U 9%e@spdZddlmZmZmZddlmZddlmZddl m Z m Z m Z m Z mZmZmZmZmZddlmZmZddlmZmZmZmZmZmZddlmZdd lm Z m!Z!m"Z"dd l#m$Z$m%Z%m&Z&m'Z'm(Z(m)Z)dd l*m+Z+m,Z,dd l-m.Z.m/Z/m0Z0m1Z1dd l2m3Z3m4Z4m5Z5ddl6m7Z7ddl8m9Z9m:Z:m;Z;ddlZ>m?Z?m@Z@ddlAmBZBmCZCddlDmEZEmFZFmGZGddlHmIZImJZJmKZKddlLmMZMmNZNddlOmPZPmQZQmRZRddlSmTZTmUZUmVZVmWZWmXZXddlYmZZZddl[m\Z\ddl]m^Z^ddl_m`Z`ddlambZbddlcmdZdddlemfZfddlgmhZhmiZimjZjddlkmlZlmmZmd d!Znd"d#Zodfd%d&Zpd'd(Zqed)d*Zrd+d,Zsd-d.Ztd/d0Zud1d2Zvd3d4Zwd5d6Zxd7d8Zydgd9d:Zzdhd;d<Z{did=d>Z|djd?d@Z}dAdBZ~dCdDZdkdEdFZGdGdHdHeQZdldIdJZdmdKdLZdMdNZedOdPZdQdRZdSdTZdUdVZdWdXZdYdZZd[d\Zdnd]d^ZGd_d`d`eQZdodbdcZdddeZdaS)pzLaplace Transforms)SpiI)Add)cacheit) AppliedUndef Derivativeexpandexpand_complex expand_mul expand_trigLambda WildFunctiondiff)Mulprod) _canonicalGeGtLt UnequalityEq)ordered)DummysymbolsWild)reimargAbs polar_liftperiodic_argument)explog)coshcothsinhasinh)MaxMinsqrt) Piecewise)cossinatan)besselibesseljbesselkbessely) DiracDelta Heaviside)erferfcEi)digammagamma lowergamma) integrateIntegral) _simplifyIntegralTransformIntegralTransformError)to_cnf conjuncts disjunctsOrAnd) MatrixBase) _lin_eq2dict)PolynomialError)roots)Poly)together)RootSum)sympy_deprecation_warningSymPyDeprecationWarningignore_warnings)debugdebugfcsfddfddfddfdd}d d }d d lm}||}||t}||tfd d}||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) cs&|kr dS|jr"|jkr"|jSdS)N)is_Powbaser")exsV/var/www/html/Darija-Ai-API/env/lib/python3.8/site-packages/sympy/integrals/laplace.pypowerHs z_simplifyconds..powercs|r|rdSt|tr,|jd}t|tr@|jd}|r\d|d|S|}|dkrpdSzT|dkrt|t|ktjkrWdS|dkrt|t|ktjkrWdSWntk rYnXdS)z_ Return True only if |ex1| > |ex2|, False only if |ex1| < |ex2|. Else return None. NrrQFT)has isinstancerargsrtrue TypeError)Zex1Zex2n)abiggerrYrVrWrXraOs$     "" z_simplifyconds..biggercsH|jst|tr |js(t|ts(||kS||}|dk r@| S||kS)z simplify x < y N) is_positiver[r)xyrrarWrXrepliees z_simplifyconds..repliecs ||}|dkrdSt||S)NTFT)r)rcrdbrfrWrXreplueos z_simplifyconds..repluecWs|dkrt|S|j|S)Nrh)boolreplace)rTr\rWrWrXreplusz_simplifyconds..replr) collect_abscs ||SNrW)rcrd)rgrWrX|z _simplifyconds..)sympy.simplify.radsimprnrrrr)exprrVr`rjrmrnrW)r`rarYrgrVrX_simplifyconds's!     rtcCst||tS)zs Expand an expression involving DiractDelta to get it as a linear combination of DiracDelta functions. )rFatomsr3rsrWrWrXexpand_dirac_deltasrwTc stdtd|ftd|f|tr2dSt|t tjtjf}td|f|t st | |tj tj fS|jstddS|jd\}}|t rtddSfd d fd d t|D}d d |D}|sdd |D}tt|}dd|jfddd|sszH_laplace_transform_integration..process_conds..cSs|dS)Nr)r as_real_imagrcrWrWrXrprqzG_laplace_transform_integration..process_conds..)z==z!