+ iSRt^RIt^RIt^RIHtHtHt^RIHt^RI H t ^RI H t ^RI HtHtHtHtHtHtHtHtHtHt^RIHtHt^RIHtHtHtHtH t H!t!H"t"H#t#^R I$H%t%^R I&H't'H(t(H)t)^R I*H+t+H,t,H-t-H.t.H/t/H0t0^R I1H2t2H3t3^R I4H5t5H6t6H7t7H8t8^RI9H:t:H;t;Ht>H?t?^RI@HAtAHBtBHCtCHDtD^RIEHFtFHGtGHHtHHItI^RIJHKtKHLtL^RIMHNtNHOtOHPtP^RIQHRtRHStSHTtTHUtU^RIVHWtW^RIXHYtYHZtZ^RI[H\t\H]t]H^t^^RI_H`t`HataHbtbHctcHdtd^RIeHftf^RIgHhth^RIiHjtj^RIkHltl^RImHntn^RIoHptp^RIqHrtr^R IsHtttHutuHvtv^R!IwHxtx^syR"tzR#t{R$t|]zR%4t}]zR&4t~]zR'4t] R(4t]zR)4t]zR*4t]zR+4t]zR,4t]zR-4t]zR.4t]zR/4t]zR04t]zR14t]zR24t]zR34t]zR44t]zR54t]zR64tR7tR8t]zR94t!R:R;]]4tROR<lt]zR=4t]zR>4t] R?4t]zR@4t]zRA4t]zRB4t]zRC4t]zRD4t]zRE4t]zRF4t]zRG4t]zRH4t]zRI4t]zRJ4t!RKRL]]4tRPRMltRNtR#)QzLaplace TransformsN)SpiI)Add)cacheit)Expr) AppliedUndef Derivativeexpandexpand_complex expand_mul expand_trigLambda WildFunctiondiffSubs)Mulprod) _canonicalGeGtLt UnequalityEqNe Relational)ordered)DummysymbolsWild)reimargAbs polar_liftperiodic_argument)explog)coshcothsinhasinh)MaxMinsqrt) Piecewisepiecewise_exclusive)cossinatansinc)besselibesseljbesselkbessely) DiracDelta Heaviside)erferfcEi)digammagamma lowergamma uppergamma)SingularityFunction) integrateIntegral) _simplifyIntegralTransformIntegralTransformError)to_cnf conjuncts disjunctsOrAnd) MatrixBase) _lin_eq2dict)PolynomialError)roots)Poly)together)RootSum)sympy_deprecation_warningSymPyDeprecationWarningignore_warnings)debugfcaV3RlpV#)c<^RIHpV'g S!V/VB#\^8Xd\R \P R7\RR\,: SP : V: 2\P R7\^, sSP R8XgSP R8XdMR\n\RR\,,\P R7S!V/VBpR \nMS!V/VBp\^,s\RR\,: R V: 2\P R7\^8Xd\R \P R7V#)  SYMPY_DEBUGfile-LT-  _laplace_transform_integration&_inverse_laplace_transform_integrationFz**** %sIntegrating ...Tz---> zO ------------------------------------------------------------------------------zO------------------------------------------------------------------------------ )sympyr\ _LT_levelprintsysstderr__name__)argskwargsr\resultfuncs*, w/Users/tonyclaw/.openclaw/workspace/skills/math-calculator/venv/lib/python3.14/site-packages/sympy/integrals/laplace.pywrapDEBUG_WRAP..wrap1s%(( ( > -cjj 1 tI~t}}dC:: Q  !AA !II %E  *d9n=CJJ O4*6*F $E 4*6*FQ  $y..&9 K > -cjj 1 )rlrnsf rm DEBUG_WRAPrr0s4 Krpc^RIHpV'd0\RR\,: V: 2\P R7R#R#)rZr[r_r`r]N)rcr\rerdrfrg)textr\s& rm_debugruNs$! T)^T2DrpcaaaaaV3RloVVVV3RloV3RloV3RlpRp^RIHpV!V4pV!V\S4pV!V\V3Rl4pV!V\V4p\ V4#)a Naively simplify some conditions occurring in ``expr``, given that `\operatorname{Re}(s) > a`. Examples ======== >>> from sympy.integrals.laplace import _simplifyconds >>> from sympy.abc import x >>> from sympy import sympify as S >>> _simplifyconds(abs(x**2) < 1, x, 1) False >>> _simplifyconds(abs(x**2) < 1, x, 2) False >>> _simplifyconds(abs(x**2) < 1, x, 0) Abs(x**2) < 1 >>> _simplifyconds(abs(1/x**2) < 1, x, 1) True >>> _simplifyconds(S(1) < abs(x), x, 1) True >>> _simplifyconds(S(1) < abs(1/x), x, 1) False >>> from sympy import Ne >>> _simplifyconds(Ne(1, x**3), x, 1) True >>> _simplifyconds(Ne(1, x**3), x, 2) True >>> _simplifyconds(Ne(1, x**3), x, 0) Ne(1, x**3) cx<VS8Xd^#VP'dVPS8Xd VP#R#)N)is_Powbaser&)exss&rmpower_simplifyconds..powervs, 7 999A66Mrpc<VPS4'dVPS4'dR#\V\4'dVP^,p\V\4'dVP^,pVPS4'dS!^V, ^V, 4#S!V4pVfR#V^8d4\V4\S4V,8*\P 8XdR#V^8d6\V4\S4V,8\P 8XdR#R#R# \ dR#i;i)zRReturn True only if |ex1| > |ex2|, False only if |ex1| < |ex2|. Else return None. NFT)has isinstancer#rirtrue TypeError)ex1ex2nabiggerr}r|s&& rmr_simplifyconds..bigger}s 771::#''!** c3  ((1+C c3  ((1+C 771::!C%3' ' #J 9 1u#c(c!fai/AFF:1u#c(c!fai/AFF:;u  s8D9;8D99 EEc<VP'g\V\4'd)VP'g\V\4'gW8#S!W4pVeV'*#W8#)zsimplify x < y ) is_positiverr#)xyrrs&& rmreplie_simplifyconds..repliesP*Q"4"4 As););EN 1L =5Lrpc<<S!W4pVR9dR#\W4#T)TF)r)rrbrs&& rmreplue_simplifyconds..replues" 1L !rpcFVR9d \V4#VP!V!#r)boolreplace)r{ris&*rmrepl_simplifyconds..repls"  8Ozz4  rp) collect_absc<S!W4#Nrq)rrrs&&rm _simplifyconds..s va|rp)sympy.simplify.radsimprrrrr) exprr|rrrrrr}rs &ff @@@rm_simplifycondsrUseB, ! 3 t D b& !D b3 4D j& )D T7Nrpc>\WP\44#)zg Expand an expression involving DiractDelta to get it as a linear combination of DiracDelta functions. )rNatomsr9rs&rmexpand_dirac_deltars jj4 55rpcaaaa\R4oVP\4'dR#\V\ S)S,4,S\ P \ P34pVP\4'g<\VPSS4V4\ P\ P3#VP'gR#VP^,wrEVP\4'dR#VV3Rlp\V4Uu.uF qv!V4NK ppVU u.uF?q^,\ P 8wgK V ^,\ PJgK=V NKA p p V 'g/VU u.uF"q^,\ P 8wgK V NK$ p p \#\%V 44pRoVP'V3RlR7V'gR#V^,wrVV3Rlp V'd\)VSV 4p\)V SV 4p \VPSS4V4V !V 4\+V !V 443#uupiuup iuup i)zThe backend function for doing Laplace transforms by integration. This backend assumes that the frontend has already split sums such that `f` is to an addition anymore. r|Nc8 <a^RIHp\Pp\Pp\ \ V44p\R\S.R7wrErgrp V\\SV,V,44,V8V\\SV,V,44,V8*\\SV,V,V,V44V8\\SV,V,V,V44V8*\\\SV,4V,V,V44V8\\\SV,4V,V,V44V8*3p VEFp \Pp .p\V 4EFpVP'd(SVP P"9d VP$pVP'd)\'V\(\*34'd VP,pV FpVP/V4oS'gKM S'd^SV,P0'dESV,SV,, \2^, 8Xd\5SSV,,4)^8pVP/V\7V\\SV ,44,V,4\SV,4V ,,, ^84oS'gkVP/\7V\\SV,V ,V44V,, 4\SV,4V ,,^84oS'gtVP/V\7\\\S4V,V ,V44V,4\SV,4V ,,, ^84oS'd\\8;QJd"V3RlWgWV 34F 'dK RM RM!V3RlWgWV 344'd\5S4SV,8pVP;\4R4P=\5S4S4pVP'd@VP>R 9g/VPAS4'gVPAS4'g W., pEKFV!VS4pVP'dVP>R 9d W., pEK~VPBS8XdR#\EVPBV 4p EK V \PJd\GW4pEK\IV\KV!4pEK Y#P'dVPL3#T3#) z6Turn ``conds`` into a strip and auxiliary conditions. _solve_inequalityzp q w1 w2 w3 w4 w5clsexcludec3J<"TFpSV,PxK R#5ir)r).0wildms& rm H_laplace_transform_integration..process_conds..s#->,TQtW00>,s #FTcLVP4P4^,#rZ)r as_real_imag)rs&rmrG_laplace_transform_integration..process_conds..s!((*"9"9";A">rpN)z==z!