7^(hSdZddlZddlZddlmZmZmZddlmZddl 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?Ze d@ZezdAZezdBZezdCZezdDZezdEZezdFZezdGZezdHZezdIZezdJZezdKZGdLdMe]ZdQdNZdOZy)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)debugfcfd}|S)Ncfddlm}|s|i|Stdk(rtdtj tddtzj |tj tdz aj dk(sj d k(rDd t_td dtzztj |i|}d t_n|i|}tdzatddtzd |tj tdk(rtdtj |S)Nr SYMPY_DEBUGzO ------------------------------------------------------------------------------file-LT-  _laplace_transform_integration&_inverse_laplace_transform_integrationFz**** %sIntegrating ...Tz---> zO------------------------------------------------------------------------------ )sympyr\ _LT_levelprintsysstderr__name__)argskwargsr\resultfuncs U/var/www/skyplay_api_hub/venv/lib/python3.12/site-packages/sympy/integrals/laplace.pywrapzDEBUG_WRAP..wrap1s%(( ( > -cjj 1 tI~t}}dC:: Q  !AA !II %E  *d9n=CJJ O4*6*F $E 4*6*FQ  $y.&9 K > -cjj 1 )rmros` rn DEBUG_WRAPrr0s4 Krpcdddlm}|r(tddtz|tj yy)Nrr[r_r`r])rdr\rfrergrh)textr\s rn_debugruNs$! T)^T2Drpcfdfdfdfd}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) c`|k(ry|jr|jk(r |jSy)Nra)is_Powbaser')exss rnpowerz_simplifyconds..powervs* 7 99A66Mrpc|jr|jryt|tr|jd}t|tr|jd}|jrd|z d|z S|}|y |dkDr,t|t|zktj k(ry|dkr-t|t|zk\tj k(ryyy#t $rYywxYw)z_ Return True only if |ex1| > |ex2|, False only if |ex1| < |ex2|. Else return None. NrraFT)has isinstancer$rjrtrue TypeError)ex1ex2nabiggerr|r{s rnrz_simplifyconds..bigger}s 771:#''!* c3 ((1+C c3 ((1+C 771:!C%3' ' #J 9 1u#c(c!fai/AFF:1u#c(c!fai/AFF:;u  s0C30C33 C?>C?c|jst|tr|jst|ts||kS||}|| S||kS)z simplify x < y ) is_positiverr$)xyrrs rnrepliez_simplifyconds..repliesL*Q"4 As);EN 1aL =5LArpc8||}|dvryt||S)NTFT)r)rrbrs rnrepluez_simplifyconds..replues& 1aL !Qrpc>|dvr t|S|j|S)Nr)boolreplace)rzrjs rnreplz_simplifyconds..repls$  8Orzz4  rpr) collect_absc||SNrq)rrrs rnz _simplifyconds..s va|rp)sympy.simplify.radsimprrrrr) exprr{rrrrrr|rs `` @@@rn_simplifycondsrUs`B, ! 3 t D b& !D b3 4D j& )D T7Nrpc@t||jtS)zs Expand an expression involving DiractDelta to get it as a linear combination of DiracDelta functions. )rOatomsr:rs rnexpand_dirac_deltars djj4 55rpctd|jtryt|t  zzt j t jf}|jts;t|j|t jt jfS|jsy|jd\}}|jtryfd}t|Dcgc] }|| }}|D cgc]0} | dt j k7s| dt jur| 2} } | s&|D cgc]} | dt j k7s| } } t#t%| }d|j'fd|sy|d\} } fd } |rt)|| }t)| | } t|j|| | t+| | fScc}wcc} wcc} w) 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{Nrcddlm}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]} tj} g}t| D]}|jr$|j