=z/[LT _l_t_i ] convergence not in half-plane.N)(sympy.solvers.inequalitiesryrNegativeInfinityr]rAr@rrrrr!r InfinityrBZ is_RelationalrhsZ free_symbolsreversedr[rrZ reversedsignmatchrbrrr,allrlsubsZrel_oprZltsrOr)r(rDrC canonical)condsryr`auxpqZw1Zw2Zw3Zw4Zw5patternscZa_Zaux_dpatZd_Zsoln)rVtrrX process_condss       ,:        z5_laplace_transform_integration..process_condscsg|] }|qSrWrW)r~r)rrWrX sz2_laplace_transform_integration..cSs,g|]$}|dtjkr|dtjk r|qS)rQr)rfalserr~rcrWrWrXrs cSsg|]}|dtjkr|qSrQ)rrrrWrWrXrscSs|dkr dS|S)Nrhr)Z count_opsrvrWrWrXcntsz+_laplace_transform_integration..cntcs|d |dfSNrrQrWr)rrWrXrprqz0_laplace_transform_integration..)keyz&[LT _l_t_i ] no convergence found.cs |Sro)rrv)rVs_rWrXsbssz+_laplace_transform_integration..sbs)rrPrZr3r;r"rZerorr<r=rrr] is_PiecewiserOr\rBlistrsortrtr) frrsimplifyFcondrZconds2r`rrrW)rrrVrrrX_laplace_transform_integrationsB  "   @    rcsT|j}t|j}t|jdkr"|Sfdd|D}|jrH||S||SdS)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)`. rcsg|]}t|qSrW_laplace_deep_collect)r~rrrWrXr sz)_laplace_deep_collect..N)funcrr\lenis_Addcollect)rrrr\rWrrXrs rcXsrtdtd}tdgd}tdgd}tdgd}tdgd}tdgd}fd d }td |||tjtj|ft||t| ||t|t t |d k|d kt |d k|d ktj |ft||td t t |d k|d kt |d k|d ktj |ft ||t| |||t |d k|d ktj|ft ||d t| |||t |d k|d ktj|ft ||d |t |d k|d ktj|ft ||d t |d k|d ktj|fd |dtjtj|fd ||t| || t | |||tt||tktj|fd t||t|t|t|||tt||||tt||tktj|f||td dd|td  ddt||td dt|||tt||||tt||tktj|ft|tt|tt|t||tt||tt|tktj|fd |tdt|td dt||tt||tjtj|f|t|d ||d |dktj|f|||t|d |||t| ||||d |t |dktt||tktj|f||||t|d t| ||t |dktt|tktj|ft||t| ||tjt||ft||t| ||dtjt||f|t|t|d |||d t|dkt||ft| dttd|t|dd|t|td|t|d ktj|ft| dd d|dttd|tdd|t|td|t|d ktj|ft| dt||td dt||t|d ktj|ftt| td dtt|dd dt||tdt||t|d ktj|ft| ttt|tdt||t|d ktj|ft| ttt|tdt||t|d ktj|f|t| d|||d dt|d dt||t|d ktj|ftdt||dtt||td dt||tt||tt|tktj|ftdt|tt|td dt||tt||tt|tktj|ft|tttj|| ||d ktj|ftd |t|| |t | |tt|tktj|ft||t|t|||||t | |||t |d ktt|tktj|ftttt| td|ttjtjtj|f|tt|d || d t|d t|t|dktj|ft|dtttj||dtdd||d ktj|ft|||d|dtjtt||ftt|||d|dtt|d||d ktj|ft|t||tjtt||ft|dtd d|d|ddtjdtt||ft|dd|td|||td d|d|ddtjdtt||ftdt|tt||t|t| |tjtj|ftdt|ttt||tjtj|ft|||d|dtjtt||ft|d|dd|d|dd|d|tjdtt||fttdt|ttd|td d|d|t| |tjtj|ftdt|ttt|t| |tjtj|ft|t|d||||d||d|d||dtjtt|tt||ft|t|||d|d|d|d||d|d||dtjtt|tt||ft|t|||d|d|d|d||d|d||dtjtt|tt||ft|||d|dtjtt||ft |||d|dtjtt||ft|dd|d|dd|d|tjdtt||ft |d|dd|d|dd|d|tjdtt||ft|t||||dtjtt||f|t|t|d d||| d ||| d |dkt||f|t |t|d d||| d ||| d |dkt||ftdt|tt||t|t||tjtj|ft dt|d |tt||t|t||tt||tjtj|fttdt|ttd d|td d|d|t||tt|||td d|dtjtj|ftt dt|ttd d|td d|d|t||tjtj|ftdt|tttd d|td  dt||tt||tjtj|ft dt|tttd d|td  dt||tjtj|ftt|dtttd dd|td  dt||d tjtj|ft t|dtttd dd|td  dt||d