o o hR@sddZddlZddlZddlmZmZmZddlmZddl m Z ddl m Z ddl mZmZmZmZmZmZmZmZmZmZddlmZmZdd lmZmZmZmZm Z m!Z!m"Z"m#Z#dd l$m%Z%dd l&m'Z'm(Z(m)Z)dd l*m+Z+m,Z,m-Z-m.Z.m/Z/m0Z0dd l1m2Z2m3Z3ddl4m5Z5m6Z6m7Z7m8Z8ddl9m:Z:m;Z;mZ>m?Z?ddl@mAZAmBZBmCZCmDZDddlEmFZFmGZGmHZHmIZIddlJmKZKmLZLddlMmNZNmOZOmPZPddlQmRZRmSZSmTZTmUZUddlVmWZWddlXmYZYmZZZddl[m\Z\m]Z]m^Z^ddl_m`Z`maZambZbmcZcmdZdddlemfZfddlgmhZhddlimjZjddlkmlZlddlmmnZnddlompZpdd lqmrZrdd!lsmtZtmuZumvZvdd"lwmxZxdayd#d$Zzd%d&Z{d'd(Z|ezd)d*Z}ezd+d,Z~ezd-d.Ze d/d0Zezd1d2Zezd3d4Zezd5d6Zezd7d8Zezd9d:Zezd;d<Zezd=d>Zezd?d@ZezdAdBZezdCdDZezdEdFZezdGdHZezdIdJZezdKdLZdMdNZdOdPZezdQdRZGdSdTdTe]ZdzdVdWZezdXdYZezdZd[Ze d\d]Zezd^d_Zezd`daZezdbdcZezdddeZezdfdgZezdhdiZezdjdkZezdldmZezdndoZezdpdqZezdrdsZGdtdudue]Zd{dvdwZdxdyZdS)|zLaplace 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)debugfcsfdd}|S)Ncsddlm}|s|i|Stdkrtdtjdtddtj|ftjdtd7ajdks7jd krPd t_td dttjd|i|}d t_n|i|}td8atd dt|ftjdtdkrstdtjd|S)Nr SYMPY_DEBUGzO ------------------------------------------------------------------------------filez -LT- %s%s%s _laplace_transform_integration&_inverse_laplace_transform_integrationFz**** %sIntegrating ...Tz-LT- %s---> %szO------------------------------------------------------------------------------ )sympyrY _LT_levelprintsysstderr__name__)argskwargsrYresultfunck/var/www/html/construction_image-detection-poc/venv/lib/python3.10/site-packages/sympy/integrals/laplace.pywrap1s*   zDEBUG_WRAP..wraprk)rjrmrkrirl DEBUG_WRAP0s rncCs2ddlm}|rtddt|ftjddSdS)NrrXz -LT- %s%sr\rZ)r`rYrbrarcrd)textrYrkrkrl_debugNs rpcsfddfddfddfdd}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&|krdS|jr|jkr|jSdS)Nr])is_Powbaser&)exsrkrlpowerws z_simplifyconds..powercs|r |r dSt|tr|jd}t|tr |jd}|r.d|d|S|}|dur8dSz.|dkrMt|t|ktjkrMWdS|dkrat|t|ktjkrdWdSWdSWdStypYdSw)z_ Return True only if |ex1| > |ex2|, False only if |ex1| < |ex2|. Else return None. Nrr]FT)has isinstancer#rfrtrue TypeError)ex1ex2n)abiggerrvrurkrlr~s(     "" z_simplifyconds..biggercsH|jst|tr|jst|ts||kS||}|dur | S||kS)z simplify x < y N) is_positiverxr#)xyrrrkrlreplies z_simplifyconds..repliecs ||}|dvr dSt||S)NTFT)r)rrbrrkrlreplues  z_simplifyconds..repluecWs|dvrt|S|j|S)Nr)boolreplace)rsrfrkrkrlrepls z_simplifyconds..replr) collect_abscs ||SNrk)rr)rrkrl z _simplifyconds..)sympy.simplify.radsimprrrrr)exprrur~rrrrk)r~rrvrrurl_simplifycondsVs !     