=)'sympy.solvers.inequalitiesrrNegativeInfinityrrIrHrrr#r"r%r$InfinityrJ is_Relationalrhs free_symbolsreversedrrr reversedsignmatchrrr r1allrsubsrel_oprltsr-r,rLrK canonical)condsrrauxpqw1w2w3w4w5patternsca_aux_dpatd_solnrr|ts& @rm process_conds5_laplace_transform_integration..process_condss6@  ff&-(#* dQC$9 bbb c#q2vqj/" "R ' c#q2vqj/" "b ( !1r6A+a-4 5 : !1r6A+a-4 5 ; !:a"f#5"9!";R@ AB F !:a"f#5"9!";R@ AR G IABDq\???qAEE,>,>'> A???z!b"X'>'>A#C Aq$1)))aeAaDjBqD.@A"I*AGGABs3qt9~$5b$8 9#ae*b. HH1LMA$5aeBh$B CB FFGArE B')*+,AC 1*Q-2CB2F JKBN!!R%j"n--/012A->,-->,---1! AYY>@@DRUAOOOqxx.cnts = ~~rpc6<V^,)S!V^,43#rrq)rrs&rmr0_laplace_transform_integration..sqteS1Y/rpkeyc(<VPSS4#r)r)rr|s_s&rmsbs+_laplace_transform_integration..sbs!syyBrp)rrr9rCr&rZerorrDrErrr is_PiecewiserirJfalselistrsortrr)frrsimplifyFcondrrrrconds2rrrrr|s&ff$ @@rmraras c AuuZ!C1I+1661::67A 55??2113E3EqvvMM >>>ffQiGAuuX=>~(1 7!]1 E 7:AA$gg#aA$a&8&88aF: "6Udaggo!!U6  !E  JJ/J0  1XFA  1a #S!Q' QVVAr]H -s1vz#c(7K KK- 8:7s$ I6II5I I+Ic\V\4'gV#VPV4;peVP4#VPpVP Uu.uFp\ WA4NK ppV!V!#uupi)z This is an internal helper function that traverses through the expression tree of `f(t)` and collects arguments. The purpose of it is that anything like `f(w*t-1*t-c)` will be written as `f((w-1)*t-c)` such that it can match `f(a*t+b)`. )rras_polyas_exprrlri_laplace_deep_collect)rrrrlr"ris&& rmrr)sk a   YYq\&yy{ 66D56VV .dcoSs,Q22rpz&_laplace_build_rules is building rulesg?)(rrrurrrr9r&r#rKrLrr:r=r"rr.r<r?rAr r7r@r' EulerGammar>r2r!r)r3r1r*r(r;r6Halfrr5r8r+) r|rrrrrrlaplace_transform_rulesrs @rm_laplace_build_rulesr:s$ c A c A S1#A S1#A S1#A uqc "C 1# &E2 34| aC  | AaCE C1QKA. CAqAv AE16 2 3  S "| AaCE AaD CAqAv AE16 2 3  S " | 1Q3q5 3r!tAv;q= QUAE AFFC )| 1Q3q5 Ac1"Q$q&kM1, QUAE AFFC )| 1Q3q5 1Q3 QUAF QVVS *| 1Q3q5 1 QUAE AFFC )|" AadF  #|& AaCES!Aa[LQBqDF+A- SX QVVS *'|* 4!A;QrT!V SQZ/T!#a%[0AA!C SX QVVS *+|. A#a%AaD57  1!uQw<2a46QqT!V,,SQZ7$tACE{:KKAM M SX QVVS */|4 a!A#RT 2d1g:c!#h#6tDI#FF SVr 1663 (5|8 Ad1gIC !2a!A$q&k>#ac(#:4QS ?#J  9|< AuQqSz!c(" R =|@ A#a%!Z!QSU+C1QK7aC@B QVSQS]R' (!&&# 7A|D AqsQT%!*_ZAC%88 QVSQ[2% & 5E|H QqSWsC4y!# A I|L 3qs3w<cTAC!8+ A M|P Ac!A#hac AC1Q3</ ARUC !Q|T aR1WtBqDF|C1QqM1$qac{2CC AAFFC !U|X 3r!Q$w< AaC48QqSAaDFO+A-d1T!A#Y;.?? ? AAFFC !Y|^ aRTAd13iK1T!#Y; 77 A!QVVS "_|b aaRT  1aR1W q49} -c"T!#Y,.? ? A!QVVS "c|h aRT47 DAJs2d13i<'88 A!QVVS "i|l aRTAd1gI RT 3r$qs)|+< < AAFFC !m|p Ac1"Q$iACAaC7++GAaC49,EE AAFFC !q|B aRQBZ!b'*Q"22  C|F aRAYj!Q// AAFFC !G|J QqSCALL)!+A-..q0 Q K|N QqsUc!#hYq[QBqD) SVr 1663 (O|R QqSUc!fSQZ\!^Br!tH44a79 QUCAK"$ %qvvs 4S|V QQ$r!t*S1S->)>%??  W|Z Ac!feAaCjRT*GAaCLQ,?@ AQVVS "[|^ QqS1s3q||,Q.q0114RU1W!3 (o|r U1WqA  tAeGAI qQqz!Q$%6!779 9 3r%y>!3 (s|@ U1WqQ$uax-( RY &A|D U1Wq1a4%( ?QT!E1H*_=a? 3r%y>!3 (E|P QqS#ac( AaCE!GQT13(]3QT13(]C RUC1J& -Q|T QqS#ac( A!tAqDyA~.1acAX >1acAX N RUC1J& -U|X QqS#ac( A!tAqDyA~.1acAX >1acAX N RUC1J& -Y|\ acA!tAqDyM RUS "]|` acA!tAqDyM RUS "a|d acAqAvq!tAadF1H}- 3r!u:s $e|h acA1Qq!tV ad1QT6!8m4 3r!u:s $i|l ac1c13+&q( RUS "m|p Ad1Q3iqsAr!t}acaRT]'BC RQ q|t Ad1Q3iqsAr!t}acaRT]'BC RQ u|b QqS3q!tQqS1H}%d1ac7m3A5 3s1v; QVVS *c|~ aC!$QT!Q$Y41QT ?1BQ0F FG ASAZ &|B Aga1o  Ad2huQqvvX &qt +QT!Q$Y1"QVV),D D RUaffW_bh 'RUS :C|H Aga1o  QqS$r( 51Q46? *14 / 11a4191Q46 2J J RURZAs $c"Q%j# 7I|V Ad1a4!8n$ % QS41QT ?" " #DAadO 3 SVr 3r!u:s ,W|\ aC!$QT!Q$Y41QT ?1BQ0F FG ASAZ &]|` Aga1o  Ad2huQqvvX &qt +QT!Q$Y1"QVV),D D RUaffW_bh 'RUS :a|f Aga1o  QqS$r( 51Q46? *14 / 11a4191Q46 2J J RURZAs $c"Q%j# 7g|p AaC"R%ac *41QT ?: RUS "q|t AaC#q41QT ?2A56QT!Q$YH "Q% u|z #Aq ((rpc\RV.R7p\R^R7pVPV4pV'dWT,P^,P V4pVP!W1,4pV'd|Ws,P 'ddWs,^8wdW\ R4\^Ws,, WT,PV4,WWs,, RR7wrp WV 3#R#) z This function applies the time-scaling rule of the Laplace transform in a straight-forward way. For example, if it gets ``(f(a*t), t, s)``, it will compute ``LaplaceTransform(f(t)/a, t, s/a)`` if ``a>0``. rrg)nargsz rule: time scaling (4.1.4)FrN) rrrricollectrru_laplace_transformrl) rrr|rrma1r"ma2rprcrs &&& rm_laplace_rule_timescaler s S1#AS"A ''!*C fkk!n$$Q'iin 36%%%#&A+ 4 5*#&Q'cfHuFIA22;  rpc\RV.R7p\R4p\R4pVP\V4V,4;p'EdWd,PW, 4;p'dWs,P'da\ R4\ We,P WWs,,4WRR7wrp \Ws,)V,4V,W3#Ws,P'd'\ R4\ We,WRR7wrp WV 3#Wd,PW1, 4;p'dWs,P'dK\ R 4\ ^\WV,, 4, We,,WRR7wrp WV 3#Ws,P'd\ R 4^^\P3#R #) a This function deals with time-shifted Heaviside step functions. If the time shift is positive, it applies the time-shift rule of the Laplace transform. For example, if it gets ``(Heaviside(t-a)*f(t), t, s)``, it will compute ``exp(-a*s)*LaplaceTransform(f(t+a), t, s)``. If the time shift is negative, the Heaviside function is simply removed as it means nothing to the Laplace transform. The function does not remove a factor ``Heaviside(t)``; this is done by the simple rules. rrrrz rule: time shift (4.1.4)Frz8 rule: Heaviside factor; negative time shift (4.1.4)z rule: Heaviside window openz" rule: Heaviside window closedN) rrr:rrurrr& is_negativerr) rrr|rrrrrrr r s &&& rm_laplace_rule_heavisider,s| S1#A S A S AggilQ&''s'&,,qu% %3 %v!!!67.FKKsv:.uF rSVGaK(1,b55v!!!NP.svqeL rr{"&,,qu% %3 %v!!!9:.11v:..