gRdZddlZddlZddlmZmZmZddlmZddl m Z ddl m Z ddl mZmZmZmZmZmZmZmZmZmZddlmZmZdd lmZmZmZmZm Z m!Z!m"Z"m#Z#dd l$m%Z%dd l&m'Z'm(Z(m)Z)dd l*m+Z+m,Z,m-Z-m.Z.m/Z/m0Z0dd l1m2Z2m3Z3ddl4m5Z5m6Z6m7Z7m8Z8ddl9m:Z:m;Z;mZ>m?Z?ddl@mAZAmBZBmCZCmDZDddlEmFZFmGZGmHZHmIZIddlJmKZKmLZLddlMmNZNmOZOmPZPddlQmRZRmSZSmTZTmUZUddlVmWZWddlXmYZYmZZZddl[m\Z\m]Z]m^Z^ddl_m`Z`maZambZbmcZcmdZdddlemfZfddlgmhZhddlimjZjddlkmlZlddlmmnZnddlompZpdd lqmrZrdd!lsmtZtmuZumvZvdd"lwmxZxdayd#Zzd$Z{d%Z|ezd&Z}ezd'Z~ezd(Ze d)Zezd*Zezd+Zezd,Zezd-Zezd.Zezd/Zezd0Zezd1Zezd2Zezd3Zezd4Zezd5Zezd6Zezd7Zd8Zd9Zezd:ZGd;dZezd?Zezd@Ze dAZezdBZezdCZezdDZezdEZezdFZezdGZezdHZezdIZezdJZezdKZezdLZGdMdNe]ZdRdOZdPZdS)SzLaplace 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)debugfcfd}|S)Ncbddlm}|s|i|Stdkrtdtjtddtzj|tjtdz ajdks jd krGd t_td dtzztj|i|}d t_n|i|}tdzatddtzd |tjtdkrtdtj|S)Nr SYMPY_DEBUGzO ------------------------------------------------------------------------------file-LT-  _laplace_transform_integration&_inverse_laplace_transform_integrationFz**** %sIntegrating ...Tz---> zO------------------------------------------------------------------------------ )sympyr\ _LT_levelprintsysstderr__name__)argskwargsr\resultfuncs S/var/www/html/ai-engine/env/lib/python3.11/site-packages/sympy/integrals/laplace.pywrapzDEBUG_WRAP..wrap1sV%%%%%% )4((( ( >> -cj 1 1 1 1 tI~~t}}ddC:    Q  !AAA !III %E  *d9n=CJ O O O OT4*6**F $E  T4*6**FQ  $y...&&9 KKKK >> -cj 1 1 1 1 )rmros` rn DEBUG_WRAPrr0s#4 Krpcjddlm}|r*tddtz|tjdSdS)Nrr[r_r`r])rdr\rfrergrh)textr\s rn_debugruNsU!!!!!!E T)^^TT2DDDDDDEErpcfdfdfdfd}d}ddlm}||}||t}||tfd}||t|}t |S) 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) cJ|krdS|jr|jkr|jSdS)Nra)is_Powbaser')exss rnpowerz_simplifyconds..powerws1 771 9 A6Mtrpc`|r|rdSt|tr |jd}t|tr |jd}|rd|z d|z S|}|dS |dkr3t|t|zktjkrdS|dkr3t|t|zktjkrdSdSdS#t $rYdSwxYw)z_ Return True only if |ex1| > |ex2|, False only if |ex1| < |ex2|. Else return None. NrraFT)has isinstancer$rjrtrue TypeError)ex1ex2nabiggerr|r{s rnrz_simplifyconds..bigger~s< 771:: #''!** 4 c3   (1+C c3   (1+C 771:: (6!C%3'' ' E#JJ 94 1uu#c((c!ffai/AF::u1uu#c((c!ffai/AF::tu::   44 s)7D"7D D-,D-c|jst|tr|jst|ts||kS||}|| S||kS)z simplify x < y ) is_positiverr$)xyrrs rnrepliez_simplifyconds..repliesg *Q"4"4   )3As);); EN F1aLL =5LArpcH||}|dvrdSt||S)NTFT)r)rrbrs rnrepluez_simplifyconds..replues2 F1aLL  4!Qrpc<|dvrt|S|j|S)Nr)boolreplace)rzrjs rnreplz_simplifyconds..repls'   88Orz4  rpr) collect_absc||SNrq)rrrs rnz _simplifyconds..svva||rp)sympy.simplify.radsimprrrrr) exprr{rrrrrr|rs `` @@@rn_simplifycondsrVsB,     !!! 322222 ;t  D 4b& ! !D 4b3333 4 4D 4j& ) )D T77NrpcRt||tS)zs Expand an expression involving DiractDelta to get it as a linear combination of DiracDelta functions. )rOatomsr:rs rnexpand_dirac_deltars djj44 5 55rpc td |trdSt|t zzt jt jf}|ts;t| |t j t j fS|j sdS|jd\}}|trdS fd fdt|D}d|D}|s d|D}t!t#|}d | fd  |sdS|d\}} fd } |r"t'| |}t'| |} t| || |t)| | fS) z The backend function for doing Laplace transforms by integration. This backend assumes that the frontend has already split sums such that `f` is to an addition anymore. r{Nrc ddlm}tj}tj}t t |}tdtg\}}}}}} } |tt|z|zz|k|tt|z|zz|ktt|z|z|z||ktt|z|z|z||kttt|z|z|z||kttt|z|z|z||kf} |D]D} tj } g}t| D]}|jr|jjvr|j}|jr#t'|t(t*fr|j}| D]}||rnrG|jr:||z t2dz krt5|z dk}||t7|tt| zz|zt|z| zzz dksc|t7|tt|z| z||zz t|z| zzdksp||t7ttt|z| z||zt|z| zzz dkr9t9fd|||| | fDrt5|k}|t4dt5}|jr3|jdvs*| s| s||gz }||}|jr |jdvr||gz }|j!krd StE|j!