tjtj|ft|t|dd|dt|d||dtt|tktj|ftt|t|t|||tjt!tjt| |ft|tt|t|t|||tjt!tjt||ftt|dd tt|| |t|d ktj|ftt|t||t|t|||tjt| |ft|tt|d |t||tjtj|ftt|dtt|| |t|d ktj|ft"||||t|d|d|t|d|d|t|dktt||f|t"||d|ttt|tj#|||d|d| tj#t t|tj# kt$||tt||f|t"||d|d ttt|tdd||||d|d| tddt t|dkt$||d tt||ft"d dt|t| ||tjtj|f|t"|dt|||d|| d t| |t t|dkt$||tj#tj|ft"d |td|t|||t|d|dt|d|dtt|tktt||ft%||||t|d|d|t|d|d|t|dktt||f|t%||d|ttt|tj#|||d|d| tj#t t|tj# kt$||tt||f|t%||d|d ttt|tdd||||d|d| tddt t|dkt$||d tt||f|t%|dt|||d|| d t||t t|dkt$||tj#tj|ft&d |dtt'||t|d|dtjtt||ftd |t|t|d|d|t|d|dtjt| |fgR}||fS)aY This is an internal helper function that returns the table of Laplace transform rules in terms of the time variable `t` and the frequency variable `s`. It is used by ``_laplace_apply_rules``. Each entry is a tuple containing: (time domain pattern, frequency-domain replacement, condition for the rule to be applied, convergence plane, preparation function) The preparation function is a function with one argument that is applied to the expression before matching. For most rules it should be ``_laplace_deep_collect``. rrVr`r|rir_tauomegacs t|SrorrrrWrXdco+rqz!_laplace_build_rules..dcoz&_laplace_build_rules is building rulesrrQr}g?)(rrrOrr]rr3r"rrCrDrr4r7rrr*r6r9r:rr1r#Z EulerGammar8r-rr%r.r5r,r&r$r(r0Halfrr/r2r')rVr`rir_rrrZlaplace_transform_rulesrWrrX_laplace_build_rulessr &$$"& 6HX@DD0&, F B 0 B 2 6 D @H$,B0: 6 4 066( 8P2LXX  08$ DD4L, R<FNN4$0* 0,& B @ T $B6 B @ T @2 >7rc Cstd|gd}tddd}||}|r||jd|}|||}|r||jr||dkrtdtd|||ftd td|||| |||||d 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``. r`rgrQ)nargsrz _laplace_apply_prog rules match:z f: %s _ %s, %s )z rule: time scaling (4.1.4)FrN) rrrr\rrbrOrP_laplace_transformr) rrrVr`rma1rma2reprcrrWrWrX_laplace_rule_timescales"     rc Cstd|gd}td}td}|t||}|r||||}|r||jrtdtd|||ftdt||||||||dd \}} } t|| ||| | fS|r||j rtdtd|||ftd t||||dd \}} } || | 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. r`rrdr _laplace_apply_prog_rules match: f: %s ( %s, %s )z rule: time shift (4.1.4)Frz9 rule: Heaviside factor, negative time shift (4.1.4)N) rrr4rbrOrPrrr" is_negative) rrrVr`rdrrrrerrrWrWrX_laplace_rule_heavisides*  rc Cstd|gd}td}td}|t||}|r|||||}|rtdtd|||ftdt||||||dd \}} } || t||| fSd S) 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)``. r`rrdzrrz% rule: multiply with exp (4.1.5)FrN)rrr"rrOrPrr) rrrVr`rdrrrrerrrWrWrX_laplace_rule_exp"s rc stdgd}tdgd}td}td|t|rts|||}|rtdtd||ftd||||}t|d krt|d krt || || ||||||}|t j t j fSd t j t j fS|rt|} tik rt| d hkrt|tfd d t| D}|t j t j fSd 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)``. r`rrirdrrrz$ rule: multiply with