j"vr |j$}|jr"t'|t(t*fr |j,}| D]}|j/|snr9|j0r*||z t2dz k(rt5|z dk}|j/|t7|tt| zz|zt|z| zzz dksN|j/t7|tt|z| z||zz t|z| zzdksW|j/|t7ttt|z| z||zt|z| zzz dkr*t9fd|||| | fDrt5|kD}|j;t4dj=t5}|jr0|j>dvs"|jAs|jAs||gz }O||}|jr|j>dvr||gz }z|jBk(ry tE|jB| } | tjurtG| |}tI|tK|}||jr |jLfS|fS) z7 Turn ``conds`` into a strip and auxiliary conditions. r_solve_inequalityzp q w1 w2 w3 w4 w5clsexcludec3<K|]}|jywr)r).0wildms rn zH_laplace_transform_integration..process_conds..s-TQtW00-scD|jjdSNr)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@  ff&-(#* dQC$9 1b"b"b 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 I- *ABDq\& +??qAEE,>,>'> A??z!b"X'>A#C A1))aeAaDjBqD.@A"I*AGGABs3qt9~$5b$8 9#ae*b. HH1LMA$5aeBh$B CB FFGArE B')*+,AC 1*Q-2CB2F JKBN!!R%j"n--/012A-BB>,--1! AYY>@@DRUAOOqxx.cnts = ~~rpc&|d |dfSNrrarq)rrs rnrz0_laplace_transform_integration..sqteS1Y/rpkeyc(|jSr)r)rr{s_s rnsbsz+_laplace_transform_integration..sbs!syyBrp)rr~r:rDr'rZerorrErFrrr is_PiecewiserjrKfalselistrsortrr)frrsimplifyFcondrrrrconds2rrrrr{s `` @@rnrbrbs c AuuZ!C1I+1661::67A 55?2113E3EqvvMM >>ffQiGAtuuX=>~(1 7!]1  7E 7:A!A$gg#A$a&8&88:F: "6adaggo!66  !E  JJ/J0  1XFAs  1a #S!Q' QVVAr]H -s1vz#c(7K KK- 8:7s9G:G?*G? H'Hct|ts|S|j|x}|jS|j}|j Dcgc]}t ||}}||Scc}w)a 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)si a  YYq\&yy{ 66D56VV .dcoSs,Q22rpz&_laplace_build_rules is building rulesrrarg?)(rr rurrrr: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| AaC  | AaCE C1QKA. CAqAv AE16 2 3  S "| AaCE AaD CAqAv AE16 2 3  S " | 1Q3q5 3r!tAv;q= QUAE AFFC )| 1Q3q5 Ac1"Q$q&kM1, QUAE AFFC )| 1Q3q5 1Q3 QUAF QVVS *| 1Q3q5 1 QUAE AFFC )|" AadF  #|& AaCES!Aa[LQBqDF+A- S1X QVVS *'|* 4!A;QrT!V S1QZ/T!A#a%[0AA!C S1X QVVS *+|. A#a%AaD57  1!uQw<2a46QqT!V,,S1QZ7$tAaCE{:KKAM M S1X QVVS */|4 a!A#RT 2d1g:c!A#h#6tD1I#FF SVr 1663 (5|8 Ad1gIC !2a!A$q&k>#ac(#:4QqS ?#J  9|< AuQqSz!ac(" R =|@ A#a%!Z!QqSU+C1QK7AaC@B QVSQqS]R' (!&&# 7A|D AqsQT%!*_ZAaC%88 QVSQ[2% & 5E|H QqSWsC4y!A# A I|L 3qs3w<cTAaC!8+ A M|P Ac!A#hac AaC1Q3</ ARUC !Q|T aR1WtBqDF|C1QqM1$qac{2CC AAFFC !U|X 3r!Q$w< AaC48QqSAaDFO+A-d1T!A#Y;.?? ? AAFFC !Y|^ aRTAd1Q3iK1T!A#Y; 77 A!QVVS "_|b aaRT  1aR1W q4!9} -c"T!A#Y,.? ? A!QVVS "c|h aRT47 DAJs2d1Q3i<'88 A!QVVS "i|l aRTAd1gI RT 3r$qs)|+< < AAFFC !m|p Ac1"Q$iAaCAaC7++GAaC4!9,EE AAFFC !q|B aRQBZ!qb'*Q"22  C|F aRAYAj!Q// AAFFC !G|J QqSCALL)!