| } | tj urtG| |}-tI|tK|}F||jr|j&n|fS) z7 Turn ``conds`` into a strip and auxiliary conditions. r_solve_inequalityzp q w1 w2 w3 w4 w5clsexcludec32K|]}|jVdSr)r).0wildms rn zH_laplace_transform_integration..process_conds..s:--TQtW0------rpcZ|dSNr)r as_real_imag)rs rnrzG_laplace_transform_integration..process_conds..s!((**"9"9";";A">rp)z==z!=N)'sympy.solvers.inequalitiesrrNegativeInfinityrrJrIrr r$r#r&r%InfinityrK is_Relationalrhs free_symbolsreversedrrr reversedsignmatchrrr!r2allrsubsrel_opr~ltsr.r-rMrL canonical)condsrrauxpqw1w2w3w4w5patternsca_aux_dpatd_solnrr{ts @rn process_condsz5_laplace_transform_integration..process_condss@@@@@@ f&--((#* dQC$9$9$9 1b"b"b c#q2vqj//"" "R ' c#q2vqj//"" "b ( !1r6A+a-44 5 5 : !1r6A+a-44 5 5 ; !:a"f#5#5"9!";R@@ A AB F !:a"f#5#5"9!";R@@ A AR G I- *- *ABDq\\& +& +?#qAE,>'>'> A?'z!b"X'>'>'A#C A+1)+aeAaDjBqD.@.@A"I*AGGABs3qt99~~$5b$8 9 9#ae**b. HH1LMM,A$5aeBh$B$B C CB FFGGArE B')*+,,A2C 1*Q--2CB2F J JKKBN!!R%jj"n--/0122A%----BB>,-----%1! AYY>>@@@DRUUAO/0xz2_laplace_transform_integration..s# 7 7 7!]]1   7 7 7rpcfg|].}|dtjkr|dtju,|/S)rar)rfalserrrs rnrz2_laplace_transform_integration..sH:::A!A$g##A$a&888888rpc>g|]}|dtjk|Sra)rrrs rnrz2_laplace_transform_integration..s#666adagoo!ooorpc6|dvrdS|S)Nrr) count_opsrs rncntz+_laplace_transform_integration..cnts" = 1~~rpc8|d |dfSNrrarq)rrs rnrz0_laplace_transform_integration..sqteSS1YY/rpkeyc0|Srr)rr{s_s rnsbsz+_laplace_transform_integration..sbs"syyBrp)rr~r:rDr'rZerorrErFrrr is_PiecewiserjrKlistrsortrr)frrsimplifyFcondrconds2rrrrrr{s `` @@@rnrbrbs c AuuZt!C1II+161:677A 55??N21113EqvMM >tfQiGAtuuXt=>=>=>=>=>=>~ 8 7 7 7y 7 7 7E:::::F 766U666  ! !E    JJ////J000 t 1XFAs      ( 1a # #S!Q'' QVVAr]]H - -ss1vvz##c((7K7K KKrpct|ts|S|x}|S|j}fd|jD}||S)a This is an internal helper function that traverses through the epression tree of `f(t)` and collects arguments. The purpose of it is that anything like `f(w*t-1*t-c)` will be written as `f((w-1)*t-c)` such that it can match `f(a*t+b)`. Nc0g|]}t|Srq_laplace_deep_collect)rr#rs rnrz)_laplace_deep_collect..7s$ < < .dcoTs,Q222rpz&_laplace_build_rules is building rulesrrarg?))rr rurrrr:r'r$rLrMrr;r>r#rr/r=r@rBr!r8rAr( EulerGammar?r3r"r*r4r<r2r+r)r-r7Halfrr6r9r,) r{rrrrrrlaplace_transform_rulesrs @rn_laplace_build_rulesr;s#$ c A c A S1#A S1#A S1#A uqc " " "C 1# & & &E22222 3444y AaC  y AaCE  C1QKKA. CAqAv  AE16 2 2 3 3 S "y AaCE  AaDD CAqAv  AE16 2 2 3 3 S " y 1Q3q5  3r!tAv;;q= QUAE  AFC )y 1Q3q5  Ac1"Q$q&kkM1, QUAE  AFC )y 1Q3q5  1Q3 QUAF  QVS *y 1Q3q5  1 QUAE  AFC )y" AadF  #y& AaCES!Aa[[LQBqDF+A- S1XX QVS *'y* 4!A;;QrT!V S1QZZ/T!A#a%[[0A0AA!C S1XX QVS *+y. A#a%AaDD57  1!uQw<2a46QqTT!V,,S1QZZ7$tAaCE{{:K:KKAM M S1XX QVS */y4 a!A#RT 2d1gg:c!A#hh#6tD1II#FF SVVr 163 (5y8 Ad1ggIC !2a!A$$q&k>#ac((#:4QqS ??