*hSSrSSKrSSKrSSKJrJrJr SSKJr SSK J r SSK J r SSK JrJrJrJrJrJrJrJrJrJr SSKJrJr SS KJrJrJrJrJ r J!r!J"r"J#r# SS K$J%r% SS K&J'r'J(r(J)r) SS K*J+r+J,r,J-r-J.r.J/r/J0r0 SS K1J2r2J3r3 SSK4J5r5J6r6J7r7J8r8 SSK9J:r:J;r;Jr>J?r? SSK@JArAJBrBJCrCJDrD SSKEJFrFJGrGJHrHJIrI SSKJJKrKJLrL SSKMJNrNJOrOJPrP SSKQJRrRJSrSJTrTJUrU SSKVJWrW SSKXJYrYJZrZ SSK[J\r\J]r]J^r^ SSK_J`r`JaraJbrbJcrcJdrd SSKeJfrf SSKgJhrh SSKiJjrj SSKkJlrl SSKmJnrn SSKoJprp SS KqJrrr SS!KsJtrtJuruJvrv SS"KwJxrx SqyS#rzS$r{S%r|\zS&5r}\zS'5r~\zS(5r\ S)5r\zS*5r\zS+5r\zS,5r\zS-5r\zS.5r\zS/5r\zS05r\zS15r\zS25r\zS35r\zS45r\zS55r\zS65r\zS75rS8rS9r\zS:5r"S;S<\]5rSPS=jr\zS>5r\zS?5r\ S@5r\zSA5r\zSB5r\zSC5r\zSD5r\zSE5r\zSF5r\zSG5r\zSH5r\zSI5r\zSJ5r\zSK5r"SLSM\]5rSQSNjrSOrg)RzLaplace 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)debugfc^U4SjnU$)Ncd>SSKJn U(dT"U0UD6$[S:Xa[S[R S9 [SS[-<TR <U<3[R S9 [S- qTR S:XdTR S :XaAS [l[S S[--[R S9 T"U0UD6nS [lOT"U0UD6n[S-q[SS[-<S U<3[R S9 [S:Xa[S[R S9 U$)Nr SYMPY_DEBUGzO ------------------------------------------------------------------------------file-LT-  _laplace_transform_integration&_inverse_laplace_transform_integrationFz**** %sIntegrating ...Tz---> zO------------------------------------------------------------------------------ )sympyr\ _LT_levelprintsysstderr__name__)argskwargsr\resultfuncs T/home/mande/repo/quber/.venv/lib/python3.13/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  $y.&9 K > -cjj 1 )rmros` rn DEBUG_WRAPrs0s4 KrqclSSKJn U(a'[SS[-<U<3[R S9 gg)Nrr[r_r`r])rdr\rfrergrh)textr\s rn_debugrvNs$! T)^T2Drqc^^^^^U4SjmUUUU4SjmU4SjmU4SjnSnSSKJn U"U5nU"U[T5nU"U[U4Sj5nU"U[U5n[ U5$) a Naively simplify some conditions occurring in ``expr``, given that `\operatorname{Re}(s) > a`. Examples ======== >>> from sympy.integrals.laplace import _simplifyconds >>> from sympy.abc import x >>> from sympy import sympify as S >>> _simplifyconds(abs(x**2) < 1, x, 1) False >>> _simplifyconds(abs(x**2) < 1, x, 2) False >>> _simplifyconds(abs(x**2) < 1, x, 0) Abs(x**2) < 1 >>> _simplifyconds(abs(1/x**2) < 1, x, 1) True >>> _simplifyconds(S(1) < abs(x), x, 1) True >>> _simplifyconds(S(1) < abs(1/x), x, 1) False >>> from sympy import Ne >>> _simplifyconds(Ne(1, x**3), x, 1) True >>> _simplifyconds(Ne(1, x**3), x, 2) True >>> _simplifyconds(Ne(1, x**3), x, 0) Ne(1, x**3) cn>UT:XagUR(aURT:Xa UR$g)Nra)is_Powbaser')exss rnpower_simplifyconds..powervs* 7 99A66MrqcB>URT5(aURT5(ag[U[5(aURSn[U[5(aURSnURT5(aT"SU- SU- 5$T"U5nUcgUS:a-[U5[T5U-:*[R :XagUS:a.[U5[T5U-:[R :Xaggg![ a gf=f)zRReturn True only if |ex1| > |ex2|, False only if |ex1| < |ex2|. Else return None. NrraFT)has isinstancer$rjrtrue TypeError)ex1ex2nabiggerr}r|s rnr_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  s)2D2D DDc>UR(d[U[5(a&UR(d[U[5(dX:$T"X5nUbU(+$X:$)zsimplify x < y ) is_positiverr$)xyrrs rnreplie_simplifyconds..repliesL*Q"4"4 As););EN 1L =5Lrqc8>T"X5nUS;ag[X5$)NTFT)r)rrbrs rnreplue_simplifyconds..replues" 1L !rqcBUS;a [U5$UR"U6$)Nr)boolreplace)r{rjs rnrepl_simplifyconds..repls"  8Ozz4  rqr) collect_absc>T"X5$Nrr)rrrs rn _simplifyconds..s va|rq)sympy.simplify.radsimprrrrr) exprr|rrrrrr}rs `` @@@rn_simplifycondsrUseB, ! 