rcCst||tS)zs Expand an expression involving DiractDelta to get it as a linear combination of DiracDelta functions. )rNatomsr9rrkrkrlexpand_dirac_deltasrc sRtd|tr dSt|t tjtjf}|ts/t | |tj tj fS|j s4dS|jd\}}|trBdSfddfddt|D}dd|D}|sdd d|D}tt|}d d |jfd d d|s|dS|d\}} fdd} |rt||}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. ruNrc sxddlm}tj}tj}tt|}tdtgd\}}}}}} } |t t |||k|t t |||kt t |||||kt t |||||kt t t |||||kt t t |||||kf} |D]3} tj } g}t| D]}|jr|jjvr|j}|jrt|ttfr|j}| D] }||rnqrψ|jrψ||tdkrt| dk}||t|t t | |t || dks |t|t t || ||t || dks0||tt t t || ||t || dkrLtfdd|||| | fDrLt|k}|tdd t}|jrp|jd vsp| sp| sv||g7}q||}|jr|jd vr||g7}q|j!krd St"|j!| } q| tj urt#| |}q{t$|t%|}q{||jr|j&fS|fS) z7 Turn ``conds`` into a strip and auxiliary conditions. r_solve_inequalityzp q w1 w2 w3 w4 w5clsexcludec3s|]}|jVqdSr)r).0wildmrkrl zH_laplace_transform_integration..process_conds..cSs|dSNr)r as_real_imagrrkrkrlrszG_laplace_transform_integration..process_conds..)z==z!=N)'sympy.solvers.inequalitiesrrNegativeInfinityryrIrHrrr#r"r%r$InfinityrJ is_Relationalrhs free_symbolsreversedrxrr reversedsignmatchrrr r1allrsubsrel_oprwltsr-r,rLrK canonical)condsrr~auxpqw1w2w3w4w5patternsca_aux_dpatd_soln)rutrrl process_condss      &:           z5_laplace_transform_integration..process_condscsg|]}|qSrkrk)rr)rrkrl z2_laplace_transform_integration..cSs,g|]}|dtjkr|dtjur|qS)r]r)rfalserrrrkrkrlrs cSsg|] }|dtjkr|qSr])rrrrkrkrlrscSs|dvrdS|S)Nrr) count_opsrrkrkrlcntsz+_laplace_transform_integration..cntcs|d |dfSNrr]rkr)rrkrlrz0_laplace_transform_integration..keycs |Srrr)rus_rkrlsbs" z+_laplace_transform_integration..sbs)rrwr9rCr&rZerorrDrErrry is_PiecewiserfrJlistrsortrr) frrsimplifyFcondrconds2r~rrrk)rrrurrrlr^s6 "  ?    $r^csJt|ts|S|}dur|S|j}fdd|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)`. Ncg|]}t|qSrk_laplace_deep_collectrr"rrkrlr7rz)_laplace_deep_collect..)rxras_polyas_exprrjrf)rrrrjrfrkrrlr*s rc s~tdtd}tdgd}tdgd}tdgd}tdgd}tdgd}fd d }td g|||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| t || t||tjtj|ft| t||t| |t|d ktj|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||ft dt|tt||t|t||tjtj|ft!dt|d |tt||t|t||tt||tjtj|ftt dt|ttd d|td d|d|t||tt|||td d|dtjtj|ftt!dt|ttd d|td d|d|t||tjtj|ft dt|tttd d|td  dt||tt||tjtj|ft!dt|tttd d|td  dt||tjtj|ft t|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| |f}||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``. rrur~rrr}tauomegacs t|SrrrrrkrldcoTrz!