#J  9y< AuQqSzz!ac(" R =y@ A#a%!Z!QqSU++C1QKK7AaC@B QVSQqS]]R' ( (!&# 7AyD AqsQT%!**_ZAaC%8%88 QVSQ[[2% & & 5EyH QqSWsC4yy!A# A IyL 3qs3w<<cTAaC!8+ A MyP Ac!A#hhac AaC1Q3</ ARUUC !QyT aR1WtBqDF||C1QqMM1$qac{2C2CC AAFC !UyX 3r!Q$w<< AaC488QqSAaDDFO+A-d1T!A#YY;.?.?? ? AAFC !Yy^ aRTAd1Q3iiK1T!A#YY; 7 77 A!QVS "_yb aaRT  1aR1W q4!99} -c"T!A#YY,.?.? ? A!QVS "cyh aRT477 DAJJs2d1Q3ii<'8'88 A!QVS "iyl aRTAd1ggI RT 3r$qs))|+<+< < AAFC !myp Ac1"Q$iiAaCAaC7++GAaC4!99,E,EE AAFC !qyt RQqS \   RbdA1aL(QqS1DacOOC C SVVr 163 (uyz RQqS \  477 "RTQqTT!V$4S1XX$=d4!99oo$M SVVr 163 ({y~ aRQBZ!qb'*Q"2"22  yB aRAYAj!Q/// AAFC !CyF QqSCAL))!+A-...q0 Q GyJ QqsUc!A#hhYq[QBqD) SVVr 163 (KyN QqSUc!ffS1QZZ\!^Br!tHH44a79 QUCAKK"$ % %qvs 4OyR QQ$r!t**S1S->->)>%?%??  SyV Ac!ffeAaCjjaRT*GAaCLLQ,?@ AQVS "WyZ QqS1s3q|,,Q.q01114RU1W>!3 (kyn U1WqA  tAeGAI qQqz!Q$%6!7!779 9 3r%yy>>!3 (oyt QtAaCyy[  41::a<Q/QBqD 9  uyx QtAaCyy[  ! RD1II.  yy| U1Wq!Q$uax-( RYY &}y@ U1Wq1a4%( ?QT!E1H*_=a? 3r%yy>>!3 (AyD aQtAaCyy[!! !488A:a1Q44%'l#:AacE#B3r!t99#L  EyH QtAaCyy[  $q'' !41::c1"Q$ii#7  IyL QqS#ac(( AaCE!GQT1Q3(]3QT1Q3(]C RUUC1JJ& -MyP QqS#ac(( Aq!tAqDyA~.1acAX >1acAX N RUUC1JJ& -QyT QqS#ac(( Aq!tAqDyA~.1acAX >1acAX N RUUC1JJ& -UyX acAq!tAqDyM RUUS "Yy\ acAq!tAqDyM RUUS "]y` acAqAvq!tAadF1H}- 3r!uu::s $ayd acA1Qq!tV ad1QT6!8m4 3r!uu::s $eyh ac1c1Q31+&&q( RUUS "iyl Ad1Q3iiqsA!r!t}acaRT]'BC RQ myp Ad1Q3iiqsA!r!t}acaRT]'BC RQ qyt aQqS k  DAJJqLa0QqS9  uyx aQqS k  AaCRT 1 T!WW 4S1XX =c$qs))nn LL  yy~ GGD4!99%% % 1aLaddU1W %qs1u - !HH ac^^ $$%!QKB$7 8 FAFC  !}yF aaQqS k"" "B1aLaddU1W$=qs1u$Ec!A#hh$N  GyJ aQqS k  477 " add1fa1Q44%'l "3qs88 +c$qs))nn < FAFC !KyP aQqS k  477 "B1aLaddU1W$=c!A#hh$F  QyT d1Q3ii! DGG #R!A$$q&\!^A1aL%@#ac((1*%M  UyX d1Q3ii! DGG #R!A$$q&\!^A1aL%@#ac((1*%M  Yy\ QqS3q!tQqS1H}%%d1ac7mm3A5 3s1vv;; QVS *]y` T!A#YYaac*1, QVbeeV$$c +ayd QqS#d1Q3ii.. $q''$q''/1Q3"7 QVRUU##S *eyh T!A#YYq[  Ac4!99*oo-q0 AAFC !iyl d1Q3ii4!99T!WW,d1Q3ii79 "Q%% myp QqS$tAaCyy// !1aQqS k?  qyt d1Q3iik  Cac OOA- AAFC !uyx AaC!Q$QT!Q$Y41QT ??1BQ0F FG ASAZZ &yy| Aga1oo  Ad2hhuQqvX &q!t +QT!Q$Y1"QV),D D RUUafW_bAhh ' 'RUUS :}yB Aga1oo  QqS$r(( 51Q446?? *1a4 / 11a4191Q446 2J J RUURZAqs $ $c"Q%%jj# 7CyH Ad1Q3iiK #qbd))A+  IyL Q1T!A#YY;'' 'QqS!qbd));C1II)E RUURZAqx ) )163 8MyP Ad1a4!8nn$ % % QqS41QT ??" " # #DAadOO 3 SVVr 3r!uu::s ,QyV AaC!Q$QT!Q$Y41QT ??1BQ0F FG ASAZZ &WyZ Aga1oo  Ad2hhuQqvX &q!t +QT!Q$Y1"QV),D D RUUafW_bAhh ' 'RUUS :[y` Aga1oo  QqS$r(( 51Q446?? *1a4 / 11a4191Q446 2J J RUURZAqs $ $c"Q%%jj# 7ayf Q1T!A#YY;'' 'QqS!qbd));C!HH)D RUURZAqx ) )163 8gyj AaC"R%ac *41QT ??: RUUS "kyn AaC#q41QT ??2A566QT!Q$YH "Q%% oyt #Aq ((rpctd|g}tdd}||}|r||jd|}|j||z}|rr||jre||dkrYt dtd||z |||z||||z d \}} } || | fSd S) z This function applies the time-scaling rule of the Laplace transform in a straight-forward way. For example, if it gets ``(f(a*t), t, s)``, it will compute ``LaplaceTransform(f(t)/a, t, s/a)`` if ``a>0``. rrgra)nargsrz rule: time scaling (4.1.4)FrN) r rrrjcollectrru_laplace_transformrm) rrr{rrma1r#ma2rprcrs rn_laplace_rule_timescaler s S1#AS"""A ''!