3 t D b& !D b3 4D j& )D T7Nrqc>[XR[55$)zg Expand an expression involving DiractDelta to get it as a linear combination of DiracDelta functions. )rOatomsr:rs rnexpand_dirac_deltars jj4 55rqcn^^^^[S5mUR[5(ag[U[ T*T-5-T[ R [ R45nUR[5(d;[URTT5U5[ R[ R4$UR(dgURSupEUR[5(agUU4Sjn[U5Vs/sH ov"U5PM nnUV s/sH5oS[ R :wdMU S[ RLdM3U PM7 n n U (d*UV s/sHoS[ R :wdMU PM n n [#[%U 55nSmUR'U4SjS9 U(dgUSupUU4S jn U(a[)UTU 5n[)U TU 5n [URTT5U5U "U 5[+U "U 554$s snfs sn fs sn f) zThe backend function for doing Laplace transforms by integration. This backend assumes that the frontend has already split sums such that `f` is to an addition anymore. r|Nrc, >^SSKJn [Rn[Rn[ [ U55n[S[T/S9upEpgpn U[[TU-U-55-U:U[[TU-U-55-U:*[[TU-U-U-U55U:[[TU-U-U-U55U:*[[[TU-5U-U-U55U:[[[TU-5U-U-U55U:*4n UGHJn [Rn /n[U 5GHnUR(a&TUR R";a UR$nUR(a'['U[([*45(a UR,nU HnUR/U5mT(dM O T(a?TUR0(a+TUTU- [2S- :Xa[5TTU-5*S:nUR/U[7U[[TU -55-U-5[TU-5U --- S:5mT(dNUR/[7U[[TU-U -U55U-- 5[TU-5U --S:5mT(dWUR/U[7[[[T5U-U -U55U-5[TU-5U --- S:5mT(a.[9U4SjXgXU 455(a[5T5TU:nUR;[4S5R=[5T5T5nUR(a<UR>S;d,URAT5(dURAT5(dX/- nGMU"UT5nUR(aUR>S;aX/- nGMURBT:Xa g [EURBU 5n GM U [RLa[GX5nGM8[IU[KU65nGMM X#R(a URL4$U4$) z6Turn ``conds`` into a strip and auxiliary conditions. r_solve_inequalityzp q w1 w2 w3 w4 w5clsexcludec3B># UHnTURv M g7fr)r).0wildms rn H_laplace_transform_integration..process_conds..s#->,TQtW00>,scDUR5R5S$Nr)r as_real_imag)rs rnrG_laplace_transform_integration..process_conds..s!((*"9"9";A">rq)z==z!=N)'sympy.solvers.inequalitiesrrNegativeInfinityrrJrIrr r$r#r&r%InfinityrK is_Relationalrhs free_symbolsreversedrrr reversedsignmatchrrr!r2allrsubsrel_oprltsr.r-rMrL canonical)condsrrauxpqw1w2w3w4w5patternsca_aux_dpatd_solnrr|ts @rn process_conds5_laplace_transform_integration..process_condss @  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>@@DRUAOOqxx.cnts = ~~rqc&>US*T"US54$Nrrarr)rrs rnr0_laplace_transform_integration..sqteS1Y/rqkeyc(>URTT5$r)r)rr|s_s rnsbs+_laplace_transform_integration..sbs!syyBrq)rrr:rDr'rZerorrErFrrr is_PiecewiserjrKfalselistrsortrr)frrsimplifyFcondrrrrconds2rrrrr|s `` @@rnrbrbs c AuuZ!C1I+1661::67A 55??2113E3EqvvMM >>ffQiGAuuX=>~(1 7!]1 E 7:AA$gg#aA$a&8&88F: "6Udaggo!U6  !E  JJ/J0  1XFA  1a #S!Q' QVVAr]H -s1vz#c(7K KK- 8:7s$ H($H-H-H--H2 H2c[U[5(dU$URU5=nbUR5$URnUR Vs/sHn[ XA5PM nnU"U6$s snf)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_exprrmrj_laplace_deep_collect)rrrrmr#rjs rnrr)sk a   YYq\&yy{ 66D56VV [UT5$r)r)rrs rndco!_laplace_build_rules..dcoSs,Q22rqz&_laplace_build_rules is building rulesrrarg?)(rr rvrrrr:r'r$rLrMrr;r>r#rr/r=r@rBr!r8rAr( EulerGammar?r3r"r*r4r2r+r)r<r7Halfrr6r9r,) r|rrrrrrlaplace_transform_rulesrs @rn_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 ((rqc[SU/S9n[SSS9nURU5nU(aXTRSR U5nUR"X1-5nU(aVXsR (aCXsS:wa;[ S5 [SXs- XTRU5-XXs- SS 9upn XU 4$g ) 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``. rrgra)nargsrz rule: time scaling (4.1.4)FrN) r rrrjcollectrrv_laplace_transformrm) rrr|rrma1r#ma2rprcrs rn_laplace_rule_timescalers S1#AS"A ''!*C fkk!n$$Q'iin 36%%#&A+ 4 5*#&Q'cfHuFIA22;  rqc[SU/S9n[S5n[S5nUR[U5U-5=n(Ga9XdRX- 5=n(aXsR(aE[ S5 [ XeR XXs-5XSS9upn [Xs*U-5U-X4$XsR(a[ S5 [ XeXSS9upn XU 4$XdRX1- 5=n(awXsR(a3[ S 5 [ S [XU- 5- Xe-XSS9upn XU 4$XsR(a[ S 5 S S [R4$g )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)Fr z8 rule: Heaviside factor; negative time shift (4.1.4)z rule: Heaviside window openraz" rule: Heaviside window closedrN) r rr;rrvr rr' is_negativerr) rrr|rrrr r rrrs rn_laplace_rule_heavisider,sp 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~% rqcH[SU/S9n[S5n[S5nUR[U5U-5nU(a]XdRU5RX1-5nU(a2[ S5 [ XeXXs- SS9upn X[ Xs5-U 4$g) a If this function finds a factor ``exp(a*t)``, it applies the frequency-shift rule of the Laplace transform and adjusts the convergence plane accordingly. For example, if it gets ``(exp(-a*t)*f(t), t, s)``, it will compute ``LaplaceTransform(f(t), t, s+a)``. rrrzz$ rule: multiply with exp (4.1.5)Fr N)r rr'r rvr r!) rrr|rrrr r rrrs rn_laplace_rule_exprVs S1#A S A S A ''#a&( C fnnQ%%ac*  9 :*361h49;IA2"SV*}b) ) rqc ~[SU/S9n[SU/S9n[S5n[S5nUR[U5U-5nU(GabXvR[5(GdEXuR U5RXA-U- 5nU(Ga[ S5 XX- n [ U 5S:a[U 5S:Xa[X*X- U-5Xv-n U R[[5(a#U R[5R5n U R5V s/sHoRXUX- 5PM sn upU S:wa(X- X- [ R"[ R$4$gS[ R"[ R$4$XuR'U5(a[)XuU5nU0:wa[+UR-55S 1:Xa[/XuU5n[1[3UR555V s/sHZn [U 5S:XdM[ U 5S:dM%[U *U-5XvRX5-URX5- PM\ sn 6nU[ R"[ R$4$gs sn fs sn f) 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 DiracDeltarNra)r rr:rr rvr!r"r'r3r2rewriter5ratsimpas_numer_denomrrrr is_polynomialrQsetvaluesrrrkeys)rrr|rrrrr r locfnrrrroslopers rn_laplace_rule_deltar#msO S1#A S1#A S A S A ''*Q-/ "C 36::j))fnnQ%%ac!e,  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$i A 11%**Q2BB.MN1--qvv66 OMs?"J5.J: J:9J:cJ[R/n[R/n[R"U5HUnUR [ [ [[[5(aURU5 MDURU5 MW [U6n[U6nXE4$)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_argsrr3r2r+r)r'append)r trigsothertermrrs rn_laplace_trig_splitr+swUUGE UUGE b! 88CdD# . . LL  LL  " U A U A 4Krqcx[SU/S9n[SU/S9n[SU/S9n/n/nUR[5R5n[R "U5HnUR U5(d#URSUSS[S[S05 M<[URSS 9U5nURU[X!-U-5-5=n bKURSX[X5-SX[[X5[[X505 MURU5 M XV4$) 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. c1rc0rkrrr')combine) r rr'r rr&rr'r!r"rpowsimpr) rrr-r.rxmxnx1r*rs rn_laplace_trig_expsumr5s 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" 6Mrqc ^/n/nSmU4SjnU4SjnU4SjnU4SjnSn [U5S:GaUR5n Sn Sn Sn [[U55HnU [X[:HnU [X[*:HnU [X[:HnU [X[*:HnU(a%U(aU [S:waU [S:waUn MU(aU(aU [S:waUn MU(dMU(dMU [S:wdMUn M U bU bU bUR U"U X S X S X S U55 UR [ [U S 555 XU /nURS S 9 UHnURU5 M GO"U bHUR U"XU S U55 UR U [5 URU 5 OU bQUR U"XU S U55 UR [ U [55 URU 5 OU bQUR U"XU S U55 UR [ U [55 URU 5 O/UR U "X55 UR U [5 [U5S:aGM[U6[U64$) z Helper function for `_laplace_rule_trig`. This function takes the list of exponentials `xm` from `_laplace_trig_expsum` and simplifies complex conjugate and real symmetric poles. It returns the result as a sum and the convergence plane. c>UR5n[[U55HtnXR5nUSR [ 5(aXR [5X'MMUS[US--R [5X'Mv U$r) copyrangelenrrr"rr2r)coeffsncr/ris rn_simpc"_laplace_trig_ltex.._simpcs{ [[]s2wA##%B!uyy}} c*A2a511#6   rqc >USUSU[U[4upVpxXa-U-U-XVU-U- U- -S[-U-U-- S[-U-U--US-U*U- U- U- -US[-U-U-S[-U-U----SUS--U--SUS--U--US-U*U- U-U--US-S[-U-U-S[-U-U--S[-U-U-- S[-U-U-- --USUS--U-SUS--U-- --/n [R[R SUS--SUS--- [R US-SUS--US---US--/n [ [T"U 5[[U 55SSS25V V s/sH upXU --PM sn n 6n [ [U [[U 55SSS25V V s/sH upXU --PM sn n 6nX- $s sn n fs sn n f)Nrr/rrrr) r!r"rrr%rrzipr9r:)t1k1k2k3r|rk0a_ra_ir<dcrrrrr>s rn _quadpole%_laplace_trig_ltex.._quadpoles`S'2c7BrFBrF:s GbL2  Bw|b !AaCGBJ .1S ;1rcBhmb()1Q3s72:!C *+,#q& Qhrk*1rcBhmb()1ac#gbj1Q3s72:-!C :QqSWRZGHI1S!V8B;36",-.   EE1661S!V8aQh. FFCFQsAvXc1f_,sAv57 !$VBZs2w"1E!F G!Fa1f!F G I !$Rs2w")=!> ?!>a1f!> ? As H ?s G> "H c >USUSU[U[4up4pVXA-U*U-X1-- S[-U-U--/n[RSU-US-US--/n[ [ T "U5[[U55SSS25V V s/sH upXU --PM sn n 6n [ [ U[[U55SSS25V V s/sH upXU --PM sn n 6n X- $s sn n fs sn n fNrr/rrr r!r"rrr%rrAr9r:)rBrCr|rrFrGrHr<rIrrrrr>s rn_ccpole#_laplace_trig_ltex.._ccpolesS'2c7BrFBrF:sgr"uqt|ac#gbj0 1eeRVS!Vc1f_ - !$VBZs2w"1E!F G!Fa1f!F G I !$Rs2w")=!> ?!>a1f!> ? As H ?s C/ C5 c >USUSU[U[4up4pVXA-X4-X1-- S[-U-U-- /n[RS[-U-US-*US-- /n[ [ T "U5[[U55SSS25V V s/sH upXU --PM sn n 6n [ [ U[[U55SSS25V V s/sH upXU --PM sn n 6n X- $s sn n fs sn n frMrN)rBrDr|rrFrGrHr<rIrrrrr>s rn_rspole#_laplace_trig_ltex.._rspolesS'2c7BrFBrF:sgqtad{QqSWRZ/ 0eeRT#XQwa/ 0 !$VBZs2w"1E!F G!Fa1f!F G I !$Rs2w")=!> ?!>a1f!