#&8!P rr{"v!!!;<1aff~% rpc\RV.R7p\R4p\R4pVP\V4V,4pV'd~Wd,PV4PW1,4pV'dJ\ R4\ We,WWs,, RR7wrp W\ Ws,4,V 3#R#) a If this function finds a factor ``exp(a*t)``, it applies the frequency-shift rule of the Laplace transform and adjusts the convergence plane accordingly. For example, if it gets ``(exp(-a*t)*f(t), t, s)``, it will compute ``LaplaceTransform(f(t), t, s+a)``. rrrzz$ rule: multiply with exp (4.1.5)FrN)rrr&rrurr ) rrr|rrrrrrr r s &&& rm_laplace_rule_exprVs S1#A S A S A ''#a&( C fnnQ%%ac*  9 :*361h49;IA2"SV*}b) ) rpc \RV.R7p\RV.R7p\R4p\R4pVP\V4V,4pV'EdWv,P\4'EgWu,P V4PWA,V, 4pV'EdZ\ R4W,W,, p \ V 4^8Ed\V 4^8Xd\W,)W,, V,4Wv,,p V P\\4'd$V P\4P4p V P4U u.uF'qPWV,W,, 4NK) up wrV ^8wd5W, W,, \ P"\ P$3#R#^\ P"\ P$3#Wu,P'V4'Ed\)Wu,V4pV/8wd\+VP-44^08Xd\/Wu,V4p\1\3VP544U u.uFlp \V 4^8XgK\ V 4^8gK'\V )V,4Wv,PW4,VPW4, NKn up !pV\ P"\ P$3#R#uup iuup i)z If this function finds a factor ``DiracDelta(b*t-a)``, it applies the masking property of the delta distribution. For example, if it gets ``(DiracDelta(t-a)*f(t), t, s)``, it will return ``(f(a)*exp(-a*s), -a, True)``. rrrrrz# rule: multiply with DiracDeltaN)rrr9rrrur r!r&r2r1rewriter4ratsimpas_numer_denomrrrr is_polynomialrPsetvaluesrrrkeys)rrr|rrrrrrlocfnrrrroslopers&&& rm_laplace_rule_deltarmsV S1#A S1#A S A S A ''*Q-/ "C s36::j))fnnQ%%ac!e, 3 8 9&-C#w!|31 #&)*36166#s##D)113B:<:K:K:MN:MQqa&-0:MN6CJ(:(:AFFCC1--qvv66 6   " "svq!BRxC ,3SVQ#BGGIM.!"Q%1*CACA!Cc1"Q$i A 11%**Q2BBB.MN1--qvv66 OMs-L6L;3 L;A L;cP\P.p\P.p\P!V4FVpVP \ \ \\\4'dVPV4KEVPV4KX \V!p\V!pWE3#)z Helper function for `_laplace_rule_trig`. This function returns two terms `f` and `g`. `f` contains all product terms with sin, cos, sinh, cosh in them; `g` contains everything else. ) rOner make_argsrr2r1r*r(r&append)rtrigsothertermrrs& rm_laplace_trig_splitr&swUUGE UUGE b! 88CdD# . . LL  LL  " U A U A 4Krpc\RV.R7p\RV.R7p\RV.R7p.p.pVP\4P4p\P !V4EFpVP V4'g$VPRVR^\^\^/4K>\VPRR7V4pVPV\W!,V,4,4;p edVPRW,\W,4,RW,\\W,4\\W,4/4KVPV4EK WV3#) a Helper function for `_laplace_rule_trig`. This function expects the `f` from `_laplace_trig_split`. It returns two lists `xm` and `xn`. `xm` is a list of dictionaries with keys `k` and `a` representing a function `k*exp(a*t)`. `xn` is a list of all terms that cannot be brought into that form, which may happen, e.g., when a trigonometric function has another function in its argument. c1rc0rkrr&)combine) rrr&r rr!rr"r r!rpowsimpr) rrr(r)rxmxnx1r%rs && rm_laplace_trig_expsumr0s dQC B dQC B S1#A B B 3   B b!xx{{ IIsD#q"aQ7 8 $T\\%\%@!DAc"$r'lN+ +A 8 IIQT#ae*_c15BquIr2ae9. / IIdO" 6Mrpc a.p.pRoV3RlpV3RlpV3RlpV3RlpRp \V4^8EdDVP4p Rp Rp Rp \\V44EFpV \,W,\,8HpV \,W,\,)8HpV \,W,\,8HpV \,W,\,)8HpV'd1V'd)V \,^8wdV \,^8wdTp KV'dV'dV \,^8wdTp KV'gKV'gKV \,^8wgEKTp EK V eV eV eVP V!V W ,R,W ,R,W ,R,V44VP \ \V R ,444WV .pVPR R 7VFpVPV4K EKV eWVP V!WV ,R,V44VP V \,4VPV 4EKXV e`VP V!WV ,R,V44VP \ V \,44VPV 4EKV e`VP V!WV ,R,V44VP \ V \,44VPV 4EKVP V !W44VP V \,4EKT\V!\V!3#) z Helper function for `_laplace_rule_trig`. This function takes the list of exponentials `xm` from `_laplace_trig_expsum` and simplifies complex conjugate and real symmetric poles. It returns the result as a sum and the convergence plane. cxVP4p\\V44FpW,P4pV^,P \ 4'd W,P \4W&KZV^,\V^,,,P \4W&K V#r) copyrangelenrrr!rr1r)coeffsncr*ris& rm_simpc"_laplace_trig_ltex.._simpcs{ [[]s2wA##%B!uyy}} c*A2a511#6   rpc <VR,VR,V\,V\,3wrVrxWa,V,V,WVV,V, V, ,^\,V,V,, ^\,V,V,,V^,V)V, V, V, ,V^\,V,V,^\,V,V,,,,^V^,,V,,^V^,,V,,V^,V)V, V,V,,V^,^\,V,V,^\,V,V,,^\,V,V,, ^\,V,V,, ,,V^V^,,V,^V^,,V,, ,,.p \P\P ^V^,,^V^,,, \P V^,^V^,,V^,,,V^,,.p \ \S!V 4\\V 44RRR1,4U U u.uFwrWV ,,NK up p !p \ \V \\V 44RRR1,4U U u.uFwrWV ,,NK up p !pW, #uup p iuup p irr*Nr) r r!rrr rrzipr4r5)t1k1k2k3r|rk0a_ra_ir7dcrrrrr9s&&&&& rm _quadpole%_laplace_trig_ltex.._quadpolesdS'2c7BrFBrF:s GbL2  Bw|b !AaCGBJ .1S ;1rcBhmb()1Q3s72:!C *+,#q& Qhrk*1rcBhmb()1ac#gbj1Q3s72:-!C :QqSWRZGHI1S!V8B;36",-.   EE1661S!V8aQh. FFCFQsAvXc1f_,sAv57 !$VBZs2w"1E!F G!Fa1ff!F G I !$Rs2w")=!> ?!>a1ff!> ? As H ?s :M2 M8 c <VR,VR,V\,V\,3wr4rVWA,V)V,W1,, ^\,V,V,,.p\PRV,V^,V^,,.p\ \ S !V4\\V44RRR1,4U U u.uFwrWV ,,NK up p !p \ \ V\\V44RRR1,4U U u.uFwrWV ,,NK up p !p W, #uup p iuup p irr*Nrr r r!rrr rr=r4r5)r>r?r|rrBrCrDr7rErrrrr9s&&& rm_ccpole#_laplace_trig_ltex.._ccpolesS'2c7BrFBrF:sgr"uqt|ac#gbj0 1eeRVS!Vc1f_ - !$VBZs2w"1E!F G!Fa1ff!F G I !$Rs2w")=!> ?!>a1ff!> ? As H ?s E #E c <VR,VR,V\,V\,3wr4rVWA,W4,W1,, ^\,V,V,, .p\PR\,V,V^,)V^,, .p\ \ S !V4\\V44RRR1,4U U u.uFwrWV ,,NK up p !p \ \ V\\V44RRR1,4U U u.uFwrWV ,,NK up p !p W, #uup p iuup p irIrJ)r>r@r|rrBrCrDr7rErrrrr9s&&& rm_rspole#_laplace_trig_ltex.._rspolesS'2c7BrFBrF:sgqtad{QqSWRZ/ 0eeRT#XQwa/ 0 !$VBZs2w"1E!F G!Fa1ff!F G I !$Rs2w")=!> ?!>a1ff!> ? As H ?s E -E c <VR,VR,rCWA,W4V, ,.p\P\PV^,).p\\ S !V4\ \ V44RRR1,4UUu.uFwrxWrV,,NK upp!p \\ V\ \ V44RRR1,4UUu.uFwrxWrV,,NK upp!p W, #uuppiuuppir<)rr rrr=r4r5) r>rAr|rrBr7rErrrrr9s &&& rm_sypole#_laplace_trig_ltex.._sypoles3C2gqr'{ #eeQVVadU # !$VBZs2w"1E!F G!Fa1ff!F G I !$Rs2w")=!> ?!>a1ff!> ? As H ?s D D cHVR,VR,r2TpW, pWE, #)rr*rq)r>r|rrBrrs&& rm _simplepole'_laplace_trig_ltex.._simplepoles$3C2  Es rpNr*rT)reverse) r5popr4r r!r"r#rrr,)r-rr|resultsplanesrFrKrNrQrTr> i_imagsym i_realsym i_pointsymireal_eqrealsymimag_eqimagsymindices_to_popr9s&&& @rm_laplace_trig_ltexrcsG F. b'A+ VVX   s2wAfb )Gfr *Gfb )Gfr *G7r"v{r"v{ WB1 WWB1  (%)*?* NN"-,bmC.@.-q2 3 MM#bCk* +(J?N     -#q $  " NN72)}S'91= > MM"R& ! FF9   " NN72)}S'91= > MM#bf+ & FF9   # NN72*~c':A> ? MM#bf+ & FF:  NN;r- . MM"R& ! =#v, &&rpc \RRR7pVP\\\\ 4'gR#\ VPW44wrE\WC4wrg\V4^8dR#VPV4'g(\WcV4wrWX,V \P3#.p .p \WSVRR7wrpVFgpV PVR,V PW"VR,, 4,4V PV \VR,4,4Ki \!V !PW14\#V !V3#) z This rule covers trigonometric factors by splitting everything into a sum of exponential functions and collecting complex conjugate poles and real symmetric poles. rTrealNFrr*r)rrr2r1r*r(r&rr0r5rcrrrr"r rr,)rt_r|rrrr-r.rrrYrXGG_planeG_condr/s&&& rm_laplace_rule_trigrk[s  cA 66#sD$ ' ' rwwr~ .DA !! 