DiracDeltarrQcsNg|]F}t|dkrt|dkrt| |qS)r)rrr"rrrrVZsloperrrWrXrZs z'_laplace_rule_delta..N)rrr3rZrrOrPrrr"rrrr] is_polynomialrHsetvaluesrrrkeys) rrrVr`rirdrlocrerorWrrX_laplace_rule_delta:s6 > rcCs`tjg}tjg}t|D],}|ttttt r<| |q| |qt|}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_argsrZr-r,r&r$r"append)fnZtrigsothertermrrrWrWrX_laplace_trig_split`s  rc Cstd}td|gd}g}g}|t}t|D]}||s`|d|ddtdt diq6|j dd}| |t|}d k r || |} d k r| } t| d kr|d||t| d d| dtt| dt t | din ||n ||q6||q6||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. rrrkr`rr")combineNr}rQ)rrewriter"r rrrZrrrZpowsimpras_poly all_coeffsr) rrrrxmxnx1rrempZmcrWrWrX_laplace_trig_expsumrs8         rcsg}g}ddfdd}fdd}fdd}fd d }d d } t|d kr|} d} d} d} tt|D]}| t||tk}| t||t k}| t||tk}| t||t k}|r|r| td kr| td kr|} qv|r|r| td kr|} qv|rv|rv| td krv|} qv| dk r| dk r| dk r||| || d|| d|| d||tt| d| | | g}|jdd|D]}||qqH| dk r||| || d||| t|| qH| dk r>||| || d||t| t|| qH| dk r||| || d||t| t|| qH|| | ||| tqHt|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. cSsj|}tt|D]P}||}|dtrF||t||<q|dt|dt||<q|Sr) copyrangerrrZrrr,r)ZcoeffsncrrirWrWrX_simpcs  z"_laplace_trig_ltex.._simpcc s|d|d|t|tf\}}}}|||||||||dt||dt|||d| ||||dt||dt||d|d|d|d||d| ||||ddt||dt||dt||dt|||d|d|d|d|g} tjtjd|dd|dtj|dd|d|d|dg} tfddt| tt | dddD} tfd dt| tt | dddD} t d | | f| | S) Nr`rr}rrcsg|]\}}||qSrWrWr~rcrdrUrWrXrsz9_laplace_trig_ltex.._quadpole..rcsg|]\}}||qSrWrWrrUrWrXrsz quadpole: (%s) / (%s)) rrrrrrrziprrrP) t1k1k2k3rVr`k0a_ra_irdcr_rrrUrX _quadpoles<$2"F" ",(z%_laplace_trig_ltex.._quadpolec s|d|d|t|tf\}}}}||| |||dt||g}tjd||d|dg}tfddt|tt|dddD} tfddt|tt|dddD} t d | | f| | S) Nr`rr}rcsg|]\}}||qSrWrWrrUrWrXrsz7_laplace_trig_ltex.._ccpole..rcsg|]\}}||qSrWrWrrUrWrXrsz ccpole: (%s) / (%s) rrrrrrrrrrP) rrrVr`rrrrrr_rrrUrX_ccpoles$*,(z#_laplace_trig_ltex.._ccpolec s|d|d|t|tf\}}}}||||||dt||g}tjdt||d |dg}tfddt|tt|dddD} tfddt|tt|dddD} t d | | f| | S) Nr`rr}rcsg|]\}}||qSrWrWrrUrWrXrsz7_laplace_trig_ltex.._rspole..rcsg|]\}}||qSrWrWrrUrWrXrsz rspole: (%s) / (%s)r) rrrVr`rrrrrr_rrrUrX_rspoles$(",(z#_laplace_trig_ltex.._rspolec s|d|d}}|||||g}tjtj|d g}tfddt|tt|dddD}tfddt|tt|dddD}td||f||S) Nr`rr}csg|]\}}||qSrWrWrrUrWrXrsz7_laplace_trig_ltex.._sypole..rcsg|]\}}||qSrWrWrrUrWrXrsz sypole: (%s) / (%s))rrrrrrrrP) rrrVr`rrrr_rrrUrX_sypoles,(z#_laplace_trig_ltex.._sypolecSs4|d|d}}|}||}td||f||S)Nr`rz simplepole: (%s) / (%s))rP)rrVr`rr_rrWrWrX _simplepoles z'_laplace_trig_ltex.._simplepolerNrr`T)reverse) rpoprrrrrrrr()rrrVresultsplanesrrrrrrZ i_imagsymZ i_realsymZ i_pointsymiZreal_eqZrealsymZimag_eqZimagsymZindices_to_poprWrrX_laplace_trig_ltexst              rc Ks&tddd}|tttts