XSR 5upg[XR 5upU"XXgU5n U bU upn U S:Xa [S5eSSS5 UR(aU$X!RUR 5:a[e[ SUR 5nUR(Gd7URUR 5n X,:a[eUR 5n[U5 [XR 5unnUW-WU-[ XR 5--nUU-n[UR5UR 5unn[UUUUU5unnSSS5 [ WR 5WR 5- UR U --UR SS9nUU- nU S- nXU-[UU5--nUR(dGM7U$!,(df  GN=f!,(df  N=f)a Poly Risch Differential Equation - Cancellation: Hyperexponential case. Explanation =========== Given a derivation D on k[t], n either an integer or +oo, ``b`` in k, and ``c`` in k[t] with Dt/t in k and ``b != 0``, either raise NonElementaryIntegralException, in which case the equation Dq + b*q == c has no solution of degree at most n in k[t], or a solution q in k[t] of this equation with deg(q) <= n. rr_Nz6is_deriv_in_field() is required to solve this problem.rFr)rfr`r/r6rr!rrrrrr.rrrr)rBrSr&r;r`etarprqrVrWrrrrtr2rErAa1aa1drrrrHrs r* cancel_expr`s+ $$((4bdd# $ , , .C  S$$' DD! R 8 =GA!Av)+*++  yy88BDD>,, Q Aiii HHRTTN 50 0 YY[ B r44(HCd(T#Xd1ddm33Cd(Cqttvrtt,HCS#sC4FB 2::< ,RTT1W4bdd5I S E sUZR( ((%iii& HS  8 sAI A8I I I+cUR(dURS:XdOURUR5[ SUR RUR5S- 5:a"U(aSSKJn U"XX#5$[XX#5$UR(dFURUR5UR RUR5S- :GaURS:Xd)UR RUR5S:aU(aSSKJ n U"XX#5$[XX#5n[U[5(aU$Uupn [U5 U RUR5U RUR5pU c [S5eU c [S 5e[!XX#5RUR5n SSS5 X-$UR RUR5S:Ga3URUR5UR RUR5S- :XaX RUR5R#5*UR RUR5R#5- :aURUR5R#5R$(d ['S 5eU(a [)S 5e[+XX#5n[U[5(aU$Uupn [!X X5n X-$UR(a [)S 5eURS :XaU(a [)S5e[-XX#5$URS:XaU(a [)S5e[/XX#5$[)SUR-5e!,(df  UW -$=f)z Solve a Polynomial Risch Differential Equation with degree bound ``n``. This constitutes step 4 of the outline given in the rde.py docstring. For parametric=False, cQ is c, a Poly; for parametric=True, cQ is Q == [q1, ..., qm], a list of Polys. r^rr)prde_no_cancel_b_larger)prde_no_cancel_b_smallNzb0 should be a non-Null valuezc0 should be a non-Null valuezResult should be a numberz0prde_no_cancel_b_equal() is not yet implemented.zWRemaining cases for Poly (P)RDE are not yet implemented (is_deriv_in_field() required).r[zIParametric RDE cancellation hyperexponential case is not yet implemented.r]zBParametric RDE cancellation primitive case is not yet implemented.zBOther Poly (P)RDE cancellation cases are not yet implemented (%s).)rrbr.r!rhr/rfrrrr isinstancerrrrdsolve_poly_rder is_number TypeErrorrrrr)rBrr&r;rrrRrRb0c0yrtrjs r*rrs! 99"''V+ HHRTTNSBDDKK$5$9: :  4)!7 7 .. ))qxx~ BDD(9A(== WW "$$++bdd"3q"8  4)!7 7 aQ + a  HIA2#BDD)2::bdd+;B:$%DEE;$%DEE"21199"$$? $5L RTT a AHHRTTNbddkk"$$6G!6K$K 244##%%bddll244&8&;&;&== =yy!!#--78 8 %'   A1 ) a  HGA!qQ+A5L 99%'BC Cww%-/HII!!//K'-/ABB'q55*+:<>GG+DEEe$#q5Ls ,A7O O.c6[XU5unup[XX#U5unupxupn [XgXX5upp[XX5n[ XUUU5upnnnUR(aUnO [XUU5nUU-U-X-4$![a [ nNRf=f)a Solve a Risch Differential Equation: Dy + f*y == g. Explanation =========== See the outline in the docstring of rde.py for more information about the procedure used. Either raise NonElementaryIntegralException, in which case there is no solution y in the given differential field, or return y in k(t) satisfying Dy + f*y == g, or raise NotImplementedError, in which case, the algorithms necessary to solve the given Risch Differential Equation have not yet been implemented. ) rIrXryrrrrrr)rLrMrNrOr;_rrVrWrTrUhnrrrirjhsr&rtrrrs r*rrs"""-KAx ,RRR @Ax"2rr6KA!  q % aB/A!UDyy  1B ' !GdNBE ""   s BBBr|)rZ)rZF)F))__doc__operatorr functoolsr sympy.corersympy.core.symbolr sympy.polysrrr r $sympy.functions.elementary.complexesr r (sympy.functions.elementary.miscellaneousr sympy.integrals.rischrrrrrrrr+r0rIrXryrrrrrrrrrr,r*rs.#--99[[[ ( V %0f"&JQht n-^ <+ ^, ^/ d9 xZz'#r,