'FB 2w{ 5588!"+sAqvv~/a%HFB NN2c7166!r#wY#77 8 MM'"RW+- . =  a $c6lF ::rpc0\RV.R7p\RV.R7p\R4pVPV\WQV34,4pV'Ed>Wd,P'Ed%We,P Uu.uFqwP V4NK pp\V4^8Xd\R4.p \Wd,4Fyp V ^8XdWe,PV^4p M#\We,W34PV^4p V PW&V,V , ^, ,V ,4K{ \We,WRR7wrpWc,W&V,,V ,\V !, ,W3#R#uupi) z This function looks for derivatives in the time domain and replaces it by factors of `s` and initial conditions in the frequency domain. For example, if it gets ``(diff(f(t), t), t, s)``, it will compute ``s*LaplaceTransform(f(t), t, s) - f(0)``. rrrrz" rule: time derivative (4.1.8)FrN)rrrr is_integerrirsumrur4rr"rr)rrr|rrrrrrrr*rrr r s&&& rm_laplace_rule_diffro|s9 S1#A S1#ASA ''!Jqa&)) *C ssv   "v{{ +{!UU1X{ + q6Q; 7 8A36]6 Aq)A"36A62771=AVAXaZ*+ # +361%HIA2FA1vIaK#q'12R< <  ,sFc VP'Ed^.p^.p\P!V4F>pVPV4'dVP V4K-VP V4K@ \ V4^8Ed\ V4p\Wa4P4p\ V4pV^8EdH\ V4p \WVRR7wrp V .p Rp\V R,V4)pV P\4'dR\V^, 4F:pV P RV^,,\WV^,4,4K< MUV'dNV P V4\V^, 4F&pV P \V R,V4)4K( V'dI\!\V4Uu.uF'pWxV, ^, ,V V,,NK) up!pVW3#\#RV.R7p\#R4pVP%VV,V,4;p'dvVV,P&'d]VV,P('dD\VV,WRR7wrp RVV,,\WVV,34,W3#R# \dRpELi;iuupi)z This function looks for multiplications with polynoimials in `t` as they correspond to differentiation in the frequency domain. For example, if it gets ``(t*f(t), t, s)``, it will compute ``-Derivative(LaplaceTransform(f(t), t, s), s)``. FrrrrNr)is_Mulrr!rr"r5rrQ all_coeffsrr ValueErrorrLaplaceTransformr4r rrrrmr)rrr|pfacofacfacpexpcNoexr_p_c_derid1r*rrrrs&&& rm_laplace_rule_sdiffrs< xxxss==#C  ## C  C $ t9q=t*Cc((*BBA1u4j/EJ ttBx++B66*++"1Q3Z R1Q3K 2!A#0F$FG(KKO"1Q3Z T$r(A%6$67(qBAb1QiQ//BCAr;& S1#A S Aggad1fos q6   Q!3!3!3+CFA5IJBBQ<RSV 55r= = )"BCsJ8 -K 8 K K c\VRR7pVP'd\W1VRR7#\V4pVP'd\W1VRR7#\V4pVP'd\W1VRR7#W08wd\W1VRR7#\\ V44pVP'd\W1VRR7#R#)aj This function tries to expand its argument with successively stronger methods: first it will expand on the top level, then it will expand any multiplications in depth, then it will try all available expansion methods, and finally it will try to expand trigonometric functions. If it can expand, it will then compute the Laplace transform of the expanded term. FdeeprN)r is_Addrr r )rrr|rs&&& rm_laplace_expandrs quAxxx!!E::1 Axxx!!E::q Axxx!!E::v!!E::{1~Axxx!!E:: rpc\\\\\\ \ .pVFpV!WV4;pfKVu# R#)z_ This function applies all program rules and returns the result if one of them gives a result. N)rrr rrkror)rrr| prog_rulesp_ruleLs&&& rm_laplace_apply_prog_rulesrsG*+>)+<$$&9;J a A -H rpc\4wr4pRpRpVFwrrp Wl8wdV !VPW/44pT pVPV4p V 'gKBV PV 4pT\ P 8XgKkT PT 4PYR/4T PT 4\ P 3u# R# \dKi;i)z^ This function applies all simple rules and returns the result if one of them gives a result. N)rrrxreplacerrr)rrr| simple_rulesrgrprep_oldprep_ft_doms_domcheckplaneprepmars&&& rm_laplace_apply_simple_rulesrs 01LbH F,8(eD  !&&!/*FH \\%  2 NN2& AFF{r*//8r*AFF44-9   sC CCcVP'g:\RRR7p\VPW/4V4PW!/4#\ V4p.pVP EFwrE\ V\4'doWP 9d_\ V\\34'dVu#VP\VPVP, 4V,4K\ V\4'd\VP 4^8XdjVP FVpVP V8Xd?VP\VPVP, 4V,4KRVuu# EK#\ V\"4'd\VP 4^8XdVP wrxVP V8XdVP V8Xd~RVP$9dYrVP\VPVP, 4\VPVP, 4, V,4EKVu#Vu# \'V!#)z This function converts a Piecewise expression to an expression written with Heaviside. It is not exact, but valid in the context of the Laplace transform. rTre>)is_realr_piecewise_to_heavisiderr0rirrrrr"r:gtsrrKr5lhsrLrr) rrrrrrc2r)r(s && rmrrs 999 #D !&qzz1&'91=FFvNNAA AFF dJ ' 'AN$R)) 488dhh#67:; b ! !c$))n&9ii66Q;HHYrvv7:;H  c " "s499~':YYFBvv{rvv{"))#rvv/rvv/01345HAB 7Nrpc \R4p\R4p\R4p\R4p\V\4'd7VP\4'gVP\ 4'gV#VP 4FwrgVP\ V!V4WC44;pe$W,W,8XdV!W,4u#VP\ V!V4W4V44;pfKvW,W,8XgKV!W,4u# VPp VPU u.uFp \W4NK p p V !V !#uup i)a This helper function takes a function `f` that is the result of a ``laplace_transform`` or an ``inverse_laplace_transform``. It replaces all unevaluated ``LaplaceTransform(y(t), t, s)`` by `Y(s)` for any `s` and all ``InverseLaplaceTransform(Y(s), s, t)`` by `y(t)` for any `t` if ``fdict`` contains a correspondence ``{y: Y}``. Parameters ========== f : sympy expression Expression containing unevaluated ``LaplaceTransform`` or ``LaplaceTransform`` objects. fdict : dictionary Dictionary containing one or more function correspondences, e.g., ``{x: X, y: Y}`` meaning that ``X`` and ``Y`` are the Laplace transforms of ``x`` and ``y``, respectively. Examples ======== >>> from sympy import laplace_transform, diff, Function >>> from sympy import laplace_correspondence, inverse_laplace_transform >>> from sympy.abc import t, s >>> y = Function("y") >>> Y = Function("Y") >>> z = Function("z") >>> Z = Function("Z") >>> f = laplace_transform(diff(y(t), t, 1) + z(t), t, s, noconds=True) >>> laplace_correspondence(f, {y: Y, z: Z}) s*Y(s) + Z(s) - y(0) >>> f = inverse_laplace_transform(Y(s), s, t) >>> laplace_correspondence(f, {y: Y}) y(t) rr|rr) rrrrrtInverseLaplaceTransformitemsrrlrilaplace_correspondence) rfdictrr|rrrYrrlr"ris "" rmrrFsH S A S A S A S A1d##EE*++566 gg.qtQ:;;HDADLQT7Ngg5adA!DEEDADLQT7N 66D:;&& A&3 "3 .&D A ; BsEc VP4Fwr4\\V44FpV^8Xd"VPV!