_laplace_build_rules..dcoz&_laplace_build_rules is building rulesrr]rg?))rrrprryrr9r&r#rKrLrr:r=r"rr.r<r?rAr r7r@r' EulerGammar>r2r!r)r3r;r1r*r(r,r6Halfrr5r8r+)rur~rr}rrrlaplace_transform_rulesrkrrl_laplace_build_rules;s@  &$$" &    6HX@D  D "0"$ && ,(F*,B0/1B2466D8:@H=$? "A$C ,EBG0I :K6M O4Q S0UW66Z (\ ^8`Pb 2d LfXhXjln0p8r$tDvDx4z L| ,  R < F N N 4$0*0, &B@&"T&$% B'")6B,.@&1T&@4"26>8 ;rc Cstd|gd}tddd}||}|rU||jd|}|||}|rU||jrU||dkrUtdtd|||||||||dd \}} } || | 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~rgr])nargsrz rule: time scaling (4.1.4)FrN) rrrrfcollectrrp_laplace_transformrj) rrrur~rma1r"ma2rprcrrkrkrl_laplace_rule_timescales  & rc CsZtd|gd}td}td}|t||}r||||}rl||jrQtdt||||||||dd\}} } t|| ||| | fS||jrltdt||||dd\}} } || | fS||||}r||jrtd td t|||||||dd\}} } || | fS||jrtd d d t j 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~rrrz rule: time shift (4.1.4)Frz8 rule: Heaviside factor; negative time shift (4.1.4)z rule: Heaviside window openr]z" rule: Heaviside window closedrN) rrr:rrprrr& is_negativerry) rrrur~rrrrrrrrkrkrl_laplace_rule_heaviside*s8         rc Cstd|gd}td}td}|t||}|rH|||||}|rHtdt||||||dd\}} } || t||| 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)``. r~rrzz$ rule: multiply with exp (4.1.5)FrN)rrr&rrprr ) rrrur~rrrrrrrrkrkrl_laplace_rule_expTs  rc stdgdtdgdtd}td|t|rԈtsԈ|rtd}t|dkrt|dkrt }|t t rn| t  }fdd |D\}}|dkr||tjtjfSd SdtjtjfS|rt|}|ikrt|d hkrt|tfd d t|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)``. r~rrrrz# rule: multiply with DiracDeltarcs$g|]}|qSrkrr)r~rrrrkrlr$z'_laplace_rule_delta..Nr]csNg|]#}t|dkrt|dkrt| |qS)r)r!r r&rr)rrusloperrrkrlrs()rrr9rwrrpr r!r&r2r1rewriter4ratsimpas_numer_denomrrry is_polynomialrPsetvaluesrrrkeys) rrrurlocfnr}rrorrk)r~rrrrurrrrl_laplace_rule_deltaks< "   rcCs`tjg}tjg}t|D]}|ttttt r| |q | |q 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_argsrwr2r1r*r(r&append)r trigsothertermrrrkrkrl_laplace_trig_splits  rc Cstd|gd}td|gd}td|gd}g}g}|t}t|D]T}||s:|d|ddtdt diq%t |j dd |}| |t|||} d urt|d| |t| |d| |tt| |t t | |iq%||q%||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. c1rc0rkr~rr&)combineN) rrr&r rrrwrr r!rpowsimpr) rrrrrxmxnx1rrrkrkrl_laplace_trig_expsums$  " rcsg}g}ddfdd}fdd}fdd}fd d }d d } t|d krM|} d} d} d} tt|D]X}| t||tk}| t||t k}| t||tk}| t||t k}|rz|rz| td krz| td krz|} q;|r|r| td kr|} q;|r|r| td kr|} q;| dur| dur| dur||| || d|| d|| d||tt| d| | | g}|jdd|D]}||qno| dur||| || d||| t|| nQ| dur||| || d||t| t|| n0| dur8||| || d||t| t|| n|| | ||| tt|d ks+t|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](}||}|dtr#||t||<q |dt|dt||<q |Sr) copyrangelenrrwr!rr1r)coeffsncrrirkrkrl_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} | | S) Nr~rrrrcg|] \}}||qSrkrkrrrrtrkrlrz9_laplace_trig_ltex.._quadpole..rcr&rkrkr'rtrkrlrr() r r!rrrrrzipr r!) t1k1k2k3rur~k0a_ra_ir#dcr}rr%rtrl _quadpoles4$2"F" &,(z%_laplace_trig_ltex.._quadpolec s|d|d|t|tf\}}}}||| |||dt||g}tjd||d|dg}tfddt|tt|dddD} tfddt|tt|dddD} | | S) Nr~rrrcr&rkrkr'rtrkrlrr(z7_laplace_trig_ltex.._ccpole..rcr&rkrkr'rtrkrlrr( r r!rrrrr)r r!) r*r+rur~r.r/r0r#r1r}rr2rtrl_ccpoles$*,(z#_laplace_trig_ltex.._ccpolec s|d|d|t|tf\}}}}||||||dt||g}tjdt||d |dg}tfddt|tt|dddD} tfddt|tt|dddD} | | S) Nr~rrrcr&rkrkr'rtrkrlrr(z7_laplace_trig_ltex.._rspole..rcr&rkrkr'rtrkrlr r(r4) r*r,rur~r.r/r0r#r1r}rr2rtrl_rspoles$(",(z#_laplace_trig_ltex.._rspolec s|d|d}}|||||g}tjtj|d g}tfddt|tt|dddD}tfddt|tt|dddD}||S)Nr~rrcr&rkrkr'rtrkrlrr(z7_laplace_trig_ltex.._sypole..rcr&rkrkr'rtrkrlrr()rrrrr)r r!) r*r-rur~r.r#r1r}rr2rtrl_sypole s,(z#_laplace_trig_ltex.._sypolecSs&|d|d}}|}||}||S)Nr~rrk)r*rur~r.r}rrkrkrl _simplepolesz'_laplace_trig_ltex.._simplepolerNrr~T)reverse) r!popr r r!rr#rrr,)rrruresultsplanesr3r5r6r7r8r* i_imagsym i_realsym i_pointsymireal_eqrealsymimag_eqimagsymindices_to_poprkr2rl_laplace_trig_ltexsn               :rFc Cstddd}|ttttsdSt|||\}}t||\}}t |dkr)dS||s>t |||\}} ||| t j fSg} g} t |||dd\} } }|D]}| |d| |||d | | t|d qOt| ||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. rTrealNrFrrr~)rrwr2r1r*r(rrrr!rFrryrrr rr,)r t_rurrrrrrrr<r;GG_planeG_condrrkrkrl_laplace_rule_trigYs"   "rMcs"tdgd}tdgd}td}||t||f}|r||jrfdd||jD}t|dkrtdg}t||D]+} | d krR|| d } n t||| f d } | |||| d| qCt |||d d \} } } |||||| t || | 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)``. r~rr}rcg|]}|qSrkrw)rrrrkrlrrz&_laplace_rule_diff..r]z" rule: time derivative (4.1.8)rFrN) rrrr is_integerrfsumrpr rrrr)rrrur~r}rrrrrrrrrrkrrl_laplace_rule_diffzs"   &rRc s|jrdg}dg}t|D]}||r||q||qt|dkrt|}t||tdkrt|}t |||dd\}} } |gd} z t d| } Wn t ycd} Ynw| t rtdD]} d| dt||| dqon| r| tdD] } t d| q| rtfddtD} | | | fStd|gd }td }||||}r||jr||jrt ||||dd\}} } d||t ||||f| | 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)``. r]Frrrcs$g|]}|d|qSrrk)rr}Nderipcrkrlrrz'_laplace_rule_sdiff..r}rrN)is_Mulrrrrr!rrQ all_coeffsrr ValueErrorrwLaplaceTransformr r rrrrPr)rrrupfacofacfacpexoexr_p_c_d1rrr}rrrkrSrl_laplace_rule_sdiffsN       $  $rdcCst|dd}|jrt|||ddSt|}|jr t|||ddSt|}|jr/t|||ddS||kr;t|||ddStt|}|jrLt|||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. FdeeprN)r