> ? As H ?s C5 C; c >USUSpCXA-X4U- -/n[R[RUS-*/n[[ T "U5[ [ U55SSS25VVs/sH upxXrU--PM snn6n [[ U[ [ U55SSS25VVs/sH upxXrU--PM snn6n X- $s snnfs snnf)Nrr/rr)rr%rrrAr9r:) rBrEr|rrFr<rIrrrrr>s rn_sypole#_laplace_trig_ltex.._sypoles3C2gqr'{ #eeQVVadU # !$VBZs2w"1E!F G!Fa1f!F G I !$Rs2w")=!> ?!>a1f!> ? As H ?s -C 1C c(USUSp2UnX- nXE- $)Nrr/rr)rBr|rrFrrs rn _simplepole'_laplace_trig_ltex.._simplepoles$3C2  Es rqrNr/rT)reverse) r:popr9r!r"r'r$rrr-)r2rr|resultsplanesrJrOrRrUrXrB i_imagsym i_realsym i_pointsymireal_eqrealsymimag_eqimagsymindices_to_popr>s @rn_laplace_trig_ltexrgsG 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& !q b'A+t =#v, &&rqc \[SSS9nUR[[[[ 5(dg[ URX55upE[XC5upg[U5S:agURU5(d#[XcU5upXX-U [R4$/n /n [XSUSS9upnUHOnU RUSU RX"US - 5-5 U RU [US 5-5 MQ [!U 6RX15[#U 6U4$) z This rule covers trigonometric factors by splitting everything into a sum of exponential functions and collecting complex conjugate poles and real symmetric poles. rTrealNrFr r/r)rrr3r2r+r)r+rr5r:rgrrr r'r!rr-)r t_r|rrrr2r3rrr]r\GG_planeG_condr4s rn_laplace_rule_trigro[s  cA 66#sD$ ' ' rwwr~ .DA !! 'FB 2w{ 5588!"+sAqvv~/a%HFB NN2c7166!r#wY#77 8 MM'"RW+- . =  a $c6lF ::rqc[SU/S9n[SU/S9n[S5nURU[XQU45-5nU(aXdR(aXeR Vs/sHowR U5PM nn[U5S:Xa[S5 /n [Xd5H\n U S:XaXeRUS5n O[XeX45RUS5n U RX&UU - S- -U -5 M^ [XeXSS 9upnXcX&U-U -[U 6- -X4$g s snf) 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)``. rrrrraz" rule: time derivative (4.1.8)rFr N)r rrr is_integerrjrsumrvr9rr'r r)rrr|rrrr rrrr/rrrrs rn_laplace_rule_diffrs|s4 S1#A S1#ASA ''!Jqa&)) *C sv  "v{{ +{!UU1X{ + q6Q; 7 8A36]6 Aq)A"36A62771=AVAXaZ*+ # +361%HIA2FA1vIaK#q'12R< <  ,s+D;c UR(GaS/nS/n[R"U5H=nURU5(aUR U5 M,UR U5 M? [ U5S:GaC[ U5n[Xa5R5n[ U5nUS:Ga [ U5n [XUSS9upn U /n Sn[U SU5*nU R[5(a=[US- 5H*nU R SUS--[XUS-5-5 M, OLU(aEU R U5 [US- 5H"nU R [U SU5*5 M$ U(a4[![U5Vs/sHnXxU- S- U U-PM sn6nUX4$[#SU/S9n[#S5nUR%UU-U-5=n(aTUUR&(a@UUR((a,[UUXSS9upn SUU-[XUU45-X4$g ![a SnGNvf=fs snf) 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)``. raFr rrrrrN)is_Mulrr&rr'r:rrR all_coeffsr r ValueErrorrLaplaceTransformr9r rr rrqr)rrr|pfacofacfacpexpcNoexr_p_c_derid1r/rrrr s rn_laplace_rule_sdiffrs3 xxxss==#C  ## C  C $ t9q=t*Cc((*BBA1u4j/EJ ttBx++B66*++"1Q3Z R1Q3K 2!A#0F$FG(KKO"1Q3Z T$r(A%6$67(qBAb1QiQ/BCAr;& S1#A S Aggad1fos q6  Q!3!3+CFA5IJBBQ<RSV 55r= = )"BCsI !I IIcl[USS9nUR(a [X1USS9$[U5nUR(a [X1USS9$[U5nUR(a [X1USS9$X0:wa [X1USS9$[[ U55nUR(a [X1USS9$g)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. Fdeepr N)r is_Addr r r)rrr|rs rn_laplace_expandrs quAxx!!E::1 Axx!!E::q Axx!!E::v!!E::{1~Axx!!E:: rqc[[[[[[ [ /nUHnU"XU5=ncMUs $ g)z_ This function applies all program rules and returns the result if one of them gives a result. N)rr#rrrorsr)rrr| prog_rulesp_ruleLs rn_laplace_apply_prog_rulesrsG*+>)+<$$&9;J a A -H rqc[5up4nSnSnUHuppn Xl:waU "URX055nU nURU5n U (dM@U RU 5nU[ R :XdMhU RU 5RXR05U RU 5[ R 4s $ g![a Mf=f)z^ This function applies all simple rules and returns the result if one of them gives a result. N)rrrxreplacerrr)rrr| simple_rulesrkrprep_oldprep_ft_doms_domcheckplaneprepmars rn_laplace_apply_simple_rulesrs 01LbH F,8(eD  !&&!/*FH \\%  2 NN2& AFF{r*//8r*AFF44-9   sC CCcUR(d6[SSS9n[URX05U5RX!05$[ U5n/nUR GHupE[ U[5(adXR ;aU[ U[[45(aUs $UR[URUR- 5U-5 M[ U[5(ay[UR 5S:Xa`UR HMnUR U:Xa6UR[URUR- 5U-5 MIUs s $ GM [ U["5(a[UR 5S:XaUR upxUR U:Xa|UR U:XalSUR$;aXpUR[URUR- 5[URUR- 5- U-5 GMUs $Us $ ['U6$)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. rTrir>)is_realr_piecewise_to_heavisiderr1rjrrrrr'r;gtsrrLr:lhsrMrr) rrrrr rc2r.r-s rnrrs 99 #D !