+A-..q0 Q K|N QqsUc!A#hYq[QBqD) SVr 1663 (O|R QqSUc!fS1QZ\!^Br!tH44a79 QUCAK"$ %qvvs 4S|V QQ$r!t*S1S->)>%??  W|Z Ac!feAaCjaRT*GAaCLQ,?@ AQVVS "[|^ QqS1s3q||,Q.q0114RU1W!3 (o|r U1WqA  tAeGAI qQqz!Q$%6!779 9 3r%y>!3 (s|@ U1Wq!Q$uax-( RY &A|D U1Wq1a4%( ?QT!E1H*_=a? 3r%y>!3 (E|P QqS#ac( AaCE!GQT1Q3(]3QT1Q3(]C RUC1J& -Q|T QqS#ac( Aq!tAqDyA~.1acAX >1acAX N RUC1J& -U|X QqS#ac( Aq!tAqDyA~.1acAX >1acAX N RUC1J& -Y|\ acAq!tAqDyM RUS "]|` acAq!tAqDyM RUS "a|d acAqAvq!tAadF1H}- 3r!u:s $e|h acA1Qq!tV ad1QT6!8m4 3r!u:s $i|l ac1c1Q31+&q( RUS "m|p Ad1Q3iqsA!r!t}acaRT]'BC RQ q|t Ad1Q3iqsA!r!t}acaRT]'BC RQ u|b QqS3q!tQqS1H}%d1ac7m3A5 3s1v; QVVS *c|~ AaC!Q$QT!Q$Y41QT ?1BQ0F FG ASAZ &|B Aga1o  Ad2huQqvvX &q!t +QT!Q$Y1"QVV),D D RUaffW_bAh 'RUS :C|H Aga1o  QqS$r( 51Q46? *1a4 / 11a4191Q46 2J J RURZAqs $c"Q%j# 7I|V Ad1a4!8n$ % QqS41QT ?" " #DAadO 3 SVr 3r!u:s ,W|\ AaC!Q$QT!Q$Y41QT ?1BQ0F FG ASAZ &]|` Aga1o  Ad2huQqvvX &q!t +QT!Q$Y1"QVV),D D RUaffW_bAh 'RUS :a|f Aga1o  QqS$r( 51Q46? *1a4 / 11a4191Q46 2J J RURZAqs $c"Q%j# 7g|p AaC"R%ac *41QT ?: RUS "q|t AaC#q41QT ?2A56QT!Q$YH "Q% u|z #Aq ((rpctd|g}tdd}|j|}|r||jdj |}|j||z}|r[||j rL||dk7rDt dtd||z ||j|z||||z d \}} } || | fSy ) 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_timescalers S1#AS"A ''!*C !fkk!n$$Q'cii!n 3q6%%#a&A+ 4 5*#a&QQ'Ac!fHuFIAr2r2;  rpctd|g}td}td}|jt||zx}r<||j||z x}r||jrOt dt ||j ||||z||d\}} } t|| |z|z| | fS||jr&t dt ||||d\}} } || | fS||j||z x}rw||jr;t d t d t|||z z ||z||d\}} } || | fS||jrt d d d tjfSy )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{rrrrrrrrs rn_laplace_rule_heavisider,s S1#A S A S AggilQ&''s'a&,,q1u% %3 %1v!!67.FKK1s1v:.1uF 2rSVGaK(1,b"551v!!NP.s1vq!eL 2r2r{"a&,,q1u% %3 %1v!!9:.1s1v:..#a&8!QP 2r2r{"1v!!;<1aff~% rpcPtd|g}td}td}|jt||z}|rc||j|j||z}|r;t dt ||||||z d\}} } || t ||z| fSy) 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{rrrrrrrrs rn_laplace_rule_exprVs S1#A S A S A ''#a&( C !fnnQ%%ac*  9 :*3q61aAh49;IAr2r"SV*}b) ) rpc ^td|g}td|g}td}td}|jt||z}|rQ||jts8||j |j||z|z }|rt d||||z } t | dk\rt| dk(rt|| ||z |z||z} | jttr#| jtj} | jD cgc]} | j|||||z c} \} } | dk7r*| | z ||z t j"t j$fSydt j"t j$fS||j'|rt)|||}|ik7rt+|j-d hk(rt/|||}t1t3|j5D cgc]V} t| dk(rFt | dk\r8t| |z||j||z|j|| z Xc} }|t j"t j$fSycc} wcc} w) 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:r~rrur!r"r'r3r2rewriter5ratsimpas_numer_denomrrrr is_polynomialrQsetvaluesrrrkeys)rrr{rrrrrrlocfnrrrroslopers rn_laplace_rule_deltarmsY S1#A S1#A S A S A ''*Q-/ "C 3q6::j)!fnnQ%%ac!e,  8 9a&Q-C#w!