dStd|||ft|||\}}td||ft ||\}} td|| ft | dkrt d dS||st |||\} } || | t jfSg} g} t|||\}}}|D]<}| |d ||||d | |t|d qt| ||t| |fS) z This rule covers trigonometric factors by splitting everything into a sum of exponential functions and collecting complex conjugate poles and real symmetric poles. rTrealNz _laplace_rule_trig: (%s, %s, %s)z f = %s g = %sz xm = %s xn = %srz# --> xn is not empty; giving up.rr`)rrZr-r,r&r$rPrrrrrOrrr]rrrrr()rt_rVdoithintsrrrrrrerrrGZG_planeZG_condrrWrWrX_laplace_rule_trig(s*   "rc sBtdgd}tdgd}td}||t||f}|r>||jr>fdd||jD} t| dkr>tdtd |||ftd g} t ||D]V} | d kr|| d } nt||| f d } | |||| d| qt |||d d \} }}|||||| t | ||fSdS)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)``. r`rr_rcsg|]}|qSrWrZ)r~rrrWrXr[sz&_laplace_rule_diff..rQ_laplace_apply_rules match: f, n: %s, %sz# rule: time derivative (4.1.8)rFrN)rrrr is_integerr\sumrOrPrrrrr)rrrVrrr`r_rrrrrrdrerrrWrrX_laplace_rule_diffNs& &rc s|jrdg}dg}t|D]$}||r8||q||qt|dkrt|}t||tdkrt dt d||ft dt|} t | ||dd\} } } | gd} zt d| } Wnt k rd} YnX| tr2tdD](}d|dt| ||dqn:| rl| tdD]}t d| qN| rtfd d tD}|| | fSd S) a This function looks for multiplications with polynoimials in `t` as they correspond to differentiation in the frequency domain. For example, if it gets ``(t*f(t), t, s)``, it will compute ``-Derivative(LaplaceTransform(f(t), t, s), s)``. rQr r z( rule: frequency derivative (4.1.6)Frrr}cs$g|]}|d|qSrrW)r~r_NZderiZpcrWrXrsz'_laplace_rule_sdiff..N)is_MulrrrrrrrIrrOrPrr ValueErrorrZLaplaceTransformrrr)rrrVrrZpfacZofacfacZpexZoexZr_Zp_Zc_Zd1rrerWrrX_laplace_rule_sdifflsD      (  rcKs|jr dSt|dd}|jr,t|||ddSt|}|jrJt|||ddSt|}|jrht|||ddS||krt|||ddStt|}|jrt|||ddSdS)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. NFdeepr)rr rr r )rrrVrrrerWrWrX_laplace_expands"   rcCs<tttttttg}|D] }||||}dk r|SqdS)zk This function applies all program rules and returns the result if one of them gives a result. N)rrrrrrr)rrrV prog_rulesp_ruleLrWrWrX_laplace_apply_prog_ruless rc Cst\}}}d}d}|D]\}} } } } || krD| |||i}| }||} | rz| | }Wntk rxYqYnX|tjkrtdtd|ftd|| ftd| f| | ||i| | tjfSqdS)zj This function applies all simple rules and returns the result if one of them gives a result. z"_laplace_apply_simple_rules match:z f: %sz rule: %s o---o %sz match: %sN) rrrxreplacer^rr]rOrP)rrrVZ simple_rulesrrZprep_oldZprep_ft_doms_domcheckplaneprepmarrWrWrX_laplace_apply_simple_ruless.       