&qzz1&'91=FFvNNAA AFF dJ ' 'AN$R)) 488dhh#67:; b ! !c$))n&9ii66Q;HHYrvv7:;H  c " "s499~':YYFBvv{rvv{"))#rvv/rvv/01345HAB 7Nrqc [S5n[S5n[S5n[S5n[U[5(a4UR[5(dUR[ 5(dU$UR 5H}upgUR[ U"U5XC55=nbXX:Xa U"X5s $UR[ U"U5X4U55=ncMgXX:XdMsU"X5s $ URn URV s/sHn [X5PM n n U "U 6$s sn f)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) r rrrrxInverseLaplaceTransformitemsrrmrjlaplace_correspondence) rfdictrr|rrrYrrmr#rjs rnrrFsH S A S A S A S A1d##EE*++566 gg.qtQ:;;HDADLQT7Ngg5adA!DEEDADLQT7N 66D:;&& A&3 "3 .&D A ; Bs#D?c UR5Hup4[[U55HnUS:XaURU"S5US5nM&US:Xa2UR[ [ U"U5U5US5US5nM^UR[ [ U"U5X45US5XE5nM M U$)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 rra)rr9r:rrr )rrrricr/s rnlaplace_initial_condsrsDs2wAAvIIadBqE*aIId:adA#61=r!uEIId:adQF#;QBBEJ   Hrqc ^[R"U5n/n/n/n/nUGH#n U RTSS9upU R[5(a[[R"U R [ 55n U H*n U RTSS9upURX-U45 M, MU R[:XarU R[T55(dS[R"[U T55n U H*n U RTSS9upURX-U45 M, GMURX45 GM& UGHupU R[5(a [U TU5[RS4nGOU R[ T55(a:U R[T55(dU R[ T5S5n [!U TU5=nc![#U TU5=nc[%U TU5=nbOz['U4SjU R)[*555(a[U TU5[RS4nO.[-U TX#S9=nbO[U TU5[RS4nUunnnURU U-5 URU5 URU5 GM [U6nU(aUR/SS9n[1U6n[3U6nUUU4$)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_AddTrac3D># UHoRT5v M g7frr)rundefrks rnr%_laplace_transform..sG0FuYYr]]0F r doit)rr&as_independentrrCrr;r'rmr0r:rrxrrrrrranyrr rbrr-rM)r rkrrterms_tterms_stermsr] conditionsffr/ft_terms_termrCf1rri_pi_ci_rlr conditions ` rnr r smmBGG 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"55QBG0FGGG&b"b113E3EtL5B33!;?@%b"b113E3EtLc3qu c#;>']Fe, LEZ I 5) ##rqc.\rSrSrSrSrSrSrSrSr g) rxia 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 >URSS5n[XX5S9nU$)NrFr )getrb)selfrrr|hintsrFLTs rn_compute_transform#LaplaceTransform._compute_transforms#IIj%0 +A! H rqcx[U[U*U-5-U[R[R45$r)rEr'rrr)rrrr|s rn _as_integralLaplaceTransform._as_integrals,#qbd) a%<==rqc URSS5nURSS5n[SURURUR45 URnURnURn[ XdXSS9nU(aUS$U$)" 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 r)rrXfunctionfunction_variabletransform_variabler )rr_nocondsrFrkrr rs rnrLaplaceTransform.doit s99Y-IIj%0 '$--*.*@*@*.*A*A*C D # #  $ $ ]] rr > Q4KHrqrrN) ri __module__ __qualname____firstlineno____doc___namerrr__static_attributes__rrrqrnrxrxs  E >rqrxc ^^^TRSS5nTRSS5n[U[5(a[US5(aTRSS5(+nU(aDU(a=Sn[ SSUS9 [ [ 5 URUUU4S j5sS S S 5 $UV s/sHn [U TT40TD6PM n n U(a9[U 6upn [U5"/URQU P76nU[U 6[U 64$[U5"/URQU P76$[UTT5RSUS 9unnnU(dUUU4$U$!,(df  N;=fs sn f) 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 >[UTT40TD6$r)laplace_transform)fijrr|rs rnr#laplace_transform..s 1#q! Eu ErqNrr)rrrNhasattrrUrWrVrrrAtypeshaper-rMrxr)rrr| legacy_matrixrrrFradtrelements_transelementsavalsr f_laplacerrrs `` ` rnrr+ssNyyE*H *e,I!Z  WQ %<%<IIi// ]7C % */+. !!89{{EG:9 012/00Q$"$/0 2.1>.B+ G7QWW7h7  #u+sJ/???Aw88881a(--e7@.BHB1 1ax ':92sE4E) E&c ^ ^^SSKJnJm SSKJn [ SSS9mU U4SjnUR U5(aURU5nUR(aL[URVs/sHn[XTX45PM sn6n [U RTU5U5S4$U"X[T*5S[R 4SS S 9upU cqU"XT5n U cgU R$(a-U RSupU R'[(5(agO[R*n U R-[.U5n U R$(aU RTU5W 4$[ S 5m[R04UU4S jjn U R-[2U 5n S n U R-[U 5n [U RTU5U5W 4$s snf!["a Sn GNf=f)z5The backend function for inverse Laplace transforms. r)meijerint_inversion_get_coeff_exp)inverse_mellin_transformrTricJ>[U5S:wa[U6$USRSRnT"UT5up#USRSnUSRSn[ S[ U5- TU-- 5U-[ TU-S[ U5- - 5U--$)z2Simplify a piecewise expression from hyperexpand. rrrra)r:r0rjargumentr;r$)rjr#coeffexponente1e2rrs rnpw_simp7_inverse_laplace_transform_integration..pw_simps t9>d# #1gll1o&&(a0 !W\\!_ !W\\!