|31 #a&Q)*3q6166#s#D)113B:<:K:K:MNQq#a&Q-0N16aCAJ(:(:AFFCC1--qvv66 q6   "s1vq!BRxC ,3SVQ#BGGIM"Q%1*A!1"Q$iA Aq 11%**Q2BBMN1--qvv66 OMs 7"J%$AJ*c:tjg}tjg}tj|D]N}|j t t tttr|j|>|j|Pt|}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_splitrs{UUGE UUGE b! 88CdD# . LL  LL   U A U A a4Krpctd|g}td|g}td|g}g}g}|jtj}t j |D]}|j |s"|jd|ddtdtdi6t|jd |}|j|t||z|zzx} O|jd| |t| |zd| |tt| |tt| |i|j|||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 r r'r rrr~rr!r"rpowsimpr) rrrr rxmxnx1rrs rn_laplace_trig_expsumr's+ dQC B dQC B S1#A B B 3   B b! xx{ IIsD#q"aQ7 8 $T\\%\%@!DAc"Q$r'lN+ +A 8 IIQqT#ae*_c1R5BquIr2ae9. / IIdO  r6Mrpc g}g}dfd}fd}fd}fd}d} t|dkDr|j} d} d} d} tt|D]}| t||tk(}| t||t k(}| t||tk(}| t||t k(}|r|r| tdk7r| tdk7r|} ||r|r| tdk7r|} |s|s| tdk7s|} | | | |j || || d || d || d ||j t t| d | | | g}|jd |D]}|j|n#| I|j || || d ||j | t|j| n| R|j || || d ||j t | t|j| n| R|j || || d ||j t | t|j| n0|j | | ||j | tt|dkDrt|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. c6|j}tt|D]q}||j}|dj t r||j t||<J|dt|dzzj t||<s|Sr) copyrangelenrr~r"r r2r)coeffsncr!ris rn_simpcz"_laplace_trig_ltex.._simpcs [[]s2w 7AA##%B!uyy}1 c*1A2a511#61  7  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} tjtj d|dzzd|dzzz tj |dzd|dzz|dzzz|dzzg} t t| tt| dddD cgc] \} } | || zzc} } } t t| tt| dddD cgc] \} } | || zzc} } }| |z Scc} } wcc} } w)Nrr!rrrr) r!r"rrrrrzipr+r,)t1k1k2k3r{rk0a_ra_ir.dcrrrrr0s rn _quadpolez%_laplace_trig_ltex.._quadpolesfS'2c7BrFBrF:2sC GbL2  rBw|b !AaCGBJ .1S ;1rcBhmb()1Q3s72:!C *+,#q& Qhrk*1rcBhmb()1ac#gbj1Q3s72:-!C :QqSWRZGHI1S!V8B;36",-.   EE1661S!V8aQh. FFCFQsAvXc1f_,sAv57 !$VBZs2w"1E!F GAa1f G I !$Rs2w")=!> ?Aa1f ? As H ?s "H %H c |d|d|t|tf\}}}}||z| |z||zz dtz|z|zzg}tjd|z|dz|dzzg}t t  |tt|dddD cgc] \} } | || zzc} } } t t |tt|dddD cgc] \} } | || zzc} } } | | z Scc} } wcc} } wNrr!rrr r!r"rrrrr2r+r,)r3r4r{rr7r8r9r.r:rrrrr0s rn_ccpolez#_laplace_trig_ltex.._ccpolesS'2c7BrFBrF:2sC2gr"uqt|ac#gbj0 1eeRVS!Vc1f_ - !$VBZs2w"1E!F GAa1f G I !$Rs2w")=!> ?Aa1f ? As H ?s C2 C8 c  |d|d|t|tf\}}}}||z||z||zz dtz|z|zz g}tjdtz|z|dz |dzz g}t t  |tt|dddD cgc] \} } | || zzc} } } t t |tt|dddD cgc] \} } | || zzc} } } | | z Scc} } wcc} } wr=r>)r3r5r{rr7r8r9r.r:rrrrr0s rn_rspolez#_laplace_trig_ltex.._rspoles S'2c7BrFBrF:2sC2gqtad{QqSWRZ/ 0eeRT#XQwa/ 0 !$VBZs2w"1E!F GAa1f G I !$Rs2w")=!