is_Addrr r )rrrurrkrkrl_laplace_expands  rhcCs<tttttttg}|D]}||||}dur|Sq dS)zk This function applies all program rules and returns the result if one of them gives a result. N)rrrrrMrRrd)rrru prog_rulesp_ruleLrkrkrl_laplace_apply_prog_rulessrlc Cst\}}}d}d}|D]F\}} } } } || kr"| |||i}| }||} | rRz| | }Wn ty9Yq w|tjkrR| | ||i| | tjfSq dS)zj This function applies all simple rules and returns the result if one of them gives a result. N)rrrxreplacerzrry)rrru simple_rulesrIrprep_oldprep_ft_doms_domcheckplaneprepmarrkrkrl_laplace_apply_simple_ruless(     rxc Cst|jstddd}t|||i|||iSt|}g}|jD]\}}t|trH||jvrHt|tt fr:|S| t |j |j |q!t|trst|jdkrs|jD]}|j|krl| t |j |j |qW|Sq!t|trt|jdkr|j\}}|j|kr|j|krd|jvr||}}| t |j |j t |j |j |q!|S|St|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. rTrGr>)is_realr_piecewise_to_heavisidernr0rfrxrrrrr:gtsrrKr!lhsrLrr) rrrrr rc2rrrkrkrlr{s>       r{c std}td}td}td}t|tr|ts!|ts!|SD]C\}}|t||||}durH||||krH|||S|t|||||}durh||||krh|||Sq%|j} fdd|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) rrurr~Ncrrk)laplace_correspondencerfdictrkrlr|rz*laplace_correspondence..) rrxrrwrZInverseLaplaceTransformitemsrrjrf) rrrrurr~rYrrjrfrkrrlrDs0$rc Cs|D]E\}}tt|D]:}|dkr||d|d}q|dkr5|tt||||d|d}q|tt||||f|d||}qq|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 rr])rr r!rrr )rrrricrrkrkrllaplace_initial_condss"$(rcs>t|}g}g}g}g}|D]c} | jdd\} } | tr?t| t} | D]} | jdd\}}|| ||fq)q| jt krk| t sktt | } | D]} | jdd\}}|| ||fqUq|| | fq|D]\} } | trt | |t jdf}ne| tr| t s| td} t| |}dust| |}dust| |}durn0tfdd| tDrt | |t jdf}nt| ||d}durn t | |t jdf}|\}}}|| |||||qut|}|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. Fas_AddTr]Nc3|]}|VqdSrrOrundefrIrkrlrrz%_laplace_transform..rdoit)rras_independentrwrBrr:rrjr/r9r{rZrrrrxrlrhanyrrr^rr,rL)r rIrrterms_tterms_stermsr< conditionsffrft_terms_termr+f1rri_pi_ci_rhru conditionrkrrlrsh         rc@s,eZdZdZdZddZddZddZd S) rZa 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^)selfrrruhintsrELTrkrkrl_compute_transforms z#LaplaceTransform._compute_transformcCs"t|t| ||tjtjfSr)rDr&rrr)rrrrurkrkrl _as_integrals"zLaplaceTransform._as_integralcKs`|dd}|dd}td|j|j|jf|j}|j}|j}t||||d}|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)rrWfunctionfunction_variabletransform_variabler)rr_nocondsrErIrr rrkrkrlr s zLaplaceTransform.doitN)re __module__ __qualname____doc___namerrrrkrkrkrlrZs   rZTc s,dd}dd}t|trt|drdd }|rJ|rJd}tdd|dtt|fd d Wd S1sDwYn5fd d |D} |rst| \} } } t |g|j | R} | t | t | fSt |g|j | RSt |jd|d\}}}|s|||fS|S)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_targetcst|fiSrlaplace_transform)fijrrurrkrlrrz#laplace_transform..Ncs g|] }t|fiqSrkr)rrrrkrlrs z%laplace_transform..rr)rrxrMhasattrrTrVrUrr)typeshaper,rLrZr)rrru legacy_matrixrrrEradtelements_transelementsavalsr f_laplacerrrrkrrlr)s< g  "  rc sddlm}mddlm}tdddfdd}|r%|}|jrBt fd d |j D}t | |dfSz||t  d tjfdd d \}} Wn tyad }Ynw|d ur||}|d urrd 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_transformrTrGcst|dkr t|S|djdj}|\}}|djd}|djd}tdt|||t|dt||S)z3 Simplify a piecewise expression from hyperexpand. rrrr])r!r/rfargumentr:r#)rfr"coeffexponente1e2)rrrkrlpw_simps z7_inverse_laplace_transform_integration..pw_simpcsg|] }t|qSrk)r_)rX)rururrrkrlrsz:_inverse_laplace_transform_integration..NF)needevalrucs||t }|rt||Sddlm}||dk}|jkr1t|j}t||St|j}t| |S)Nrr) rr&rwr:rrrr'r|)r"H0r~rrelr)rrrkrlsimp_heavisides      z>_inverse_laplace_transform_integration..simp_heavisidecSs tt|Sr)r r&)r"rkrkrlsimp_exprz8_inverse_laplace_transform_integration..simp_exp)sympy.integrals.meijerintrrsympy.integrals.transformsrris_rational_functionapartrgrrfrErr&rrrGrrwrDryrr/rr:) rrurIrurrrrrrrrrk)rrururrrrlr_sN         r_c Cs~ddlm}||\}}||r;||}t|dkr;|\}}}|||d|d|||d|d}||S)Nr)fractionrr)rrrrrXr!) rrurr}rcfr~rrrkrkrl_complete_the_square_in_denom s     0rc CsTtd}td}td|gd}td|gd}td|gd}tddd }d d }|||tj|d f|||| ||d t| |t|tj|d fd |d |d d t||||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`. rurr~rrrz._inverse_laplace_build_rules is building rulescSs$z||WSty|YSwr)factorrO)rrurkrkrl_frac&s   z+_inverse_laplace_build_rules.._fraccSs|Srrkrrkrkrlsame,sz*_inverse_laplace_build_rules..samer]rr) rrrprryr&r?r2r1r@)rurr~rrrr _ILT_rulesrkrkrl_inverse_laplace_build_ruless*0@&. rcs|dkrtdt|tjfSt\}}}d}|||i}|D]N\}} } } } || | fkr7| || } | | f}| |rn| }|tjurVfdd|dD}|d|}|tjkrnt|| ||itjfSq dS)@ Helper function for the class InverseLaplaceTransform. r]z rule: 1 o---o DiracDelta()rmcrNrk)rnrrwrkrlrUrz7_inverse_laplace_apply_simple_rules..rN) rpr9rryrrrr:rn)rrurrrrI_prepfsubsrsrrrtrvr]_Frrfrkrrl#_inverse_laplace_apply_simple_rules@s(       &rc Cstd|gd}td|gd}td}||t|||f}|rB||jrBtdt|||||ddd\}} | |||| fSdS) rr~rr}rz3 rule: t**n*f(t) o---o (-1)**n*diff(F(s), s, n)Fr dorationalN)rrr rPrp_inverse_laplace_transform) rrurrur~r}rrwrrrkrkrl_inverse_laplace_diff]s rcCstd|gd}td}||s|t|tjfS|ts dS|t||}|rI||jr?tdt|||tjfSt ||||tjfS|t|||}|ry||jrotdt |||||||ddd