**C !fk!n$$Q''ci!nn  3q6% #a&A++ 4 5 5 5*#a&QQ'Ac!fHuFFFIAr2r2;  4rpctd|g}td}td}|t||zx}rx||||z x}r||jrht dt ||||||z||d\}} } t|| |z|z| | fS||jr1t dt ||||d\}} } || | fS||||z x}r||jrMt d t d t|||z z ||z||d\}} } || | fS||jrt d d d tj fSd S)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 openraz" rule: Heaviside window closedrN) r rr;rrurrr' is_negativerr) rrr{rrrrr rr r s rn_laplace_rule_heavisider*s S1#A S A S AggillQ&'''s&a&,,q1u%% %3 #1v! 66777.FKK1s1v:..1uFFF 2rSVGaK((1,b"551v! #NPPP.s1vq!eLLL 2r2r{"a&,,q1u%% %3 &1v! #9:::.1s1v:...#a&8!QPPP 2r2r{"1v! &;<<<1af~% 4rpctd|g}td}td}|t||z}|r|||||z}|rPt dt ||||||z d\}} } || t ||z| fSdS) 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)r rr'rrurr!) rrr{rrrrr rr r s rn_laplace_rule_exprTs S1#A S A S A ''#a&&(  C *!fnnQ%%ac**  * 9 : : :*3q61aAh49;;;IAr2r"SV**}b) ) 4rpct tdg tdg td}td|t|z rE ts$ | z z  r)t d  z }t |dkrt|dkrt  z z z}|ttr,| t } fd|D\}}|dkr%||z z tjtjfSd SdtjtjfS |rt' |}|ikrt)|d hkrgt- |t/ fd t1|D} | tjtjfSd S) 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 DiracDeltarcZg|]'}|z (Srqr)rrrrr rs rnrz'_laplace_rule_delta..s3NNNQq#a&Q-00NNNrpNracg|]o}t|dkt|dk(t| zz|z pS)r)r"r!r'r)rrrr{sloperrs rnrz'_laplace_rule_delta..svMMM"Q%%1**A!1"Q$iiA Aq 1 11%**Q2B2BBAKrp)r rr:r~rrur!r"r'r3r2rewriter5ratsimpas_numer_denomrrr is_polynomialrQsetvaluesrrrkeys)rrr{rlocfnrrrorrrrr rrs `` @@@@@@rn_laplace_rule_deltar!ks S1#A S1#A S A S A ''*Q--/ " "C 73q6::j))7!fnnQ%%ac!e,,  7 8 9 9 9a&Q-C#ww!||31 #a&Q)**3q6166#s##4D))1133BNNNNNNN":K:K:M:MNNN166aCAJ(:AFCC41-qv66 q6   " " 7s1vq!!BRxxC ,,33SVQMMMMMMMM#BGGIIMMMN1-qv66 4rpcJtjg}tjg}tj|D]_}|t t tttr| |J| |`t|}t|}||fS)z Helper function for `_laplace_rule_trig`. This function returns two terms `f` and `g`. `f` contains all product terms with sin, cos, sinh, cosh in them; `g` contains everything else. ) rOner make_argsr~r3r2r+r)r'append)rtrigsothertermrrs rn_laplace_trig_splitr)sUGE UGE b!! 88CdD# . .  LL     LL     U A U A a4Krpc0td|g}td|g}td|g}g}g}|t}t j|D]}||s(|d|ddtdtdi@t| d |}| |t||z|zzx} q|d| |t| |zd| |tt| |tt| |i||||fS) a Helper function for `_laplace_rule_trig`. This function expects the `f` from `_laplace_trig_split`. It returns two lists `xm` and `xn`. `xm` is a list of dictionaries with keys `k` and `a` representing a function `k*exp(a*t)`. `xn` is a list of all terms that cannot be brought into that form, which may happen, e.g., when a trigonometric function has another function in its argument. c1rc0rkrrr')combine) r rr'r rr$r~r%r!r"rpowsimpr) rrr+r,rxmxnx1r(rs rn_laplace_trig_expsumr3st dQC B dQC B S1#A B B 3   B b!!  xx{{  IIsD#q"aQ7 8 8 8 $T\\%\%@%@!DDAc"Q$r'llN++ +A 8 IIQqT#ae**_c1R5BquIIr2ae99. / / / / IIdOOOO r6Mrpc fg}g}dfd}fd}fd}fd}d} t|dkrm|} d} d} d} tt|D]}| t||tk}| t||t k}| t||tk}| t||t k}|r'|r%| tdkr| tdkr|} |r|r| tdkr|} |r|r| tdkr|} | | | ||| || d || d || d ||t t| d | | | g}|d |D]}||n| c||| || d ||| t|| n!| o||| || d ||t | t|| n| o||| || d ||t | t|| n?