^4V^,4pK+V^8Xd7VP\ \ V!V4V4V^4V^,4pKhVP\ \ V!V4W34V^4WE,4pK K V#)a This helper function takes a function `f` that is the result of a ``laplace_transform``. It takes an fdict of the form ``{y: [1, 4, 2]}``, where the values in the list are the initial value, the initial slope, the initial second derivative, etc., of the function `y(t)`, and replaces all unevaluated initial conditions. Parameters ========== f : sympy expression Expression containing initial conditions of unevaluated functions. t : sympy expression Variable for which the initial conditions are to be applied. fdict : dictionary Dictionary containing a list of initial conditions for every function, e.g., ``{y: [0, 1, 2], x: [3, 4, 5]}``. The order of derivatives is ascending, so `0`, `1`, `2` are `y(0)`, `y'(0)`, and `y''(0)`, respectively. Examples ======== >>> from sympy import laplace_transform, diff, Function >>> from sympy import laplace_correspondence, laplace_initial_conds >>> from sympy.abc import t, s >>> y = Function("y") >>> Y = Function("Y") >>> f = laplace_transform(diff(y(t), t, 3), t, s, noconds=True) >>> g = laplace_correspondence(f, {y: Y}) >>> laplace_initial_conds(g, t, {y: [2, 4, 8, 16, 32]}) s**3*Y(s) - 2*s**2 - 4*s - 8 )rr4r5rrr )rrrricr*s""" rmlaplace_initial_condsrsDs2wAAvIIadBqE*aIId:adA#61=r!uEIId:adQF#;QBBEJ   Hrpca\P!V4p.p.p.p.pVEF7p V PSRR7wrV P\4'dc\P!V P \ 44p V F1p V PSRR7wrVPW,V34K3 KV P\8Xd{V P\S44'g[\P!\V S44p V F1p V PSRR7wrVPW,V34K3 EK%VPW34EK: VEFwrV P\4'd!\V SV4\PR3pEMHV P\ S44'd<V P\S44'gV P\ S4^4p \!V SV4;pf$\#V SV4;pf\%V SV4;peM\&;QJd2V3RlV P)\*44F 'gK RM' RM#!V3RlV P)\*444'd \V SV4\PR3pM2\-V SW#R7;peM\V SV4\PR3pVwpppVPV V,4VPV4VPV4EK \V!pV'dVP/RR7p\1V!p\3V!pVVV3#)z Front-end function of the Laplace transform. It tries to apply all known rules recursively, and if everything else fails, it tries to integrate. Fas_AddTc3D<"TFqPS4xK R#5irr)rundefrgs& rmr%_laplace_transform..sG0FuYYr]]0F rdoit)rr!as_independentrrBrr:r"rlr/r9rrtrrrrrranyrrrarr,rL)rrgrrterms_tterms_stermsrY conditionsffr*ft_terms_termr?f1rri_pi_ci_rkr conditions&f&$ rmrrsmmBGG E FJ!!"U!3 66% & &]]2::i#89F--b-? adBZ( WW !"&&B*@*@]]#:2r#BCF--b-? adBZ(  LL! ! 66% & &!"b"-q/A/A4HAvvim$$RVVJrN-C-CWWYr]A.5b"bAAQ 3BB??Q )"b"55QBG0FGG0FGGG&b"b113E3EtL5B33!;?@%b"b113E3EtLc3qu c#;>']Fe, LEZ I 5) ##rpc:a]tRtRtoRtRtRtRtRtRt Vt R#) rtia Class representing unevaluated Laplace transforms. For usage of this class, see the :class:`IntegralTransform` docstring. For how to compute Laplace transforms, see the :func:`laplace_transform` docstring. If this is called with ``.doit()``, it returns the Laplace transform as an expression. If it is called with ``.doit(noconds=False)``, it returns a tuple containing the same expression, a convergence plane, and conditions. Laplacec DVPRR4p\WW5R7pV#)rFr)getra)selfrrr|hintsrELTs&&&&, rm_compute_transform#LaplaceTransform._compute_transforms#IIj%0 +A! H rpc\V\V)V,4,V\P\P34#r)rDr&rrr)rrrr|s&&&&rm _as_integralLaplaceTransform._as_integrals,#qbd) a%<==rpc ,VPRR4pVPRR4p\RVPVPVP34VPpVPpVPp\ WdWSR7pV'd V^,#V#)" Try to evaluate the transform in closed form. Explanation =========== Standard hints are the following: - ``noconds``: if True, do not return convergence conditions. The default setting is `True`. - ``simplify``: if True, it simplifies the final result. The default setting is `False`. nocondsTrFz[LT doit] (%s, %s, %s)r)rrWfunctionfunction_variabletransform_variabler)rr_nocondsrErgrrrs&, rmrLaplaceTransform.doit s99Y-IIj%0 '$--*.*@*@*.*A*A*C D # #  $ $ ]] rr > Q4KHrprqN) rh __module__ __qualname____firstlineno____doc___namerrr__static_attributes____classdictcell__ __classdict__s@rmrtrts%  E >rprtc aaaSPRR4pSPRR4p\V\4'Ed\VR4'dSPRR4'*pV'dRV'dJRp\ RRVR7\ \ 4;_uu_4VPVVV3R l4uuR R R 4#VU u.uFp \V SS3/SBNK p p V'd>\V !wrp \V4!.VPOV N5!pV\V !\V !3#\V4!.VPOV N5!#\VSS4PRVR 7wpppV'gVVV3#V# +'giLA;iuup i) a Compute the Laplace Transform `F(s)` of `f(t)`, .. math :: F(s) = \int_{0^{-}}^\infty e^{-st} f(t) \mathrm{d}t. Explanation =========== For all sensible functions, this converges absolutely in a half-plane .. math :: a < \operatorname{Re}(s) This function returns ``(F, a, cond)`` where ``F`` is the Laplace transform of ``f``, `a` is the half-plane of convergence, and `cond` are auxiliary convergence conditions. The implementation is rule-based, and if you are interested in which rules are applied, and whether integration is attempted, you can switch debug information on by setting ``sympy.SYMPY_DEBUG=True``. The numbers of the rules in the debug information (and the code) refer to Bateman's Tables of Integral Transforms [1]. The lower bound is `0-`, meaning that this bound should be approached from the lower side. This is only necessary if distributions are involved. At present, it is only done if `f(t)` contains ``DiracDelta``, in which case the Laplace transform is computed implicitly as .. math :: F(s) = \lim_{\tau\to 0^{-}} \int_{\tau}^\infty e^{-st} f(t) \mathrm{d}t by applying rules. If the Laplace transform cannot be fully computed in closed form, this function returns expressions containing unevaluated :class:`LaplaceTransform` objects. For a description of possible hints, refer to the docstring of :func:`sympy.integrals.transforms.IntegralTransform.doit`. If ``noconds=True``, only `F` will be returned (i.e. not ``cond``, and also not the plane ``a``). .. deprecated:: 1.9 Legacy behavior for matrices where ``laplace_transform`` with ``noconds=False`` (the default) returns a Matrix whose elements are tuples. The behavior of ``laplace_transform`` for matrices will change in a future release of SymPy to return a tuple of the transformed Matrix and the convergence conditions for the matrix as a whole. Use ``legacy_matrix=False`` to enable the new behavior. Examples ======== >>> from sympy import DiracDelta, exp, laplace_transform >>> from