> ?Aa1f ? As H ?s C9 C? c |d|d}}||z|||z zg}tjtj|dz g}tt  |t t |dddDcgc] \}}|||zzc}}} tt |t t |dddDcgc] \}}|||zzc}}} | | z Scc}}wcc}}w)Nrr!rr)rrrrr2r+r,) r3r6r{rr7r.r:rrrrr0s rn_sypolez#_laplace_trig_ltex.._sypoles3C22gq"r'{ #eeQVVadU # !$VBZs2w"1E!F GAa1f G I !$Rs2w")=!> ?Aa1f ? As H ?s 0C 3C c.|d|d}}|}||z }||z S)Nrr!rq)r3r{rr7rrs rn _simplepolez'_laplace_trig_ltex.._simplepoles*3C2  Es rprNr!rT)reverse) r,popr+r!r"rr$rrr-)r$rr{resultsplanesr;r?rArCrEr3 i_imagsym i_realsym i_pointsymireal_eqrealsymimag_eqimagsymindices_to_popr0s @rn_laplace_trig_ltexrSsG F. b'A+ VVX   s2w Af1b )GfAr *Gf1b )GfAr *G7r"v{r"v{ WB1 WB1  (%)*?* NN"Y-,bmC.@Z.-q2 3 MM#bCk* +(J?N     -# q    " NN72r)}S'91= > MM"R& ! FF9   " NN72r)}S'91= > MM#bf+ & FF9   # NN72r*~c':A> ? MM#bf+ & FF:  NN;r1- . MM"R& !q b'A+t =#v, &&rpc dtdd}|jtttt syt |j||\}}t||\}}t|dkDry|j|s&t|||\}} ||z| tjfSg} g} t|||d\} } }|D]O}| j|d| j|||d z z| j| t|d zQt!| j||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)rrr'r,rSrrrrr!rr-)rt_r{rrrr$r%rrrIrHGG_planeG_condr&s rn_laplace_rule_trigr[[s# cA 66#sD$ ' rwwr1~ .DAq !!Q 'FB 2w{ 558!"a+1sAqvv~/1a%H7F /B NN2c7166!Qr#wY#77 8 MM'"RW+- . / =  a $c6lF ::rpctd|g}td|g}td}|j|t|||fz}|r||jr||j Dcgc]}|j |}}t|dk(rtdg} t||D]^} | dk(r||j|d} n!t|||| fj|d} | j|||| z dz z| z`t||||d \} } }|||||z| zt| z z| |fSy cc}w) 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)``. rrrrraz" rule: time derivative (4.1.8)rFrN)r rrr is_integerrjr~sumrur+rrrr)rrr{rrrrrrrr!rrrrs rn_laplace_rule_diffr_|sV S1#A S1#ASA ''!Jq1a&)) *C s1v  "1v{{ +!QUU1X + + q6Q; 7 8A3q6] ,6A Aq)A"3q6Aq62771=ASVAXaZ*+  , +3q61a%HIAr2FAs1vIaK#q'12R< <  ,s*Ec |jrdg}dg}tj|D]6}|j|r|j |&|j |8t |dkDr4t |}t||j}t |}|dkDrt |} t| ||d\} } } | g} d} t| d| }| jtr)+<$$&9;J 1a A -H rpct\}}}d}d}|D]\}} } } } || k7r| |j||i}| }|j|} | s= | j| }|t j k(sc| j| j||i| j| t j fcSy#t$rYwxYw)zj This function applies all simple rules and returns the result if one of them gives a result. N)rrrxreplacerrr)rrr{ simple_rulesrWrprep_oldprep_ft_doms_domcheckplaneprepmars rn_laplace_apply_simple_rulesrs 01L"bH F,84(ueUD t !&&!R/*FH \\%   NN2& AFF{r*//Q8r*AFF444   sB>> C  C cR|js;tdd}t|j||i|j||iSt |}g}|j D]\}}t |tr]||j vrOt |ttfr|cS|jt|j|jz |ztt |trst|j dk(r[|j D]K}|j |k(r5|jt|j|jz |zG|ccSt |t"rt|j dk(r|j \}}|j |k(rz|j |k(rkd|j$vr||}}|jt|j|jz t|j|jz 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. rTrUr>)is_realr_piecewise_to_heavisider~r1rjrrrrrr;gtsrrLr,lhsrMrr) rrrrrrc2r rs rnrrs 99 #D !