_ aE lQ[0 1" 4 akAc%jL0 1" 4 5 6rqNF)needevalrucV>UR"[T*5T5nURT5(a [X5$SSKJn U"US:T5nUR T:Xa$[UR5n[TU-U5$[UR 5n[TU-*U5$)Nrr) rr'rr;rrrr(r)r#H0rrrelr/rrs rnsimp_heaviside>_inverse_laplace_transform_integration..simp_heavisides HHS!Wa  5588S% %@Aq) 77a<CGG AQUB' 'CGG Aq1uXr* *rqc*[[U55$r)r r')r#s rnsimp_exp8_inverse_laplace_transform_integration..simp_expsc#h''rq)sympy.integrals.meijerintrrsympy.integrals.transformsrris_rational_functionapartrrrjrcrFrr'rrrHrrrErrr0rr;)rr|rkrrrrrXrrrrrrrs @@@rnrcrcsNC cA 6 a  GGAJxx vv!5Q1eN 21477*1aR4:L48%I  y a ( 9 >>ffQiGAuuX66D IIi )~~vva}d"" c A vv + + )^,A( #x A QVVAr]H -t 33c " s.G(&G G,+G,cSSKJn U"U5up4URU5(aTURU5R 5n[ U5S:Xa&UupgnXaUSU-- -S-X- -USU-- S-- -nX4- $)Nr)fractionrr)rrrrrvr:) rr|rrrcfrrrs rn_complete_the_square_in_denomr s/ a[FQq YYq\ $ $ & r7a<GA!a1gI>!#%q!A#wl23A 3Jrqc [S5n[S5n[SU/S9n[SU/S9n[SU/S9n[S5 SnS nX - U[RUS 4X0U-U*--XS - -[ U*U-5-[ U5- [RUS 4S US -US --S -- [X!-5X!-[X!-5-- S US --- [RUS 4S X-- XS - -[ U5- [RUS 4S XU-U--- [X2U-5X#-[ U5-- [RUS 4/nXpU4$) z This is an internal helper function that returns the table of inverse Laplace transform rules in terms of the time variable `t` and the frequency variable `s`. It is used by `_inverse_laplace_apply_rules`. r|rrrrrz._inverse_laplace_build_rules is building rulescJURU5$![a Us$f=fr)factorrP)rr|s rn_frac+_inverse_laplace_build_rules.._frac(s) 88A;  H s  ""cU$rrr)rs rnsame*_inverse_laplace_build_rules..same.srqrarr) rr rvrrr'r@r3r2rA)r|rrrrrr _ILT_ruless rn_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 ! rqcFUS:Xa&[S5 [U5[R4$[ 5up4nSnUR X05nUHuppn XkU 4:wa U "X|-5n X4nW R U5nU(dM5U nU[RLa,USVs/sHnURU5PM nnUS"U6nU[R:XdM[U5U RU5R XR05-[R4s $ gs snf)8 Helper function for the class InverseLaplaceTransform. raz rule: 1 o---o DiracDelta()rrN) rvr:rrrrrrr;)rr|rrrrk_prepfsubsrrrrr{_Frrrrjs rn#_inverse_laplace_apply_simple_rulesr Bs  Av01!}aff$$57JB E FFA7OE*4&e3 3K eiBKE XXe_ 2A01!51 25aD$KAFF{ |ENN2$6$;$;RG$DDaffLL+5  6s!Dc  [SU/S9n[SU/S9n[S5nURU[XaU45-5nU(a<XuR(a)[ S5 [ XvXUSSS9upU*Xu-U-U 4$g) rrrrrz3 rule: t**n*f(t) o---o (-1)**n*diff(F(s), s, n)Fr dorationalN)r rr rqrv_inverse_laplace_transform) rr|rrrrrrrrs rn_inverse_laplace_diffr%_s S1#A S1#A S A :aQ(( )B beDE) E15BRU{1}a rqc [SU/S9n[S5nURU5(dU[U5-[R4$UR[ 5(dgUR [ XA-55nU(aZXdR(a+[S5 [X&U-5[R4$[XX#5[R4$UR [ XA-5U-5nU(aNXdR(a[S5 [XeXXd-USSS 9$[XX#5[R4$g) 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") r rr:rrr'rrrvrr$)rr|rrrrr s rn_inverse_laplace_time_shiftr'ps S1#A S A 5588A&& 55:: ''#ac( C 6   ? @aAh'/ /*1:AFFB B ''#ac(1* C 6   J K-SV8UUtM M+1:AFFB B rqcURU5(dU[U5-[R4$[ UR =n5S:Xa[ SU/S9nUSRX- 5=n(ae[Xe5R(aI[S5 [Xe*U-5[URU5XU5-[R4$g)rrarrrz& rule: F(s-a) o---o exp(-a*t)*f(t)N)rr:rrr:rjr rr!rrvr'rrm)rr|rrrjrrs rn_inverse_laplace_freq_shiftr)s 5588A&& 166>4a qc "q'--$ $B $"RU)*?*? ; <RUF1H 'q 1?@ABI I rqc [SU/S9n[S5nURX-U-5nU(aXdR(aXdR(a}[ S5 [ XeXUSSS9upxUR [U5S5nUR[5(a[XrXd5U4$[U5[XrXd5-U4$g ) rrrrz+ rule: s**n*F(s) o---o diff(f(t), t, n)FTr"raN) r rrqrrvr$rr;rrr) rr|rrrrr rrs rn_inverse_laplace_time_diffr+s S1#A S A ''!$q&/C sv  SV%7%7<=) FA%%DB IIilA & 55( ) )cf%q( (Q<Q36 22A5 5 rqc @-^^[SU/S9n[SU/S9m[SU/S9n[SU/S9mSn[RnUR5nUV s/sHoR XAU--T-T-5PM n n SU ;ag[R n /n /n /nU HpnXS:XaX-n MUTR (aU RU5 M8UTR(aU RU5 M_URU5 Mr [U UU4SjS 9n [U UU4S jS 9n [U5S:wag[U 5S :XGa[U 5S:XGaU STS :XaU SU[R:XaU STU SU- nS U SU- U -nUR (aYU[[5- [U5- UU-[US -U-5-[U[U5-5-- n[!S5 GOyU STS:XaU SU[R:XaU STU SU- nUS -nS U SUS -- U -nUR (azUS S [[5- [U5-[U5-- S S U-U-- [UU-5-[#[U5[U5-5S - ---n[!S5 GOU STS:XaU SU[R:XaU STU SU- nS U SUS-- U -nUR (awUS [[5- US -U-S --[U5-UU-[US -U-5-S US --U-S--[U[U5-5-- -n[!S5 GOU STS:XaU SU[R:XaU STU SU- nS U SUS-- U -S- nUR (aUUSUS--US --SUS --U--S--[US -U-5-[U[U5-5-S [[5- US--U[S5S - --S US --U-S--- -n[!S5 GOU ST[R*:XaZU SUS :XaN[U STU SU- 5nS [U SU5- U -nU[%SUU-5-n[!S5 GOf[U 5S :XGa[U 5S :XGaU STS:XaU SU[R:XaU ST[R:XaU STS:XaU STn[U SU5U SU- U -nUS US--US --SUS --U--S --[US -U-5-[U[U5-5-S [[5- U-US -U-S --[U5-- n[!S5 U STS :XaU SUS :XaU ST[R:XaU SUS :XaU STU SU- nU STU SU- n[U SU5U SU- U -nU[U*U-5[U5- [[5- [UU- 5[U*U-5-[#[UU- 5[U5-5---n[!S5 GOW[U 5S :XGaL[U 5S:XGa<U STS :XaU SUS :XaU S T[R*:XaU S US :XaU S TS:XaU ST*U SU- nS [U S U5- U SU- U -nUR (aIU[U5- [UU-5-[#[U5[U5-5-n[!S5 GO^U STS :XaU SUS :XaU STS:XaU S TS :XaU S U[R:XaxU S TU S U- nS U SU- U S U- U- U -nUR (a:US [US -U-5[U[U5-5-- -n[!S5 G OU STS :XaU SU[R:XaU S T[R*:XaU S US :XaU S TS:Xa{U STU SU- nS U SU[U S U5-- U -nUR (a7U[US -U-5-[U[U5-5-n[!S5 G OU ST[S5*S - :XaU SUS :XaU STS:XaU S TS :XaU S U[R:XaU S TU S U- nS U SU[S5S - -U S U-- US -- U -nUR (a\US [[5- U-[U5-[US -U-5[U[U5-5--S - -n[!S5 G OU STS:XaU SU[R:XaU S TS :XaU S US :XaU S TS:XaU STU SU- nUS -nS U SUS -- U S U- U -nUR (azUS U- S U-S U- - [UU-5-[[U5[U5-5--S [[5- [U5- [U5-- -n[!S 5 G OU STS:XaU SU[R:XaU S T[R*:XaU S US :XaU S TS:XaU STU SU- nS U SUS -- [U S U5- U -nUR (a_US [[5- [U5-S U-U-[US -U-5-[U[U5-5-- -n[!S!5 G OU STS:XaU SU[R:XaU S T[R*:XaU S US :XaU S TS:XaU STnU [U S U5- U SU- nUS US --U-S -U-[US -U-5-[U[U5-5-S [[5- U-U[S5S - --- -n[!S"5 GOU STS :XaU SUS :XaU S T[R*:XaU S US :XaU STU SU- nU S TU S U- nU [U S U5- U SU- nUS [UU- 5- [U*U-5-[#[UU- 5[U5-5--n[!S#5 GO[U 5S :XGa [U 5S :XGaU STS :XGa|U SUS :XGaoU S TS :XGabU S U[R:XGaGU ST[R:XGa,U SUS :XGaU STS:XGaU S TU S U- nUS -nU ST*U SU- n[U SU5U SU- U S U- UU- - U -nUR (aUR (aUU[UU-5-[[U5[U5-5-[U5[U5-[UU-5-[[U5[U5-5--U[UU-5-- -n[!S$5 GORU STS :XaU SUS :XaU STS:XaU S TS :XaU S U[R:XaU STS :XaU SU[R:XaU STU SU- nU S TU S U- nUU-S:XaZU SUU SU- U S U- U -nUS [US -U-5-[U[U5-5-S - -n[!S%5 GO[U S TS :XGaU S US :XGaU S TS:XGaU STS:XaU SU[R:XaU STS :XaU SU[R:XaU STU SU- nU STU SU- nUU-S:XaU SUS -U S U- U SUS -- U -nUS S&US --U-[US -U-5-[U[U5-5--S&[[5- U-[U5-- -n[!S'5 GO0U S TS :XGaBU S US :XGa5U S TS:XGa(U STS:XGaU SU[R:XGaU STS:XaU SU[R:XaU STU SU- nU STU SU- nUU-S:XaU SUS-U S U- U SUS-- U -nUS S&US--US --S&US --U--S --[US -U-5-[U[U5-5-S&[[5- U-[U5-S US --U-S --- S - -n[!S(5 GO[U 5S:XGa[U 5S:XGaU STS :XGaU STS:XGa U SUS :XaU S TS :XaU S US :XaU S T[R*:XaU S US :XaU S T*U S U- nS U SU- U S U- [U S U5- U -nUR (arUU[S5*S - -[UU-5-[#[U5[U5-5-S U- [[5- [U5-- -n[!S)5 GOU STS :XGaU SUS :XGaU S TS :XGasU S U[R:XGaXU S T[R*:XGa<U S US :XGa/U S TS:XGa"U S TU S U- nUS -nU ST*U SU- nS U SU- U S U- [U S U5- [U5UU- -- nUR (aUR (aU[U5[UU-5-[[U5[U5-5-[U5[UU-5-[#[U5[U5-5--[U5[UU-5-- -n[!S*5 Ucg['U5U-U4$s sn f)+rrrrrrNrc&>UTUTS:gUT4$rrrrrrs rnr-_inverse_laplace_irrational..1qtqy!A$(?rqrc&>UTUTS:gUT4$rrrr.s rnrr/r0rqrarrz rule 5.3.4rz rule 5.3.10rz rule 5.3.13r z rule 5.3.16z rule 5.3.35/44z rule 5.3.14z rule 5.3.22z rule 5.3.1z rule 5.3.5z rule 5.3.7z rule 5.3.8z rule 5.3.11z rule 5.3.12z rule 5.3.15z rule 5.3.23z rule 5.3.6z rule 5.3.17z rule 5.3.18z rule 5.3.19z rule 5.3.2z rule 5.3.9)r rras_ordered_factorsrr%rr'rsortedr:rr/rr'r=rvr<r7r;)r r|rrrrrlrfarr constantszerospolesrestr*rk_a_sqb_a_numrrs @@rn_inverse_laplace_irrationalrBs S1#A S1#A S1#A S1#A FI   B*, -"Q''1T6!8a- "B - rzI E E D 7a<!I !W LL  !W LL  KK  5? @E 5? @E 4yA~ 5zQ3u:? 8A;" q!!6q!U1Xa[(B58A;y(B~~tBxKQ'rE#b!eAg,&tBtAwJ'778() 1Xa[B 58A;!&8A;uQx{*DqB58A;>!)+BAd2hJtBx/Q771R463r!t9,c$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{:BrEB9LBBNNs2a4yLd2htAw&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$%%&tBxZ]47%:Ab!eGAIaK%HIIJKL)* UqSZ1_a r!eAhqkQ&658A;!;Ka r!eAhqkQ&6a w&58A;!