sympy.abc import t, s, a >>> laplace_transform(t**4, t, s) (24/s**5, 0, True) >>> laplace_transform(t**a, t, s) (gamma(a + 1)/(s*s**a), 0, re(a) > -1) >>> laplace_transform(DiracDelta(t)-a*exp(-a*t), t, s, simplify=True) (s/(a + s), -re(a), True) There are also helper functions that make it easy to solve differential equations by Laplace transform. For example, to solve .. math :: m x''(t) + d x'(t) + k x(t) = 0 with initial value `0` and initial derivative `v`: >>> from sympy import Function, laplace_correspondence, diff, solve >>> from sympy import laplace_initial_conds, inverse_laplace_transform >>> from sympy.abc import d, k, m, v >>> x = Function('x') >>> X = Function('X') >>> f = m*diff(x(t), t, 2) + d*diff(x(t), t) + k*x(t) >>> F = laplace_transform(f, t, s, noconds=True) >>> F = laplace_correspondence(F, {x: X}) >>> F = laplace_initial_conds(F, t, {x: [0, v]}) >>> F d*s*X(s) + k*X(s) + m*(s**2*X(s) - v) >>> Xs = solve(F, X(s))[0] >>> Xs m*v/(d*s + k + m*s**2) >>> inverse_laplace_transform(Xs, s, t) 2*v*exp(-d*t/(2*m))*sin(t*sqrt((-d**2 + 4*k*m)/m**2)/2)*Heaviside(t)/sqrt((-d**2 + 4*k*m)/m**2) References ========== .. [1] Erdelyi, A. (ed.), Tables of Integral Transforms, Volume 1, Bateman Manuscript Prooject, McGraw-Hill (1954), available: https://resolver.caltech.edu/CaltechAUTHORS:20140123-101456353 See Also ======== inverse_laplace_transform, mellin_transform, fourier_transform hankel_transform, inverse_hankel_transform rFr applyfuncz#deprecated-laplace-transform-matrixz Calling laplace_transform() on a Matrix with noconds=False (the default) is deprecated. Either noconds=True or use legacy_matrix=False to get the new behavior. z1.9)deprecated_since_versionactive_deprecations_targetc <\VSS3/SB#r)laplace_transform)fijrr|rs&rmr#laplace_transform..s 1#q! Eu ErpNrr)rrrMhasattrrTrVrUrrr=typeshaper,rLrtr)rrr| legacy_matrixrrrEradtrelements_transelementsavalsr f_laplacerrrs&ff&l rmrr+svNyyE*H *e,I!Z  WQ %<%<IIi// ]7C % */+. !!899{{EG:9 012/00Q$"$/0 2.1>.B+ G7QWW7h7  #u+sJ/???Aw88881a(--e7@.BHB1 1ax ':92sE2F2 F c a aa^RIHpHo ^RIHp\ RRR7oV V3RlpVP V4'dVPV4pVP'dM\VPUu.uFp\WSW44NK up!p \V PSV4V4R3#V!W\S)4R\P 3RRR 7wrV f{V!WS4p V fR#V P$'d4V P^,wrV P'\(4'dR#M\P*p V P-\.V4p V P$'dV PSV4X 3#\ R 4o\P03VV3R llp V P-\2V 4p R p V P-\V 4p \V PSV4V4X 3#uupi \"dRp EL)i;i) z5The backend function for inverse Laplace transforms. )meijerint_inversion_get_coeff_exp)inverse_mellin_transformrTrec<\V4^8wd \V!#V^,P^,PpS!VS4wr#V^,P^,pV^,P^,p\ ^\ V4, SV,, 4V,\ SV,^\ V4, , 4V,,#)z2Simplify a piecewise expression from hyperexpand. )r5r/riargumentr:r#)rir"coeffexponente1e2rrs* rmpw_simp7_inverse_laplace_transform_integration..pw_simps t9>d# #1gll1o&&(a0 !W\\!_ !W\\!_ aE lQ[0 1" 4 akAc%jL0 1" 4 5 6rpNF)needevalrucj<VP!\S)4S4pVPS4'd \W4#^RIHpV!V^8S4pVP S8Xd)\VP4p\SV,V4#\VP 4p\SV,)V4#)rZr) rr&rr:rrrr'r)r"H0rrrelr*rrs&& rmsimp_heaviside>_inverse_laplace_transform_integration..simp_heavisides HHS!Wa  5588S% %@Aq) 77a<CGG AQUB' 'CGG Aq1uXr* *rpc*\\V44#r)r r&)r"s&rmsimp_exp8_inverse_laplace_transform_integration..simp_expsc#h''rp)sympy.integrals.meijerintrrsympy.integrals.transformsrris_rational_functionapartrrrirbrErr&rrrGrrrDrrr/rr:)rr|rgrrrrrXrrrrrrrs&&&&$ @@@rmrbrbsNC cA 6 a  GGAJxxx vv!5Q1eN 21477*1aR4:L48%I  y a ( 9 >>>ffQiGAuuX66D IIi )~~~vva}d"" c A vv + + )^,A( #x A QVVAr]H -t 33c " s3G*-)G// H?Hcb^RIHpV!V4wr4VPV4'dVPV4P 4p\ V4^8XdSVwrgpWaV^V,, ,^,W, ,V^V,, ^,, ,pW4, #)rZ)fraction)rr rrrrr5) rr|r rrcfrrrs && rm_complete_the_square_in_denomr s/ a[FQq YYq\ $ $ & r7a<GA!a1gI>!#%q!A#wl23A 3Jrpc \R4p\R4p\RV.R7p\RV.R7p\RV.R7p\R4RpR pW , V\PV^3W0V,V),,W^, ,\ V)V,4,\ V4, \PV^3^V^,V^,,^,, \W!,4W!,\W!,4,, ^V^,,, \PV^3^W,, W^, ,\ V4, \PV^3^WV,V,,, \W2V,4W#,\ V4,, \PV^3.pWpV3#) z This is an internal helper function that returns the table of inverse Laplace transform rules in terms of the time variable `t` and the frequency variable `s`. It is used by `_inverse_laplace_apply_rules`. r|rrrrrz._inverse_laplace_build_rules is building rulescLVPV4# \dTu#i;ir)factorrO)rr|s&&rm_frac+_inverse_laplace_build_rules.._frac(s) 88A;  H s  ##cV#rrq)rs&rmsame*_inverse_laplace_build_rules..same.srp) rrrurrr&r?r2r1r@)r|rrrrrr _ILT_ruless rm_inverse_laplace_build_rulesrsp c A c A S1#A S1#A S1#A ;<  aq! sqbkM1s8C1I-eAh6 FFD!  AqDAI> CHqs3qs8|3a1f= q  AD11u:eAh&a8 AsQhJAs+QT%(]; q J ! rpcrV^8Xd'\R4\V4\P3#\ 4wr4pRpVP W/4pVFwrrp WkV 38wdV !W|,4p W3pX P V4pV'gK;T pV\PJd5V^,Uu.uFpVPV4NK ppV^,!V!pV\P8XgK\V4V PV4P WR/4,\P3u# R#uupi)8 Helper function for the class InverseLaplaceTransform. z rule: 1 o---o DiracDelta()rN) rur9rrrrrrr:)rr|rrrrg_prepfsubsrrrrrw_Frrrris&&& rm#_inverse_laplace_apply_simple_rulesrBs  Av01!}aff$$57JB E FFA7OE*4&e3 3K eiBKE XXe_ 2A01!51 25aD$KAFF{ |ENN2$6$;$;RG$DDaffLL+5  6s-D4c R\RV.R7p\RV.R7p\R4pVPV\WaV34,4pV'dUWu,P'd=\ R4\ Wv,WVRRR7wrV)Wu,,V,V 3#R#) rrrrrz3 rule: t**n*f(t) o---o (-1)**n*diff(F(s), s, n)Fr dorationalN)rrr rmru_inverse_laplace_transform) rr|rrrrrrrrs &&&& rm_inverse_laplace_diffr$_s S1#A S1#A S A :aQ(( )B beDE) E15BRU{1}a rpc $\RV.R7p\R4pVPV4'g#V\V4,\P3#VP\ 4'gR#VP \ WA,44pV'dhWd,P'd4\R4\W&V,,4\P3#\WW#4\P3#VP \ WA,4V,4pV'dcWd,P'd/\R4\We,WWd,,VRRR 7#\WW#4\P3#R#) rrrrNz* rule: exp(-a*s) o---o DiracDelta(t-a)z5 rule: exp(-a*s)*F(s) o---o Heaviside(t-a)*f(t-a)FTr!) rrr9rrr&rr rurr#)rr|rrrrrs&&&& rm_inverse_laplace_time_shiftr&ps S1#A S A 5588A&& 55:: ''#ac( C 6    ? @aAh'/ /*1:AFFB B ''#ac(1* C 6    J K-SV8UUtM M+1:AFFB B rpcVPV4'g#V\V4,\P3#\ VP ;p4^8Xd\ RV.R7pV^,PW, 4;p'dw\We,4P'dV\R4\We,)V,4\VPV4WV4,\P3#R#)rrrz& rule: F(s-a) o---o exp(-a*t)*f(t)N)rr9rrr5rirrr rrur&rrl)rr|rrrirrs&&&& rm_inverse_laplace_freq_shiftr(s 5588A&& 166>4a qc "q'--$ $B $"RU)*?