r%csDtd||ft|}g}g}g}|D]}|jdd\} } t| |} dk rVnt| |} dk rlnvt| |} dk rn`tfdd| t Drt | |t j df} n.t | ||d} dk rnt | |t j df} | \} } }|| | || ||q*t|}|r*|jdd }t|}t|}|||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. z[LT _l_t] (%s, %s, %s)FZas_AddNc3s|]}|VqdSror r~ZundefrrWrXrsz%_laplace_transform..Trr)rPrras_independentr%rranyrurrrrrrrr(rD)rrrrtermsZterms_sr conditionsffrftreri_Zpi_ci_resultr" conditionrWr(rXrsF     rc@s4eZdZdZdZddZddZddZd d Zd S) 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. LaplacecKs |dd}t||||d}|S)NrFr)getr)selfrrrVrr=LTrWrWrX_compute_transform"s z#LaplaceTransform._compute_transformcCs"t|t| ||tjtjfSro)r<r"rrr)r6rrrVrWrWrX _as_integral'szLaplaceTransform._as_integralcCsXg}g}|D]\}}||||q t|}t|}|tjkrPtddd||fS)Nr4zNo combined convergence.)rrDr(rrr?)r6extrarrr"rrWrWrX_collapse_extra*s    z LaplaceTransform._collapse_extracKsd|dd}|dd}td|j|j|jf|j}|j}|j}t||||d}|r\|dS|SdS) 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)rrN)r5rPfunctionfunction_variabletransform_variabler)r6r_nocondsr=rrrrerWrWrXr7s  zLaplaceTransform.doitN) __name__ __module__ __qualname____doc___namer8r9r;rrWrWrWrXrs   rc sdd}dd}t|trt|drdd }|r|rd}tdd|dtt$|fd d W5QRSQRXnbfd d |D} |rt| \} } } t ||j | f} | t | t | fSt ||j | fSt |jd|d }|s |S|dSdS)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) 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 r=Fr 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)Zdeprecated_since_versionZactive_deprecations_targetcst|fSrolaplace_transform)fijrrVrrWrXrprqz#laplace_transform..csg|]}t|fqSrWrH)r~rJrKrWrXrsz%laplace_transform..)r=rrN)r5r[rEhasattrrLrNrMrGrtypeshaper(rDrr)rrrVZ legacy_matrixrrAr=rZadtZelements_transelementsZavalsr-Z f_laplacer7rWrKrXrIWs6O   rIc sddlm}mddlm}tdddfdd}|rJ|}|jrt fd d |j D}t | |dfSz(||t  d tjfdd d \}} Wntk rd }YnX|d kr"||}|d krd S|jr|j d\}} |trd Sntj} |t|}|jr:| || fStdtjffdd } |t| }dd} |t | }t | || fS)z6 The backend function for inverse Laplace transforms. r)meijerint_inversion_get_coeff_exp)inverse_mellin_transformrTrcst|dkrt|S|djdj}|\}}|djd}|djd}tdt|||t|dt||S)z3 Simplify a piecewise expression from hyperexpand. rr}rrQ)rr+r\argumentr4r)r\rcoeffexponente1e2)rQrrWrXpw_simps z7_inverse_laplace_transform_integration..pw_simpcsg|]}t|qSrW)&_inverse_laplace_transform_integration)r~X)r"rVrrrWrXrsz:_inverse_laplace_transform_integration..NF)Zneedevalr=ucs|t }|r&t||Sddlm}||dk}|jkrbt|j}t||St|j}t| |SdS)Nrrx) rr"rZr4rryrr#Zgts)rZH0r`ryrelr)rr[rWrXsimp_heavisides      z>_inverse_laplace_transform_integration..simp_heavisidecSs tt|Sro)r r")rrWrWrXsimp_expsz8_inverse_laplace_transform_integration..simp_exp)Zsympy.integrals.meijerintrPrQsympy.integrals.transformsrRris_rational_functionapartrrr\r=rr"rrr?rrZr<r]rlr+rr4) rrVrr"rrPrRrXrrr]r^rW)rQr"rVrrr[rXrYsL           rYc Cs~ddlm}||\}}||rv||}t|dkrv|\}}}|||d|d|||d|d}||S)Nr)fractionrr})rrrbrrrr) rrVrbr_rcfr`rirrWrWrX_complete_the_square_in_denom"s     