St 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rSdSt|||fS).rr~rrrr}Ncs&g|]}|qSrk)rrr~rrr}rurkrlrs&z/_inverse_laplace_irrational..rc||dk|fSrrkrrr}rkrlrz-_inverse_laplace_irrational..rcrrrkrrrkrlrrr]rrz rule 5.3.4rz rule 5.3.10rz rule 5.3.13r rz rule 5.3.16z rule 5.3.35/44z rule 5.3.14z rule 5.3.22z rule 5.3.1z rule 5.3.5z rule 5.3.7z rule 5.3.8z rule 5.3.11z rule 5.3.12z rule 5.3.15z rule 5.3.23z rule 5.3.6z rule 5.3.17z rule 5.3.18z rule 5.3.19z rule 5.3.2z rule 5.3.9)rrryas_ordered_factorsrrrrsortedr!rr.rr&r<rpr;r6r:)r rurrurhrfarw constantszerospolesrestrrk_a_sqb_a_numrkrrl_inverse_laplace_irrationals>        "&& 2&$:&<6(&&  :&$& ( $($,6&$(&($$.$4>&$$8&((*&(  6 $( 4$&&4".$$&(&$$&02$$&0:*6$(00$&(*&& rcCs2tg}|D]}|||||}dur|SqdSrN)rrrurrurirjrrkrkrl!_inverse_laplace_early_prog_ruless rcCs:tttttg}|D]}|||||}dur|Sq dSr)rrrrrrrkrkrl!_inverse_laplace_apply_prog_rulessrcCs|jrdSt|dd}|jrt||||dddSt|}|jr)t||||dddSt|}|jr:t||||dddS||rF||}|jrSt||||dddSdS)rNFreTr)rgr rr rrr)r rurrurrkrkrl_inverse_laplace_expand s0      rc std}||}t|}g}tjg} |D]&} | \} } | |} | dfdd| D} fdd| |D}t | dkrU|dt |}| |qt | dkrr|dt | d |}| t ||qt | dkr'| dd}| d|d}t |dkrtjg|}t|\}}|dkr|||d||t | |}njd }|jr| }d }tt|d||d}t|}|r|t | |t||||||t | |t||}n$|t | |t||||||t | |t||}| t ||qt| ||||d d \}}| || |qt|}|rK|jd d }|t| fS) rx_rcg|]}|qSrkrkrdc_leadrkrlr4rz-_inverse_laplace_rational..crrkrkrrrkrlr5rr]rrFTrr)rrrrrryrrrXr!r9rr&r:rrtuplerrrPr r.rr(r*r1r2rrL)r rurrurrrrrrrr}rr1r#rr~rlrhypb2bsrrrhrkrrl_inverse_laplace_rational&sf           ( &  H     r cs~t|}g}g}|D]} | tr |   } | jdd\} } |r=| r=t| |||d} dusht | |} dusht | ||} dusht | ||} dusht | ||} durin1t fdd| tDrt| ||tjf} nt| |||d} durn t| ||tjf} | \} }|| | ||q t|}|r|jdd}t|}||fS)z Front-end function of the inverse Laplace transform. It tries to apply all known rules recursively. If everything else fails, it tries to integrate. FrrNc3rrrOrrrkrlrrz-_inverse_laplace_transform..r)rrrwr&rrRrrr rrrrrrrrrryr_rrrL)r rrIrurrrrrrrrrrrrhrrkr rlr^s\      rc@sPeZdZdZdZedZedZddZe ddZ d d Z d d Z d dZ dS)rz Class representing unevaluated inverse Laplace transforms. For usage of this class, see the :class:`IntegralTransform` docstring. For how to compute inverse Laplace transforms, see the :func:`inverse_laplace_transform` docstring. zInverse LaplaceNonercKs(|durtj}tj|||||fi|Sr)r_none_sentinelrF__new__)rrrurruoptsrkrkrlrszInverseLaplaceTransform.