&qzz1a&'91=FF1vNNAA AFF D dJ 'AN$R) 488dhh#67:; b !c$))n&9ii 66Q;HHYrvv7:;H   c "s499~':YYFBvv{rvv{"))#Brvv/rvv/01345HA B 7Nrpc jtd}td}td}td}t|tr*|jts|jt s|S|j D]\}}|jt ||||x}||||k(r |||cS|jt |||||x} i||||k(su|||cS|j} |jD cgc]} t| |} } | | Scc} w)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 rrr~rdInverseLaplaceTransformitemsrrmrjlaplace_correspondence) rfdictrr{rrrYrrmr#rjs rnrrFs(H S A S A S A S A1d#EE*+56 1gg.qtQ:;;HaDAaDLQqT7Ngg5adAq!DEEaDAaDLQqT7N  66D:;&& A3 "3 . AD A ; BsD0c z|jD]\}}tt|D]}|dk(r|j|d|d}$|dk(r1|jt t ||||d|d}Z|jt 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)rr+r,rrr )rrrricr!s rnlaplace_initial_condsrsDK2s2w KAAvIIadBqE*aIId:adA#61=r!uEIId:adQF#;QBBqEJ  KK Hrpctj|}g}g}g}g}|D]} | jd\} } | jtr\tj| j t } | D].} | jd\}}|j| |z|f0| jtk(rn| jtsTtjt| } | D].} | jd\}}|j| |z|f0 |j| | f!|D]l\} } | jtrt| |tjdf}n| jt r5| jts| jt d} t!| |x} t#| |x} t%| |x}nwt'fd| j)t*Drt| |tjdf}n1t-| ||x} nt| |tjdf}|\}}}|j| |z|j||j|ot|}|r|j/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_AddTrac3@K|]}|jywrr~)rundefrWs rnrz%_laplace_transform..sGuUYYr]Grdoit)rras_independentr~rCr r;rrmr0r:rrdrrrrr{rvanyrr rbrr-rM)rrWrrterms_tterms_stermsrI conditionsffr!ft_terms_termr4f1rri_pi_ci_rlr conditions ` rnrrsmmBGG E FJ "!!"U!32 66% &]]2::i#89F )--b-?B adBZ( )WW !"&&B*@]]#:2r#BCF )--b-?B adBZ( ) LL!R ! "2 66% &!"b"-q/A/A4HAvvim$RVVJrN-CWWYr]A.5b"bAAQ 3BB??Q )"b"55QBG0FGG&b"b113E3EtL5B33!;?@%b"b113E3EtLc3qu c#;>']Fe, LEZ I 5) ##rpc&eZdZdZdZdZdZdZy)rda 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 H|jdd}t||||}|S)NrFr)getrb)selfrrr{hintsrFLTs rn_compute_transformz#LaplaceTransform._compute_transforms'IIj%0 +Aq!i H rpcxt|t| |zz|tjtjfSr)rEr'rrr)rrrr{s rn _as_integralzLaplaceTransform._as_integrals,#qbd) a%<==rpc |jdd}|jdd}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_nocondsrFrWrrrs rnrzLaplaceTransform.doit s99Y-IIj%0 '$--*.*@*@*.*A*A*C D # #  $ $ ]] r2rI > Q4KHrpN)ri __module__ __qualname____doc___namerrrrqrprnrdrds  E >rprdc jdd}jdd}t|trt|drʉjdd }|rA|r?d}t dd|t t 5|jfd cd d d S|D cgc]} t| fi} } |r:t| \} } } t|g|j| }|t| t| fSt|g|j| St|jd| \}}}|s|||fS|S#1swY4xYwcc} w) 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 t|fiSr)laplace_transform)fijrr{rs rnrz#laplace_transform..s 1#q! Eu ErpNrr)rrrNhasattrrUrWrVrrr2typeshaper-rMrdr)rrr{ legacy_matrixrrrFradtrelements_transelementsavalsr f_laplacerrrs `` ` rnrr+s~NyyE*H *e,I!Z WQ %<IIi// ]7C % */+. !!89 G{{EG G G 012(+0Q$"$2N2.1>.B+%#DG7QWW7h7  #u+sJ/???tAw88881a(--e7@.BHB1 1ax ' G G2s:D9E9Ec  ddlm}m ddlm}t dd fd}|j |r|j|}|jrKt|jDcgc]}t||||c}} t| j||dfS |||t dtj fdd \} } | g|||} | y| j$r(| jd\} } | j't(rytj*} | j-t.