|| | ||| tt|dkmt|t|fS) a 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|}tt|D]}||}|dt r$||t||<`|dt|dzzt||<|Sr) copyrangelenrr~r"rr2r)coeffsncr-ris rn_simpcz"_laplace_trig_ltex.._simpcs [[]]s2ww 7 7AA##%%B!uyy}} 71 c**1A2a511#661 rpc *|d|d|t|tf\}}}}||z|z|z|||z|z |z zdtz|z|zz dtz|z|zz|dz| |z |z |z z|dtz|z|zdtz|z|zzzzd|dzz|zzd|dzz|zz|dz| |z |z|zz|dzdtz|z|zdtz|z|zzdtz|z|zz dtz|z|zz zz|d|dzz|zd|dzz|zz zzg} tjtjd|dzzd|dzzz tj|dzd|dzz|dzzz|dzzg} t fdt | tt| dddD} t fdt| tt| dddD} | | z S) Nrr-rrrc&g|] \}}||zzSrqrqrrrr{s rnrz9_laplace_trig_ltex.._quadpole..% G G GAa1f G G Grprc&g|] \}}||zzSrqrqr?s rnrz9_laplace_trig_ltex.._quadpole..% ? ? ?Aa1f ? ? ?rp) r!r"rrr#rrzipr7r8)t1k1k2k3r{rk0a_ra_ir:dcrrr<s ` rn _quadpolez%_laplace_trig_ltex.._quadpoleshS'2c7BrFBrF:2sC GbL2  rBw|b !AaCGBJ .1S ;1rcBhmb()1Q3s72:!C *+,#q& Qhrk*1rcBhmb()1ac#gbj1Q3s72:-!C :QqSWRZGHI1S!V8B;36",-.   E161S!V8aQh. FCFQsAvXc1f_,sAv57  G G G GVVBZZs2ww"1E!F!F G G G I  ? ? ? ?Rs2ww")=!>!> ? ? ? 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As rpc |d|d}}||z|||z zg}tjtj|dz g}tfdt  |t t |dddD}tfdt |t t |dddD}||z S)Nrr-rc&g|] \}}||zzSrqrqr?s rnrz7_laplace_trig_ltex.._sypole..r@rprc&g|] \}}||zzSrqrqr?s rnrz7_laplace_trig_ltex.._sypole..rBrp)rr#rrrCr7r8) rDrGr{rrHr:rKrrr<s ` rn_sypolez#_laplace_trig_ltex.._sypole s3C22gq"r'{ #eQVadU #  G G G GVVBZZs2ww"1E!F!F G G G I  ? ? ? ?Rs2ww")=!>!> ? ? ? As rpc:|d|d}}|}||z }||z S)Nrr-rq)rDr{rrHrrs rn _simplepolez'_laplace_trig_ltex.._simplepoles*3C2  Es rprNr-rT)reverse) r8popr7r!r"r%r$rrr-)r0rr{resultsplanesrLrQrUrYr[rD i_imagsym i_realsym i_pointsymireal_eqrealsymimag_eqimagsymindices_to_popr<s @rn_laplace_trig_ltexrisG F. b''A++ VVXX   s2ww  Af1b )GfAr *Gf1b )GfAr *G 7 r"v{{r"v{{  W B1  W B1 %)*?* NN "Y-,bmC.@Z.-q22 3 3 3 MM#bCkk** + + +(J?N     - - -#  q    " NN772r)}S'91== > > > MM"R& ! ! ! FF9      " NN772r)}S'91== > > > MM#bf++ & & & FF9      # NN772r*~c':A>> ? ? ? MM#bf++ & & & FF:     NN;;r1-- . . . MM"R& ! ! !q b''A++t =#v, &&rpc tdd}|tttt sdSt |||\}}t||\}}t|dkrdS||s&t|||\}} ||z| tj fSg} g} t|||d\} } }|D]h}| |d| |||d z z| | t|d zit!| ||t#| |fS) z This rule covers trigonometric factors by splitting everything into a sum of exponential functions and collecting complex conjugate poles and real symmetric poles. rTrealNrFrr-r)rr~r3r2r+r)r)rr3r8rirrrr%r!rr-)rt_r{rrrr0r1rrr_r^GG_planeG_condr2s rn_laplace_rule_trigrqYsd cA 66#sD$ ' 't rwwr1~~ . .DAq !!Q ' 'FB 2ww{{t 5588 /!"a++1sAqv~/1a%HHH7F / /B NN2c7166!Qr#wY#7#77 8 8 8 MM'"RW++- . . . . =  a $ $c6lF ::rpctdg}tdg}td}||t||fz}|r||jr fd||jD}t |dkrtdg}t||D]x} | dkr|| d} n,t||| f d} | |||| z dz z| zyt|||d \} } } |||||z| zt|z z| | fSd S) a  This function looks for derivatives in the time domain and replaces it by factors of `s` and initial conditions in the frequency domain. For example, if it gets ``(diff(f(t), t), t, s)``, it will compute ``s*LaplaceTransform(f(t), t, s) - f(0)``. rrrrc:g|]}|Srqr~)rrrs rnrz&_laplace_rule_diff..s# + + +!QUU1XX + + +rpraz" rule: time derivative (4.1.8)rFrN) r rrr is_integerrjsumrur7rr%rr)rrr{rrrrrrr-rrr r s ` rn_laplace_rule_diffrwzs S1#A S1#ASA ''!Jq1a&))) * *C =s1v = + + + +s1v{ + + + q66Q;; 7 8 8 8A3q6]] , ,66A Aq))AA"3q6Aq622771==ASVAXaZ*++++*3q61a%HHHIAr2FAs1vIaK#q'12R< < 4rpc 2|jrdg}dg}tj|D]B}||r||-||Ct |dkr~t |}t||t dkr7t |}t|||d\}} } |gd} td| } n#t$rd} YnwxYw| trFtdz D]2} d| dzzt||| dzz3nV| rT| tdz D],} td| -| r)t!fdtD} | | | fSt#d|g}t#d }|||z|zx}r^||jrQ||jrDt||||d\}} } d||zt||||fz| | fSd S) a 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)``. raFrrrc>g|]}|z dz |zSrrq)rrNderipcs rnrz'_laplace_rule_sdiff..s.BBBAb1QiQ/BBBrprrrN)is_Mulrr$rr%r8rrR all_coeffsrr ValueErrorr~LaplaceTransformr7r rr rrur)rrr{pfacofacfacpexoexr_p_c_d1r-rrrrrzr{r|s @@@rn_laplace_rule_sdiffrs x'ss=## ! !C  ## ! C     C    t99q==t**Cc1((**BBA1uu4jj/QEJJJ BttBx+++BB!BBB66*++8"1Q3ZZHH R1Q3K 2q!A#0F0F$FGGGGH8KKOOO"1Q3ZZ88 T$r(A%6%6$67777'BBBBBBqBBBCAr2;& S1#A S Aggad1foos> q6  >Q!3 >+CFAq5IIIJBBQ<R!SV 5 55r2= = 4s0D DDct|d}|jrt|||dSt|}|jrt|||dSt|}|jrt|||dS||krt|||dStt |}|jrt|||dSdS)a 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 avilable 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 rn_laplace_expandrs quAx;!!QE::::1 Ax;!!QE::::q Ax;!!QE::::Avv!!QE::::{1~~Ax;!!QE:::: 4rpctttttt t g}|D]}||||x}|cSdS)zk This function applies all program rules and returns the result if one of them gives a result. N)rr!r rrqrwr)rrr{ prog_rulesp_ruleLs rn_laplace_apply_prog_rulesrs[*+>)+<$$&9;J 1a A -HHH . 4rpct\}}}d}d}|D]\}} } } } || kr"| |||i}| }||} | r | | }n#t$rYjwxYw|t jkrL| | ||i| | t jfcSdS)zj This function applies all simple rules and returns the result if one of them gives a result. N)rrrxreplacerrr)rrr{ simple_rulesrmrprep_oldprep_ft_doms_domcheckplaneprepmars rn_laplace_apply_simple_rulesrs 011L"bH F,844(ueUD t  T!&&!R//**FH \\%  4 NN2&&    AF{{r**//Q88r**AF4444 4s!A77 BBc2|jsKtdd}t|||i|||iSt |}g}|jD]\}}t |tr\||jvrSt |ttfr|cS| t|j |j z |zwt |trft|jdkrN|jD]E}|j|kr3| t|j |j z |z@|ccSt |t"rt|jdkr|j\}}|j|kri|j|kr^d|jvr||}}| t|j |j z t|j |j z z |z|cS|cSt'|S)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. rTrkr>)is_realr_piecewise_to_heavisiderr1rjrrrrr%r;gtsrrLr8lhsrMrr) rrrrrrc2r,r+s rnrrs' 9O #D ! ! !&qzz1a&'9'91==FF1vNNNAA AF  D dJ ' ' ANN$R)) < 48dh#677:;;;; b ! ! c$)nn&9&9i  6Q;;HHYrv77:;;;;HHHHH   c " " s49~~':':YFBv{{rv{{")##Brv//rv//01345555HHH 7Nrpc td}td}td}td}t|tr4|ts|t s|SD]\}}|t ||||x}%||||kr|||cS|t |||||x} %||||kr|||cS|j} fd|j D} | | S)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{rrNc0g|]}t|Srq)laplace_correspondence)rr#fdicts rnrz*laplace_correspondence..|s$ A A A3 "3 . . A A Arp) r rrr~rInverseLaplaceTransformitemsrrmrj) rrrr{rrrYrrmrjs ` rnrrDslH S A S A S A S A1d##EE*++566  1gg.qqttQ::;;;HaDAaDLL1QqT77NNNgg5aaddAq!DDEEEaDAaDLL1QqT77NNN 6D A A A A!& A A AD 4;rpc |D]\}}tt|D]}|dkr&||d|d}.|dkrC|t t ||||d|d}w|t t ||||f|d||}|S)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)rr7r8rrr )rrrricr-s rnlaplace_initial_condsrsDKK2s2ww K KAAvvIIaaddBqE**aIId:aaddA#6#61==r!uEEIId:aaddQF#;#;QBBBqEJJ  K Hrpctj|}g}g}g}g}|D]C} | d\} } | trftj| t } | D]6} | d\}}|| |z|f7| jtkr| ts]tjt| } | D]6} | d\}}|| |z|f7,|| | fE|D]\} } | tr t| |tjdf}n*| t rE| ts#| t d} t!