0rdc CsVtd}td}td|gd}td|gd}td|gd}tddd }d d }|||tj|d f|||| ||d t| |t||d k|d fd |d|ddt||||t||d|dtj|d fd ||||d t|tj|d fd ||||t |||||t|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`. rVrr`rrirz._inverse_laplace_build_rules is building rulescSs*z ||WStk r$|YSXdSro)factorrG)rrVrWrWrX_frac<s z+_inverse_laplace_build_rules.._fraccSs|SrorWrrWrWrXsameBrqz*_inverse_laplace_build_rules..samerQrr}r) rrrOrr]r"r9r-r,r:)rVrr`rirrfrg _ILT_rulesrWrWrX_inverse_laplace_build_rules-s.<@&. ric Cs|dkr4tdtddtd|ft|tjfSt\}}}d}|||i}|D]\}} } } } || | fkr| || } | | f}| |}|rVz| |}Wnt k rYqVYnX|tjkrVtdtd|ftd|| ftd|ft || |||itjfSqVd S) @ Helper function for the class InverseLaplaceTransform. rQz*_inverse_laplace_apply_simple_rules match: f: %srz" rule: 1 o---o DiracDelta(%s)rz rule: %s o---o %s ma: %sN) rOrPr3rr]rirrrr^r4)rrVrrhrrZ_prepZfsubsr rr!r#rZ_Fr$rrWrWrX#_inverse_laplace_apply_simple_rulesTs2          (rmcCs6td|gd}td}||s2|t|tjfS|t||}|r||jrtdt d|ftdt d|ft|||tjfStdt ||||tjfS|t|||}|r2||jrtdt d|ftd t d|ft |||||||Stdt ||||tjfSd S) rjr`rrz"_inverse_laplace_time_shift match:rkz+ rule: exp(-a*s) o---o DiracDelta(t-a)rlz6_inverse_laplace_time_shift match: negative time shiftz6 rule: exp(-a*s)*F(s) o---o Heaviside(t-a)*f(t-a)N) rrZr3rr]rr"rrOrPInverseLaplaceTransform_inverse_laplace_transform)rrVrr"r`rrrWrWrX_inverse_laplace_time_shiftvs2       rpc Cstd|gd}td}||||}|r||jr||jrtdtd|ftdtd|ft|||||\}}|t|d}| t rt |||||fSt|t |||||fSd S) rjr_rrz!_inverse_laplace_time_diff match:rkz, rule: s**n*F(s) o---o diff(f(t), t, n)rlrQN) rrr rbrOrProrlr4rZrnr) rrVrr"r_rrrerrWrWrX_inverse_laplace_time_diffs   rqcCs4ttg}|D]"}|||||}dk r |Sq dS)rjN)rprq)rrVrr"rrrerWrWrX!_inverse_laplace_apply_prog_ruless rrcCs|jr dSt|dd}|jr*t||||St|}|jrFt||||St|}|jrbt||||S||rz||}|jrt||||SdS)rjNFr)rr ror r`rar)rrVrr"rerWrWrX_inverse_laplace_expands   rsc std|||ftd}||}t|}g}tjg} |D]T} | \} } | | } | dfdd| D} fdd| | D}t | dkr|dt |}| |q.csg|] }|qSrWrWrrurWrXrsrQr}rFT)r dorationalr)z[ILT _i_l_r] returns %s)rPrrarrrr]Zas_numer_denomrrrr3rr"r4rertuplerrrHrr*rr$r&r,r-rorD)rrVrr"rrtrr,terms_tr-rr_rrrrer`rilrZhypb2bsr/rr2rWrurX_inverse_laplace_rationalst           ( &  H     r|csZt|}g}g}td||f|D]} | jdd\} } |rf| rft| |||} dk rfnt| |} dk r|nt| ||} dk rnzt| ||} dk rnbt fdd| t Drt | ||t jf} n0t| |||d} dk rnt | ||t jf} | \} }|| | ||q&t|}|rJ|jdd}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. z[ILT _i_l_t] (%s, %s, %s)Fr&Nc3s|]}|VqdSror r'rrWrXr'sz-_inverse_laplace_transform..rr))rrrPr*r`r|rmrsrrr+rurrnrr]rYrrrD)rrrr"rrvr,rxr-rrrrer0r1r2r3rWr}rXro sf    roc@sPeZdZdZdZedZedZddZe ddZ d d Z d d Z d dZ dS)rnz 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 LaplaceNonercKs$|dkrtj}tj|||||f|Sro)rn_none_sentinelr>__new__)r{rrVrcr"optsrWrWrXrLszInverseLaplaceTransform.