*?*? ; <RUF1H 'q 1?@ABI I rpc \RV.R7p\R4pVPW,V,4pV'dWd,P'dWd,P'd\ R4\ We,WVRRR7wrxVP \V4^4pVP\4'd\WrWd,4V3#\V4\WrWd,4,V3#R#) rrrrz+ rule: s**n*F(s) o---o diff(f(t), t, n)FTr!N) rrrmrrur#rr:rrr) rr|rrrrrrrs &&&& rm_inverse_laplace_time_diffr*s S1#A S A ''!$q&/C sv   SV%7%7%7<=) FA%%DB IIilA & 55( ) )cf%q( (Q<Q36 22A5 5 rpc 4Gaa\RV.R7p\RV.R7o\RV.R7p\RV.R7oRp\PpVP4pVU u.uF.qP WAV,,S,S,4NK0 p p RV 9dR#\P p .p .p .pV FpW,^8Xd W,p KVS,P 'dV PV4KFVS,P'dV PV4KrVPV4K \V VV3RlR7p \V VV3R lR7p \V4^8wdR#\V 4^8XEd#\V 4^8XEdV ^,S,R8XdV ^,V,\P8XdV ^,S,V ^,V,, p^V ^,V,, V ,pVP 'd~V\\4, \V4, VV,\V^,V,4,\V\V4,4,, p\!R 4E MV ^,S,R 8XEd>V ^,V,\P8XEdV ^,S,V ^,V,, pV^,p^V ^,V,^,, V ,pVP 'dV^^\\4, \V4,\V4,, ^^V,V,, \VV,4,\#\V4\V4,4^, ,,,p\!R 4EMV ^,S,R!8XEdFV ^,V,\P8XEd"V ^,S,V ^,V,, p^V ^,V,^,, V ,pVP 'dV^\\4, V^,V,^,,\V4,VV,\V^,V,4,^V^,,V,^,,\V\V4,4,, ,p\!R 4EM*V ^,S,R"8XEdV ^,V,\P8XEdiV ^,S,V ^,V,, p^V ^,V,^,, V ,^, pVP 'EdVV^V^,,V^,,^ V^,,V,,^,,\V^,V,4,\V\V4,4,^\\4, V^,,V\^4^, ,,^V^,,V,^,,, ,p\!R 4EMV ^,S,\P)8XdV ^,V,^8Xd{\V ^,S,V ^,V,, 4p^\V ^,V,4, V ,pV\%^VV,4,p\!R4EM\V 4^8XEd9\V 4^8XEd(V ^,S,R!8XEdV ^,V,\P8XEd_V ^,S,\P8XEd;V ^,S,^8XEd%V ^,S,p\V ^,V,4V ^,V,, V ,pV^V^,,V^,,^V^,,V,,^,,\V^,V,4,\V\V4,4,^\\4, V,V^,V,^,,\V4,, p\!R4V ^,S,R8XEdxV ^,V,^8XEdbV ^,S,\P8XEd>V ^,V,^8XEd(V ^,S,V ^,V,, pV ^,S,V ^,V,, p\V ^,V,4V ^,V,, V ,pV\V)V,4\V4, \\4, \VV, 4\V)V,4,\#\VV, 4\V4,4,,,p\!R4EM\V 4^8XE d\V 4^8XE dV ^,S,R8XEd8V ^,V,^8XEd"V ^,S,\P)8XdV ^,V,^8XdV ^,S,^8XdV ^,S,)V ^,V,, p^\V ^,V,4, V ^,V,, V ,pVP 'd^V\V4, \VV,4,\#\V4\V4,4,p\!R4EMV ^,S,R8XEd)V ^,V,^8XEdV ^,S,^8XdV ^,S,R8XdV ^,V,\P8XdV ^,S,V ^,V,, p^V ^,V,, V ^,V,, V, V ,pVP 'dSV^\V^,V,4\V\V4,4,, ,p\!R4EMV ^,S,R8XEd3V ^,V,\P8XEdV ^,S,\P)8XdV ^,V,^8XdV ^,S,^8XdV ^,S,V ^,V,, p^V ^,V,\V ^,V,4,, V ,pVP 'dLV\V^,V,4,\V\V4,4,p\!R4EMV ^,S,\^4)^, 8XEd|V ^,V,^8XEdfV ^,S,^8XEdPV ^,S,R8XEd:V ^,V,\P8XEdV ^,S,V ^,V,, p^V ^,V,\^4^, ,V ^,V,,, V^,, V ,pVP 'dV^\\4, V,\V4,\V^,V,4\V\V4,4,,^, ,p\!R4EMV ^,S,R 8XEdV ^,V,\P8XEdqV ^,S,R8XEd[V ^,V,^8XEdEV ^,S,^8XEd/V ^,S,V ^,V,, pV^,p^V ^,V,^,, V ^,V,, V ,pVP 'dV^V, ^V,^V, , \VV,4,\\V4\V4,4,,^\\4, \V4, \V4,, ,p\!R4EMIV ^,S,R 8XEd}V ^,V,\P8XEdYV ^,S,\P)8XEd4V ^,V,^8XEdV ^,S,^8XEdV ^,S,V ^,V,, p^V ^,V,^,, \V ^,V,4, V ,pVP 'dV^\\4, \V4,^V,V,\V^,V,4,\V\V4,4,, ,p\!R4EMV ^,S,R!8XEdqV ^,V,\P8XEdMV ^,S,\P)8XEd(V ^,V,^8XEdV ^,S,^8XdV ^,S,pV \V ^,V,4, V ^,V,, pV^V^,,V,^,V,\V^,V,4,\V\V4,4,^\\4, V,V\^4^, ,,, ,p\!R4EM1V ^,S,R8XEdCV ^,V,^8XEd-V ^,S,\P)8XEdV ^,V,^8XdV ^,S,V ^,V,, pV ^,S,V ^,V,, pV \V ^,V,4, V ^,V,, pV^\VV, 4, \V)V,4,\#\VV, 4\V4,4,,p\!R4E M\V 4^8XEdh\V 4^8XEdWV ^,S,R8XEdEV ^,V,^8XEd/V ^,S,R8XEdV ^,V,\P8XEdV ^,S,\P8XEdV ^,V,^8XEdV ^,S,^8XEdV ^,S,V ^,V,, pV^,pV ^,S,)V ^,V,, p\V ^,V,4V ^,V,, V ^,V,, VV, , V ,pVP 'dVP 'dVV\VV,4,\\V4\V4,4,\V4\V4,\VV,4,\\V4\V4,4,,V\VV,4,, ,p\!R4E M[V ^,S,R8XEdV ^,V,^8XEd~V ^,S,^8XEdhV ^,S,R8XEdRV ^,V,\P8XEd.V ^,S,^8XEdV ^,V,\P8XdV ^,S,V ^,V,, pV ^,S,V ^,V,, pVV,^8XdV ^,V,V ^,V,, V ^,V,, V ,pV^\V^,V,4,\V\V4,4,^, ,p\!R4EMV ^,S,R8XEdV ^,V,^8XEdV ^,S,^8XEdV ^,S,R 8XEdV ^,V,\P8XEdV ^,S,^8XEdnV ^,V,\P8XEdJV ^,S,V ^,V,, pV ^,S,V ^,V,, pVV,^8XdV ^,V,^,V ^,V,, V ^,V,^,, V ,pV^^V^,,V,\V^,V,4,\V\V4,4,,^\\4, V,\V4,, ,p\!R4EMV ^,S,R8XEd=V ^,V,^8XEd'V ^,S,^8XEdV ^,S,R!8XEdV ^,V,\P8XEdV ^,S,^8XEdV ^,V,\P8XEdV ^,S,V ^,V,, pV ^,S,V ^,V,, pVV,^8XEdDV ^,V,^,V ^,V,, V ^,V,^,, V ,pV^^V^,,V^,,^V^,,V,,^,,\V^,V,4,\V\V4,4,^\\4, V,\V4,^V^,,V,^,,, ^, ,p\!R4EM_\V 4^8XEdO\V 4^8XEd>V ^,S,R8XEdV ^,S,^8XEdV ^,V,^8XEdV ^,S,R8XEd{V ^,V,^8XEdeV ^,S,\P)8XEd@V ^,V,^8XEd*V ^,S,)V ^,V,, p^V ^,V,, V ^,V,, \V ^,V,4, V ,pVP 'dVV\^4)^, ,\VV,4,\#\V4\V4,4,^V, \\4, \V4,, ,p\!R4EMkV ^,S,R8XEdVV ^,V,^8XEd@V ^,S,R8XEd*V ^,V,\P8XEdV ^,S,\P)8XEdV ^,V,^8XEdV ^,S,^8XEdV ^,S,V ^,V,, pV^,pV ^,S,)V ^,V,, p^V ^,V,, V ^,V,, \V ^,V,4, \V4VV, ,, pVP 'dVP 'dV\V4\VV,4,\\V4\V4,4,\V4\VV,4,\#\V4\V4,4,,\V4\VV,4,, ,p\!R4VfR#\'V4V,V3#uup i)#rrrrrrNc><VS,VS,^8gVS,3#rrqrrrs&rmr-_inverse_laplace_irrational..1qtqy!A$(?rprc><VS,VS,^8gVS,3#rrqr-s&rmrr.r/rpz rule 5.3.4z rule 5.3.10z rule 5.3.13z rule 5.3.16z rule 5.3.35/44z rule 5.3.14z rule 5.3.22z rule 5.3.1z rule 5.3.5z rule 5.3.7z rule 5.3.8z rule 5.3.11z rule 5.3.12z rule 5.3.15z rule 5.3.23z rule 5.3.6z rule 5.3.17z rule 5.3.18z rule 5.3.19z rule 5.3.2z rule 5.3.9rr)rrras_ordered_factorsrr rr"r sortedr5rr.rr&r<rur;r6r:)rr|rrrrrkrfarr constantszerospolesrestr%rk_a_sqb_a_numrrs&&&& @@rm_inverse_laplace_irrationalr>s S1#A S1#A S1#A S1#A FI   B*, -"Q''1T6!8a- "B - rzI E E D 7a<!I !W LL  !W LL  KK  5? @E 5? @E 4yA~ 5zQ3u:? 8A;" q!!6q!U1Xa[(B58A;y(B~~~tBxKQ'rE#b!eAg,&tBtAwJ'778() 1Xa[B 58A;!&8A;uQx{*DqB58A;>!)+BAd2hJtBx/Q771R463r!t9,c$r(472B.CA.EFGH)* 1Xa[B 58A;!&q!U1Xa[(B58A;>!)+B~~~$r( BE!GAI.tAw61SQq\)1RU719Q;7RQZ8HHIJ)* 1Xa[B 58A;!&q!U1Xa[(B58A;>!)+A-B~~~1RU71a4<2q5 2145c"a%'lBRQZ()$r( 2q5(QqT!V4aAgaikBCD)* 1Xa[QVVG #a q(8eAhqk%(1+-.B4a $$Y.B'!RT*+F ( ) UqSZ1_a r!eAhqkQVV&;a qvv%%(1+*:q!BeAhqk"58A;.y8BAb!eGAqDL2q5*1,-c"a%'l:RQZ !#$T"X:b="a%'!)#!%(1+-i7B"!AbD#bd)3Db$q'9I4JJJ$r( 48+DG345)*a r!eAhqkQVV&;a w&58A;!+;a q q!U1Xa[(B58A;>!$uQx{"33I=B~~~$r( 47*"Qs2q57|+DDG,<<=>)*a r!eAhqkQVV&;a w&58A;!+;a q q!B4a ,,U1Xa[8B2q51aBE!G ,T"T!W*-==$r( 2 a!A$q&k)*+F % &a r!eAhqkQ&6a w&58A;!