| |x} 't#| |x} t%| |x}nt'fd| t*Drt| |tjdf}n5t-| ||x} nt| |tjdf}|\}}}|| |z||||t|}|r|d}t1|}t3|}|||fS) 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_AddTraNc3BK|]}|VdSrrt)rundefrms rnrz%_laplace_transform..s-GGuUYYr]]GGGGGGrprdoit)rr$as_independentr~rCrr;r%rmr0r:rrrrrrrranyrr rbrr-rM)rrmrrterms_tterms_stermsr_ conditionsffr-ft_terms_termrEf1rri_pi_ci_rlr conditions ` rnrrsmBGG E FJ " "!!"U!332 66% & & "]2::i#8#899F ) )--b-??B adBZ(((( )W ! !"&&B*@*@ !]#:2r#B#BCCF ) )--b-??B adBZ(((( ) LL!R ! ! ! !2 66% & & M!"b"--q/A4HAAvvimm$$ /RVVJrNN-C-C /WWYr]]A..5b"bAAAQ 3BB???Q )"b"555QBGGGG0F0FGGGGG M&b"b1113EtL5B3333!;?@%b"b1113EtLc3qu c# ']F-e,, LEZ I 5) ##rpc(eZdZdZdZdZdZdZdS)ra 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 X|dd}t||||}|S)NrFr)getrb)selfrrr{hintsrFLTs rn_compute_transformz#LaplaceTransform._compute_transforms0IIj%00 +Aq!i H H H rpcxt|t| |zz|tjtjfSr)rEr'rrr)rrrr{s rn _as_integralzLaplaceTransform._as_integrals-#qbd)) a%<===rpc |dd}|dd}td|j|j|jf|j}|j}|j}t ||||}|r|dS|S)j 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)rr)rrXfunctionfunction_variabletransform_variabler)rr_nocondsrFrmrrrs rnrzLaplaceTransform.doit s99Y--IIj%00 '$-*.*@*.*A*C D D D #  $ ] r2rI > > >  Q4KHrpN)ri __module__ __qualname____doc___namerrrrqrprnrrsR   E >>>rprTc dd}dd}t|trt|drdd }|r]|r[d}t dd|t t 5|fd cd d d S#1swxYwYnjfd |D} |r>> 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"t|fiSrlaplace_transform)fijrr{rs rnrz#laplace_transform..s 1#q! E Eu E ErpNc.g|]}t|fiSrqr)rrrr{rs rnrz%laplace_transform..sG222(+0Q$$"$$222rprr)rrrNhasattrrUrWrVrrCtypeshaper-rMrr)rrr{ legacy_matrixrrrFradtelements_transelementsavalsr f_laplacerrrs `` ` rnrr)s:NyyE**H *e,,I!Z  9WQ %<%<9IIi///  9] 97C % */+.    !!899 G G{{EEEEEEGG G G G G G G G G G G G G G G G G G222222/0222N 9.1>.B+%#DGG7QW7h777  #u+sJ/???tAww8888881a((--e7@.BBHB1 1ax sB??CCc ddlm}m ddlm}t dd fd}|r|}|jrCt fd|j D}t| |dfS ||t d tjfdd \}} n#t $rd }YnwxYw|l|| }|d S|jr-|j d\}} |t&rd Sn tj} |t,|}|jr| || fSt d tjf fd } |t0| }d} |t| }t| || fS)z6 The backend function for inverse Laplace transforms. r)meijerint_inversion_get_coeff_exp)inverse_mellin_transformrTrkcxt|dkr t|S|djdj}|\}}|djd}|djd}t dt |z |zz |zt |zdt |z z |zzS)z3 Simplify a piecewise expression from hyperexpand. rrrra)r8r0rjargumentr;r$)rjr#coeffexponente1e2rrs rnpw_simpz7_inverse_laplace_transform_integration..pw_simps t99>>d# #1gl1o&(.a00x !W\!_ !W\!_ aE lQ[0 1 1" 4 akAc%jjL0 1 1" 4 5 6rpc 6g|]}t|Srq)rc)rXrr{rrs rnrz:_inverse_laplace_transform_integration..s95Q1eXNNrpNF)needevalrucl|jt }|rt||Sddlm}||dk}|jkr't|j}t|z|St|j}t|z |S)Nrr) rr'r~r;rrrr(r)r#H0rrrelr-rrs rnsimp_heavisidez>_inverse_laplace_transform_integration..simp_heavisides CHS!WWa  5588 &S"%% %@@@@@@Aq)) 7a<<CG AQUB'' 'CG Aq1uXr** *rpc:tt|Sr)r r')r#s rnsimp_expz8_inverse_laplace_transform_integration..simp_expsc#hh'''rp)sympy.integrals.meijerintrrsympy.integrals.transformsrris_rational_functionapartrrrjrFrr'rrrHrr~rErrr0rr;)rr{rmrrrrrrrrrrrrs ` `` @@@rnrcrcsNMMMMMMMCCCCCC cA 6 6 6 6 6 6 a   GGAJJx8 v 211477**1aaR4:L48%III44 !  