__new__cCs|jd}|tjkrd}|S)Nr)r\rnr)r6r"rWrWrXfundamental_planeQs  z)InverseLaplaceTransform.fundamental_planecKst||||jf|Sro)rYr)r6rrVrrrWrWrXr8Xsz*InverseLaplaceTransform._compute_transformcCsL|jj}tt|||||tjtj|tjtjfdtjtjS)Nr}) __class___cr<r"rZ ImaginaryUnitrPi)r6rrVrrrWrWrXr9\s z$InverseLaplaceTransform._as_integralc Ksl|dd}|dd}td|j|j|jf|j}|j}|j}|j}t|||||d}|rd|dS|SdS) r<r=TrFz[ILT doit] (%s, %s, %s)rrN)r5rPr>r?r@rro) r6rrAr=rrrr"rerWrWrXrcs  zInverseLaplaceTransform.doitN)rBrCrDrErFrrrrpropertyrr8r9rrWrWrWrXrn>s  rnNc sBt|tr,t|dr,|fddSt|jfS)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 rGcst|fSro)inverse_laplace_transform)ZFijrr"rVrrWrXrprqz+inverse_laplace_transform..)r[rErLrGrnr)rrVrr"rrWrrXrs -rcsltdtgd\fddfddfdd fd d fd d |S) zEFast inverse Laplace transform of rational function including RootSumza, b, nrzcsR|s|S|jr|S|jr*|S|jr8|St|trJ|StdSro)rZrrrRr[rKNotImplementedErrore)_ilt_add_ilt_mul_ilt_pow _ilt_rootsumrVrWrX_ilts  z#_fast_inverse_laplace.._iltcs|jt|jSro)rmapr\rrrWrXrsz'_fast_inverse_laplace.._ilt_addcs$|\}}|jrt||Sro)r*rr)rrTrs)rrVrWrXrsz'_fast_inverse_laplace.._ilt_mulcs|}|dk r|||}}}|jr||dkr|| dt|| || t| S|dkrt|| |StdSr)rZ is_Integerr"r9r)rrnmamZbm)r`rir_rVrrWrXrs4z'_fast_inverse_laplace.._ilt_powcs,|jj}|jj\}t|jt|t|Sro)Zfunrs variablesrKZpolyr rJ)rrsvariablerrWrXrs z+_fast_inverse_laplace.._ilt_rootsum)rr)rrVrrW) rrrrrr`rir_rVrrX_fast_inverse_laplaces  r)T)T)T)T)T)T)T)T)TT)N)rEZ sympy.corerrrZsympy.core.addrZsympy.core.cacherZsympy.core.functionrrr r r r r rrZsympy.core.mulrrZsympy.core.relationalrrrrrrZsympy.core.sortingrZsympy.core.symbolrrrZ$sympy.functions.elementary.complexesrrrrr r!Z&sympy.functions.elementary.exponentialr"r#Z%sympy.functions.elementary.hyperbolicr$r%r&r'Z(sympy.functions.elementary.miscellaneousr(r)r*Z$sympy.functions.elementary.piecewiser+Z(sympy.functions.elementary.trigonometricr,r-r.Zsympy.functions.special.besselr/r0r1r2Z'sympy.functions.special.delta_functionsr3r4Z'sympy.functions.special.error_functionsr5r6r7Z'sympy.functions.special.gamma_functionsr8r9r:Zsympy.integralsr;r<r_r=r>r?Zsympy.logic.boolalgr@rArBrCrDZsympy.matrices.matricesrEZsympy.polys.matrices.linsolverFZsympy.polys.polyerrorsrGZsympy.polys.polyrootsrHZsympy.polys.polytoolsrIZsympy.polys.rationaltoolsrJZsympy.polys.rootoftoolsrKZsympy.utilities.exceptionsrLrMrNZsympy.utilities.miscrOrPrtrwrrrrrrrrrrrrrrrr%rrrIrYrdrirmrprqrrrsr|rornrrrWrWrWrXs  ,           Z v U$&% &  .  +E x S  &"# : 3F 3