|} | j$r| j| fSt d tj0ffd } | j-t2| } d } | j-t| } t| j|| fScc}w#t"$rd} Y wxYw)z6 The backend function for inverse Laplace transforms. r)meijerint_inversion_get_coeff_exp)inverse_mellin_transformrTrUcJt|dk7rt|S|djdj}|\}}|djd}|djd}t dt |z |zz |zt |zdt |z z |zzS)z3 Simplify a piecewise expression from hyperexpand. rrrra)r,r0rjargumentr;r$)rjr#coeffexponente1e2rrs rnpw_simpz7_inverse_laplace_transform_integration..pw_simps t9>d# #1gll1o&&(a0x !W\\!_ !W\\!_ aE lQ[0 1" 4 akAc%jL0 1" 4 5 6rpNF)needevalrucL|jt }|jr t||Sddlm}||dkD}|j k(r$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 CHHS!Wa  558S"% %@Aq) 77a<CGG AQUB' 'CGG Aq1uXr* *rpc*tt|Sr)r r')r#s rnsimp_expz8_inverse_laplace_transform_integration..simp_expsc#h''rp)sympy.integrals.meijerintrrsympy.integrals.transformsrris_rational_functionapartrurrjrcrFrr'rrrHrr~rErrr0rr;)rr{rWrrrrrXrrrrrrrs @@@rnrcrcsNC cA 6 a GGAJxx vv5Q1eXN 21477*1aaR4:L48%I4  y 1a ( 9 >>ffQiGAtuuX66D IIi )~~vva}d"" c A vv + )^,A( #x A QVVAr]H -t 33c " s&G+G GGcddlm}||\}}|j|rV|j|j }t |dk(r)|\}}}|||d|zz zdz||z z|d|zz dzz z}||z S)Nr)fractionrr)rrr rrbr,) rr{rrrcfrrrs rn_complete_the_square_in_denomr s/ a[FQq YYq\ $ $ & r7a<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`. It is used by `_inverse_laplace_apply_rules`. r{rrrrrz._inverse_laplace_build_rules is building rulescH |j|S#t$r|cYSwxYwr)factorrP)rr{s rn_fracz+_inverse_laplace_build_rules.._frac(s) 88A;  H s  !!c|Srrq)rs rnsamez*_inverse_laplace_build_rules..same.srprarr) rr rurrr'r@r3r2rA)r{rrrrrr _ILT_ruless rn_inverse_laplace_build_rulesrs c A c A S1#A S1#A S1#A ;<  1aq! qsqbkM1qs8C1I-eAh6 FFD!  AqDAI> C!Hqs3qs8|3a1f= q  AqD1q1u:eAh&a8 AqsQhJAqs+QT%(]; q J q! rpc8|dk(r&tdt|tjfSt \}}}d}|j ||i}|D]\}} } } } || | fk7r| || z} | | f} j |}|s3| }|tjur)|dDcgc]}|j|}}|d|}|tjk(st|| j|j ||iztjfcSycc}w)@ Helper function for the class InverseLaplaceTransform. raz rule: 1 o---o DiracDelta()r}rN) rur:rrrrrr~r;)rr{rrrrW_prepfsubsrrrrrg_Frrrrjs rn#_inverse_laplace_apply_simple_rulesrBs  Av01!}aff$$57JB E FFAr7OE*4 M&ueT3 T3K eCiB3KE XXe_ A01!51 255AaD$KAFF{ |ENN2$6$;$;RG$DDaffLL M  6sDctd|g}td|g}td}|j|t|||fz}|r?