y  1a ( ( 94 > fQiGAtuuX t 6D IIi ) )~#vva}}d"" c A v + + + + + + + )^,,A((( #x  A QVVAr]]H - -t 33s!.C CCc$ddlm}||\}}||rc||}t |dkr)|\}}}|||d|zz zdz||z z|d|zz dzz z}||z S)Nr)fractionrr)rr rrr~r8) rr{r rrcfrrrs rn_complete_the_square_in_denomr  s////// Xa[[FQq4 YYq\\ $ $ & & r77a<<GAq!Aa1gI>!A#%q!A#wl23A Q3Jrpc td}td}td|g}td|g}td|g}tdd}d }||z |tj|d f|||z| zz||d z zt | |zzt |z tj|d fd |d z|d zzd zz t||z||zt||zzz d |d zzz tj|d fd ||zz ||d z zt |z tj|d fd |||z|zzz t|||z||zt |zz tj|d fg}|||fS) 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`. 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EG&J&$&$$$&&A YYq\\ $ $ & &Q% $ $ $ $ $ $ $ ; ; ; ;1!8!8!:!: ; ; ; r77a<<1jmm#A NN1     WW\\1c2a5&(mm#A NN9Q<<> * * * * WW\\1aAAq!t##%%A2ww!||fX]99DAqAvvqSAacE]C1II-=AC%Aa,,113344Q7!WW%%''O#qbd)) DAJJ.!AaC%2r!t992%%)"Q$ZZ200A#qbd)) C1II-1Q3 3r!t990DSAYY0NNA NN9Q<<> * * * *1aEHHHHHB NN2      d # # # # ']F-e,, 3 # ##rpctj|}g}g}|D]} | tr>|   } | d\} } |r,| rt| |||x} St| |x} ?t| ||x} *t| ||x} t| ||x} ntfd| tDrt!| ||t"jf} n6t'| |||x} nt!| ||t"jf} | \} }|| | z||t|}|r|d}t-|}||fS)z Front-end function of the inverse Laplace transform. It tries to apply all known rules recursively. If everything else fails, it tries to integrate. FrrNc3BK|]}|VdSrrt)rrrs rnrz-_inverse_laplace_transform..s-BB52BBBBBBrpr)rr$r~r'rrSrrrQrrArErCrrr rrrrcr%rrM)rrrmrrrrrrr(r-rrrrrlrs ` rnr r ^sb M"  EGJ%% 88C== ?99R"%%..0055b2#>>D""2e"441 D#88<< D/r2ux9999:!RDDD72r5III-aR???72r5III  BBBBAGGL,A,ABBB B B D)BE::AFCAA;r2ux9999AEF (BE::AFCA cqu# ']F-e,,Z I 9 rpcpeZdZdZdZedZedZdZe dZ dZ dZ d Z d S) rz 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 J| tj}tj|||||fi|Sr)r_none_sentinelrG__new__)rrr{rroptss rnrXzInverseLaplaceTransform.__new__s0 =+:E (aAuEEEEErpc@|jd}|tjurd}|S)Nr)rjrrW)rrs rnfundamental_planez)InverseLaplaceTransform.fundamental_planes& !  +: : :E rpc ,t||||jfi|Sr)rcr[)rrr{rrs rnrz*InverseLaplaceTransform._compute_transforms-5 q!T+66/466 6rpc|jj}tt||z|z||tjtjzz |tjtjzzfdtjztjzz S)Nr) __class___crEr'r ImaginaryUnitrPi)rrr{rrs rnrz$InverseLaplaceTransform._as_integralsn N  S1XXaZ!Q)C%C"#aoaj&@"@"B C C qtVAO # % &rpc |dd}|dd}td|j|j|jf|j}|j}|j}|j}t |||||d}|r|dS|S)rrTrFz[ILT doit] (%s, %s, %s)rr)rrXrrrr[r ) rrrrFrrmrrrs rnrzInverseLaplaceTransform.doits99Y--IIj%00 (4=+/+A+/+B+D E E E #  $ ]& & B d D D D  Q4KHrpN)rirrrrrrWr_rXpropertyr[rrrrqrprnrrs EU6]]N sBFFF X 666&&&rprc Fdd}dd}t|tr+t|dr|fdSt |d|\}}|r|S||fS)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$t|fiSr)inverse_laplace_transform)Fijrrr{rs rnrz+inverse_laplace_transform.. s1#q!ULLeLLrpr)rrrNrrrr) rr{rrrrrFrrs ```` rnrfrfsZyyD))H *e,,I!Z  NWQ %<%<N{{ L L L L L L LNN N #1aE 2 2 7 7  8 + +DAq!t rpc tdtg\ fdfdfd fdfd|S)zEFast inverse Laplace transform of rational function including RootSumza, b, nrc|s|S|jr |S|jr |S|jr |St |t r |St r)r~rr}rxrrTNotImplementedError)e_ilt_add_ilt_mul_ilt_pow _ilt_rootsumr{s rn_iltz#_fast_inverse_laplace.._ilt suuQxx &H X &8A;;  X &8A;;  X &8A;;  7 # # &<?? 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