__new__cCs|jd}|tjur d}|S)Nr)rfrr)rrurkrkrlfundamental_planes  z)InverseLaplaceTransform.fundamental_planecKst||||jfi|Sr)r_r)rrrurrrkrkrlrs  z*InverseLaplaceTransform._compute_transformcCsL|jj}tt|||||tjtj|tjtjfdtjtjS)Nr) __class___crDr&r ImaginaryUnitrPi)rrrurrrkrkrlrs z$InverseLaplaceTransform._as_integralc Ksj|dd}|dd}td|j|j|jf|j}|j}|j}|j}t|||||dd}|r3|dS|S)rrTrFz[ILT doit] (%s, %s, %s)rr)rrWrrrrr) rrrrErrIr rurrkrkrlrs  zInverseLaplaceTransform.doitN)rerrrrrrrrpropertyrrrrrkrkrkrlrs   rc  spdd}dd}t|tr"t|dr"|fddSt|jd|d\}}|r4|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 rTrFrcst|fiSr)inverse_laplace_transform)Fijrrururrkrlr rz+inverse_laplace_transform..r)rrxrMrrrr) rrurrurrrErrrkrrlrs -  rcsltdtgd\fddfddfdd fd d fd d |S) zEFast inverse Laplace transform of rational function including RootSumza, b, nrcsN|s|S|jr|S|jr|S|jr|St|tr%|Str)rwrgrWrqrxrSNotImplementedErrore)_ilt_add_ilt_mul_ilt_pow _ilt_rootsumrurkrl_ilt s  z#_fast_inverse_laplace.._iltcs|jt|jSr)rjmaprfrr!rkrlr- sz'_fast_inverse_laplace.._ilt_addcs$|\}}|jr t||Sr)rrWr)rrr)r!rurkrlr0 s z'_fast_inverse_laplace.._ilt_mulcs|}|durM|||}}}|jr>|dkr>| dt|| || t| S|dkrMt|| |Str)r is_Integerr&r?r)rrnmambm)r~rr}rurrkrlr6 s4z'_fast_inverse_laplace.._ilt_powcs,|jj}|jj\}t|jt|t|Sr)funr variablesrSpolyrrR)rrvariabler#rkrlr @ s z+_fast_inverse_laplace.._ilt_rootsum)rr)rrurrk) r!rrrr r~rr}rurrl_fast_inverse_laplace s  r,)Tr)rrcr` sympy.corerrrsympy.core.addrsympy.core.cachersympy.core.exprrsympy.core.functionrr r r r r rrrrsympy.core.mulrrsympy.core.relationalrrrrrrrrsympy.core.sortingrsympy.core.symbolrrr$sympy.functions.elementary.complexesr r!r"r#r$r%&sympy.functions.elementary.exponentialr&r'%sympy.functions.elementary.hyperbolicr(r)r*r+(sympy.functions.elementary.miscellaneousr,r-r.$sympy.functions.elementary.piecewiser/r0(sympy.functions.elementary.trigonometricr1r2r3r4sympy.functions.special.besselr5r6r7r8'sympy.functions.special.delta_functionsr9r:'sympy.functions.special.error_functionsr;r<r='sympy.functions.special.gamma_functionsr>r?r@rA-sympy.functions.special.singularity_functionsrBsympy.integralsrCrDrrErFrGsympy.logic.boolalgrHrIrJrKrLsympy.matrices.matrixbaserMsympy.polys.matrices.linsolverNsympy.polys.polyerrorsrOsympy.polys.polyrootsrPsympy.polys.polytoolsrQsympy.polys.rationaltoolsrRsympy.polys.rootoftoolsrSsympy.utilities.exceptionsrTrUrVsympy.utilities.miscrWrarnrprrr^rrrrrrrrrFrMrRrdrhrlrxr{rrrrZrr_rrrrrrrrrrrr rrrr,rkrkrkrls   0(           [  o  Y  )  ,      2    /<- C 8 P  (      <    7 8 G =