+;q!U1Xa[(Bq!U1Xa[(B4a ,,U1Xa[8B$r"u+ c2#a%j(T"R%[a-@)AACF % & UqSZ1_a r!eAhqkQ&6a r!eAhqkQVV&;a qvv%%(1+*:a q 8A;uQx{*DqB(1+eAhqk)BeAhqk"58A;.uQx{:BrEB9LBBNNNs2a4yLd2htAw&6!77HT"X%c"Q$i/T"Xd1g5E0FFGs2a4yL!"()a r!eAhqkQ&6a q U1Xa[B%6a qvv%%(1+*:a qvv%!HQKa +Eq!U1Xa[(B%x1}1Xa[q!,U1Xa[8Bc"a%'lN447 #33A57)*a r!eAhqkQ&6a q U1Xa[B%6a qvv%%(1+*:a qvv%!HQKa +Eq!U1Xa[(B%x1}1Xa[!^E!HQK/a Q>yH"a% #b!eAg,.tBtAwJ/???d2hJrM$q')*+)*a r!eAhqkQ&6a q U1Xa[B%6a qvv%%(1+*:a qvv%!HQKa +Eq!U1Xa[(B%x1}1Xa[!^E!HQK/a Q>yHqQwq!t|Ab!eGAI-a/0RU1W=DG$%%&tBxZ]47%:Ab!eGAIaK%HIIJKL)* UqSZ1_a r!eAhqkQ&658A;!;Ka r!eAhqkQ&6a w&58A;!+;(1+eAhqk)B58A;uQx{*4a +<rr|rrrrrs&&&& rm!_inverse_laplace_early_prog_rulesrBs2 ..Ja' 'A 4H rpcp\\\\\.pVFpV!WW#4;pfKVu# R#r@)r&r(r*r$r>rAs&&&& rm!_inverse_laplace_apply_prog_rulesrDsB ./J,.C-/Ja' 'A 4H rpc VP'dR#\VRR7pVP'd\WAW#RRR7#\V4pVP'd\WAW#RRR7#\V4pVP'd\WAW#RRR7#VP V4'd VP V4P 4pVP'd\WAW#RRR7#R#)rNFrTr!)rr r#r r r r)rr|rrrs&&&& rm_inverse_laplace_expandrF s  yyyrAxxx) !Ut= =2Axxx) !Ut= =r Axxx) !Ut= = q!! HHQK   xxx) !Ut= = rpc \R4pVPV4p\P!V4p.p\P .p VEFp V P 4wrV PV4P4p V ^,pV Uu.uF qV, NK p pV PV4P4Uu.uF qV, NK pp\V 4^8Xd`\V4^, p\V4F<pV^,\VVV^,, 4,pVPV4K> K\V 4^8XdMV^,\V ^,)V,4,pVP\V4V,4EKT\V 4^8XEdAV ^,^, pV ^,V^,, P4p\V4^8Xd\P .V,p\#V4wppV^8Xd@VV,V^VV,, ,,\V)V,4,pEMkRpVP$'dV)pRp\'\)V^,V, V4P+44^,p\-V4P/4pV'd~V\V)V,4,\1VV,4,VVV,, V, \V)V,4,\3VV,4,,pM|V\V)V,4,\5VV,4,VVV,, V, \V)V,4,\7VV,4,,pVP\V4V,4EK\9WW#VRR7wppVPV4V PV4EK \V!pV'dVP/RR7pV\;V !3#uupiuupi)rx_FTr!r)rr rr!rrrrrrr5 enumerater9r"r&r:rrtupler rrPrr.rr(r*r1r2r#rL)rr|rrrrHrrrrr%rrrEdc_leadrr7rzrrrrlrhypb2bsrrrks&&&&$ rm_inverse_laplace_rationalrP(sH B  A MM! EG&&J$$& YYq\ $ $ &Q%!# $Aii $!"1!8!8!: ;!:Aii!: ; r7a<B Ar]aDAq1v..q!#W\1c2a5&(m#A NN9Q<> * W\1aAAq!t##%A2w!|ffX]9DAqAvqSAacE]C1I-===AC%Aa,1134Q7!W%%'#qbd) DAJ.!AaC%2r!t92%%)"Q$Z200#qbd) C1I-1Q3 3r!t90DSAY0NNA NN9Q<> *1HHHB NN2    d #QT']Fe, 3 # ##S% ;s Q>Qc .a\P!V4p.p.pVEFp V P\4'd3V P SS)4P 4P SS)4p V P SRR7wrV'd,V PS4'd\V SW#VR7;p fI\V SV4;p f7\V SW#4;p f%\V SW#4;p f\V SW#4;p eM\;QJd2V3RlV P\44F 'gK RM' RM#!V3RlV P\444'd\!V SW#4\"P$3p M2\'V SW#VR7;p eM\!V SW#4\"P$3p V wrVP)W,4VP)V4EK \V!pV'dVP+RR7p\-V!pVV3#)z Front-end function of the inverse Laplace transform. It tries to apply all known rules recursively. If everything else fails, it tries to integrate. Frrc3D<"TFqPS4xK R#5irr)rrrs& rmr-_inverse_laplace_transform..sB,A52,ArTr)rr!rr&rrRrr rPrrBrFrDrrrrrrrbr"rrL)rrrgrrr"rrrr%r*rrrrrkrs&f&&$$ rmr#r#bs MM" EGJ 88C==99R"%..055b2#>D""2e"4t88<</r2x99:!RDD72rII-aR??72rII  SBAGGL,ABSSSBAGGL,AB B B)B:AFFCA;r2x99AEF (B:AFFCA qu#KN']Fe,Z I 9 rpcpa]tRtRtoRtRt]!R4t]!R4tRt ] R4t Rt R t R tR tVtR #) riz Class representing unevaluated inverse Laplace transforms. For usage of this class, see the :class:`IntegralTransform` docstring. For how to compute inverse Laplace transforms, see the :func:`inverse_laplace_transform` docstring. zInverse LaplaceNonerc \Vf\Pp\P!WW#V3/VB#r)r_none_sentinelrF__new__)rrr|rroptss&&&&&,rmrXInverseLaplaceTransform.__new__s, =+::E ((uEEErpcXVP^,pV\PJdRpV#)N)rirrW)rrs& rmfundamental_plane)InverseLaplaceTransform.fundamental_planes( !  +:: :E rpc 0\WW0P3/VB#r)rbr])rrr|rrs&&&&,rmr*InverseLaplaceTransform._compute_transforms"5 !++6/46 6rpcVPPp\\W#,4V,W$\P \P ,, V\P \P ,,34^\P,\P ,, #)) __class___crDr&r ImaginaryUnitrPi)rrr|rrs&&&& rmr$InverseLaplaceTransform._as_integralss NN   SXaZ!)C%C"#aooajj&@"@"B C qttVAOO # % &rpc HVPRR4pVPRR4p\RVPVPVP34VPpVPpVPpVP p\ WdWWVRR7pV'd V^,#V#)rrTrFz[ILT doit] (%s, %s, %s)r!)rrWrrrr]r#) rrrrErrgrrrs &, rmrInverseLaplaceTransform.doits99Y-IIj%0 (4==+/+A+A+/+B+B+D E # #  $ $ ]]&& & B d D Q4KHrprqN)rhrrrrrrrWrdrXpropertyr]rrrrrrs@rmrrsQ E6]N sBF  6&rprc .aaaaSPRR4pSPRR4p\V\4'd+\VR4'dVP VVVV3Rl4#\ VSSS4P RVR7wrxV'dV#Wx3#)a Compute the inverse Laplace transform of `F(s)`, defined as .. math :: f(t) = \frac{1}{2\pi i} \int_{c-i\infty}^{c+i\infty} e^{st} F(s) \mathrm{d}s, for `c` so large that `F(s)` has no singularites in the half-plane `\operatorname{Re}(s) > c-\epsilon`. Explanation =========== The plane can be specified by argument ``plane``, but will be inferred if passed as None. Under certain regularity conditions, this recovers `f(t)` from its Laplace Transform `F(s)`, for non-negative `t`, and vice versa. If the integral cannot be computed in closed form, this function returns an unevaluated :class:`InverseLaplaceTransform` object. Note that this function will always assume `t` to be real, regardless of the SymPy assumption on `t`. For a description of possible hints, refer to the docstring of :func:`sympy.integrals.transforms.IntegralTransform.doit`. Examples ======== >>> from sympy import inverse_laplace_transform, exp, Symbol >>> from sympy.abc import s, t >>> a = Symbol('a', positive=True) >>> inverse_laplace_transform(exp(-a*s)/s, s, t) Heaviside(-a + t) See Also ======== laplace_transform hankel_transform, inverse_hankel_transform rTrFrc"<\VSSS3/SB#r)inverse_laplace_transform)Fijrrr|rs&rmr+inverse_laplace_transform.. s1#q!ULeLrpr)rrrMrrrr) rr|rrrrrErrs &fffl rmrmrmsZyyD)H *e,I!Z  WQ %<%<{{ LN N #1aE 2 7 7  8 +DAt rpcaaaaaaaaa a \R\S.R7woo o VVVVV3RloV3RloVV3RloVV V VV3RloV3RloS!V4#)zEFast inverse Laplace transform of rational function including RootSumza, b, nrc<VPS4'gV#VP'd S!V4#VP'd S!V4#VP'd S!V4#\ V\ 4'd S!V4#\ hr)rrrqryrrSNotImplementedError)e_ilt_add_ilt_mul_ilt_pow _ilt_rootsumr|s&rm_ilt#_fast_inverse_laplace.._ilt# skuuQxxH XXXA;  XXXA;  XXXA;  7 # #? 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