+;(1+eAhqk)B58A;uQx{*4a +< * W\1aAAq!t##%A2w!|ffX]9DAqAvqSAacE]C1I-==AC%Aa,1134Q7!W%%'#qbd) DAJ.!AaC%2r!t92%%)"Q$Z200#qbd) C1I-1Q3 3r!t90DSAY0NNA NN9Q<> *1HHHB NN2    d #QT']Fe, 3 # ##S% ;s M-7M2c ^[R"U5n/n/nUGHun U R[5(a2U R TT*5R 5R TT*5n U R TSS9upU(a&U RT5(a[U TX#US9=n cD[U TU5=n c3[U TX#5=n c"[U TX#5=n c[U TX#5=n bOy[U4SjU R[555(a[!U TX#5["R$4n O.['U TX#US9=n bO[!U TX#5["R$4n U upUR)X-5 UR)U5 GMx [U6nU(aUR+SS9n[-U6nUU4$)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. Frr c3D># UHoRT5v M g7frr)rrrs rnr-_inverse_laplace_transform..sB,A52,Arr)rr&rr'rrSrr rTr rFrJrHrrr rrrrcr'rrM)r rrkrrr#rrrr*r/rrrrrlrs ` rnr$r$bs MM" EGJ 88C==99R"%..055b2#>D""2e"4t88<</r2x99:!RDD72rII-aR??72rII  BAGGL,AB B B)B:AFFCA;r2x99AEF (B:AFFCA qu#KN']Fe,Z I 9 rqcd\rSrSrSrSr\"S5r\"S5rSr \ S5r Sr S r S rS rg ) 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 ZUc[Rn[R"XX#U40UD6$r)r_none_sentinelrG__new__)rrr|rroptss rnr\InverseLaplaceTransform.__new__s, =+::E ((uEEErqcNURSnU[RLaSnU$)Nr)rjrr[)rrs rnfundamental_plane)InverseLaplaceTransform.fundamental_planes( !  +:: :E rqc 0[XX0R40UD6$r)rcr`)rrr|rrs rnr*InverseLaplaceTransform._compute_transforms"5 !++6/46 6rqc8URRn[[X#-5U-X$[R [R -- U[R [R --45S[R-[R -- $)Nr) __class___crEr'r ImaginaryUnitrPi)rrr|rrs rnr$InverseLaplaceTransform._as_integralss NN   SXaZ!)C%C"#aooajj&@"@"B C qttVAOO # % &rqc 8URSS5nURSS5n[SURURUR45 URnURnURnUR n[ XdXWUSS9nU(aUS$U$)rrTrFz[ILT doit] (%s, %s, %s)r"r)rrXrrrr`r$) rrrrFrrkr rrs rnrInverseLaplaceTransform.doits99Y-IIj%0 (4==+/+A+A+/+B+B+D E # #  $ $ ]]&& & B d D Q4KHrqrrN)rirrrrrrr[rfr\propertyr`rrrrrrrqrnrrsI E6]N sBF  6&rqrc "^^^^TRSS5nTRSS5n[U[5(a)[US5(aUR UUUU4Sj5$[ UTTT5R SUS9upxU(aU$Xx4$)a Compute the inverse Laplace transform of `F(s)`, defined as .. math :: f(t) = \frac{1}{2\pi i} \int_{c-i\infty}^{c+i\infty} e^{st} F(s) \mathrm{d}s, for `c` so large that `F(s)` has no singularites in the half-plane `\operatorname{Re}(s) > c-\epsilon`. Explanation =========== The plane can be specified by argument ``plane``, but will be inferred if passed as None. Under certain regularity conditions, this recovers `f(t)` from its Laplace Transform `F(s)`, for non-negative `t`, and vice versa. If the integral cannot be computed in closed form, this function returns an unevaluated :class:`InverseLaplaceTransform` object. Note that this function will always assume `t` to be real, regardless of the SymPy assumption on `t`. For a description of possible hints, refer to the docstring of :func:`sympy.integrals.transforms.IntegralTransform.doit`. Examples ======== >>> from sympy import inverse_laplace_transform, exp, Symbol >>> from sympy.abc import s, t >>> a = Symbol('a', positive=True) >>> inverse_laplace_transform(exp(-a*s)/s, s, t) Heaviside(-a + t) See Also ======== laplace_transform hankel_transform, inverse_hankel_transform rTrFrc">[UTTT40TD6$r)inverse_laplace_transform)Fijrrr|rs rnr+inverse_laplace_transform.. s1#q!ULeLrqr)rrrNrrrr) rr|rrrrrFrrs ```` rnrorosZyyD)H *e,I!Z  WQ %<%<{{ LN N #1aE 2 7 7  8 +DAt rqc^^^^^^^^^ ^ [S[T/S9umm m UUUUU4SjmU4SjmUU4SjmUU U UU4SjmU4SjmT"U5$)zEFast inverse Laplace transform of rational function including RootSumza, b, nrc>URT5(dU$UR(aT"U5$UR(aT"U5$UR(aT"U5$[ U[ 5(aT"U5$[ er)rrruryrrTNotImplementedError)e_ilt_add_ilt_mul_ilt_pow _ilt_rootsumr|s rn_ilt#_fast_inverse_laplace.._ilt# seuuQxxH XXA;  XXA;  XXA;  7 # #? 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