||jr0t dt |||||dd\}} | ||z|z| fSy) rrrrrz3 rule: t**n*f(t) o---o (-1)**n*diff(F(s), s, n)Fr dorationalN)r rr r]ru_inverse_laplace_transform) rr{rrrrrrrrs rn_inverse_laplace_diffr_s S1#A S1#A S A :a!Q(( )B beDE) qE1a5B1RU{1}a rpctd|g}td}|j|s|t|ztjfS|jt sy|j t ||z}|rY||jr,tdt|||ztjfSt||||tjfS|j t ||z|z}|rR||jr%tdt||||||z|dd St||||tjfSy) 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 r~r:rrr'rrrurr)rr{rrrrrs rn_inverse_laplace_time_shiftrps* S1#A S A 558A&& 55: ''#ac( C q6   ? 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@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<> *1aEHHHB NN2    d #Q($T']Fe, 3 # ##S% ;s  M5 Mcjtj|}g}g}|D]l} | jtr2| j  j j  } | j d\} } |r$| jrt| |||x} Ct| |x} 3t| ||x} "t| ||x} t| ||x} nxtfd| jtDrt!| ||t"j$f} n2t'| |||x} nt!| ||t"j$f} | \} }|j)| | z|j)|ot|}|r|j+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. Frrc3@K|]}|jywrr)rrrs rnrz-_inverse_laplace_transform..sB52Brr)rrr~r'rrSrrr3rr%r)r'rrr rrrrcrrrM)rrrWrrrrrrrr!rrrrrlrs ` rnrrbs MM" EGJ% 88C=99R"%..055b2#>D""2e"41t88</r2ux99:!RDD72r5II-aR??72r5II  BAGGL,AB B)BE:AFFCA;r2ux99AEF (BE:AFFCA cqu#K%N']Fe,Z I 9 rpc\eZdZdZdZedZedZdZe dZ dZ dZ d Z y ) 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 Z|tj}tj|||||fi|Sr)r_none_sentinelrG__new__)rrr{rroptss rnr:zInverseLaplaceTransform.__new__s0 =+::E ((aAuEEErpcL|jd}|tjurd}|S)Nr)rjrr9)rrs rnfundamental_planez)InverseLaplaceTransform.fundamental_planes( !  +:: :E rpc 4t||||jfi|Sr)rcr=)rrr{rrs rnrz*InverseLaplaceTransform._compute_transforms&5 q!T++6/46 6rpc<|jj}tt||z|z||tj tj zz |tj tj zzfdtjztj zz S)Nr) __class___crEr'r ImaginaryUnitrPi)rrr{rrs rnrz$InverseLaplaceTransform._as_integralsw NN   S1XaZ!Q)C%C"#aooajj&@"@"B C qttVAOO # % &rpc 8|jdd}|jdd}td|j|j|jf|j}|j}|j}|j }t |||||d}|r|dS|S)rrTrFz[ILT doit] (%s, %s, %s)rr)rrXrrrr=r) rrrrFrrWrrrs rnrzInverseLaplaceTransform.doits99Y-IIj%0 (4==+/+A+A+/+B+B+D E # #  $ $ ]]&& & B d D Q4KHrpN)rirrrrrr9rAr:propertyr=rrrrqrprnrrsI E6]N sBF  6&rprc jdd}jdd}t|tr#t|dr|j fdSt |j 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!ULeLrpr)rrrNrrrr) rr{rrrrrFrrs ```` rnrHrHsZyyD)H *e,I!Z WQ %<{{ LN N #1aE 2 7 7  8 +DAq!t rpc tdtg\ fdfdfd fdfd|S)zEFast inverse Laplace transform of rational function including RootSumza, b, nrc|js|S|jr|S|jr|S|jr|St |t r|St r)r~rurarxrrTNotImplementedError)e_ilt_add_ilt_mul_ilt_pow _ilt_rootsumr{s rn_iltz#_fast_inverse_laplace.._ilt# s`uuQxH XXA;  XXA;  XXA;  7 #? 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