o *ήcHe@s6dZddlmZmZddlmZddlmZmZm Z ddl m Z ddl m Z mZmZmZmZmZmZmZmZddlmZddlmZmZdd lmZmZmZmZm Z m!Z!dd l"m#Z#m$Z$dd l%m&Z&m'Z'm(Z(dd l)m*Z*dd l+m,Z,m-Z-ddl.m/Z/m0Z0m1Z1m2Z2m3Z3ddl4m5Z5m6Z6m7Z7ddl8m9Z9m:Z:m;Z;mm?Z?ddl@mAZAmBZBmCZCddlDmEZEmFZFddlGmHZHmIZImJZJmKZKmLZLddlMmNZNmOZOmPZPmQZQddlRmSZSmTZTddlUmVZVmWZWmXZXddlYmZZZm[Z[m\Z\ddl]m^Z^ddl_m`Z`maZaddlbmcZcddldmeZemfZfmgZgmhZhmiZiddljmkZkddllmmZmmnZnddlompZpdd lqmrZrmsZsdd!ltmuZudd"lvmwZwmxZxdd#lymzZzm{Z{m|Z|dd$l}m~Z~dd%lmZGd&d'd'eZGd(d)d)eZd*d+Zd,d-Zed.Zd/d0Zeed1fd2d3ZGd4d5d5eZd6d7Zd8d9ZGd:d;d;eZdd?Zd@aGdAdBdBeZdCdDZdEdFZdGdHZeddIdJZdKdLZdMdNZdOdPZddQdRZddSdTZddUdVZddWdXZddYdZZdd[d\ZGd]d^d^eZdd_d`Zed1ddadbZGdcddddeZddedfZdgdhZed1ddidjZGdkdldleZGdmdndneZdodpZGdqdrdreZdsdtZed1ddudvZGdwdxdxeZGdydzdzeZd{d|ZGd}d~d~eZddZGdddeZddZGdddeZddZed1dddZGdddeZGdddeZddZGdddeZddZd@S)z Integral Transforms )reducewraps)repeat)SpiI)Add) AppliedUndef count_ops Derivativeexpandexpand_complex expand_mulFunctionLambda WildFunction)Mul)igcdilcm) _canonicalGeGtLt UnequalityEq)default_sort_keyordered)DummysymbolsWild)postorder_traversal) factorialrf)reargAbs polar_liftperiodic_argument)explog exp_polar)coshcothsinhtanhasinh)ceiling)MaxMinsqrt) Piecewisepiecewise_fold)coscotsintanatan)besselibesseljbesselkbessely) DiracDelta Heaviside)erferfcEi)digammagamma lowergamma)meijerg) integrateIntegral)_dummy)to_cnf conjuncts disjunctsOrAnd) MatrixBase) _lin_eq2dictPolyNonlinearError)roots)factorPoly)together)CRootOfRootSum)sympy_deprecation_warningSymPyDeprecationWarningignore_warnings)iterable)debugcs eZdZdZfddZZS)IntegralTransformErrora Exception raised in relation to problems computing transforms. Explanation =========== This class is mostly used internally; if integrals cannot be computed objects representing unevaluated transforms are usually returned. The hint ``needeval=True`` can be used to disable returning transform objects, and instead raise this exception if an integral cannot be computed. cstd||f||_dS)Nz'%s Transform could not be computed: %s.)super__init__function)self transformramsg __class__A/tmp/pip-target-vg8gfxp4/lib/python/sympy/integrals/transforms.pyr`@s  zIntegralTransformError.__init__)__name__ __module__ __qualname____doc__r` __classcell__rgrgrerhr^2s r^c@s|eZdZdZeddZeddZeddZedd Zd d Z d d Z ddZ ddZ ddZ eddZddZdS)IntegralTransforma} Base class for integral transforms. Explanation =========== This class represents unevaluated transforms. To implement a concrete transform, derive from this class and implement the ``_compute_transform(f, x, s, **hints)`` and ``_as_integral(f, x, s)`` functions. If the transform cannot be computed, raise :obj:`IntegralTransformError`. Also set ``cls._name``. For instance, >>> from sympy import LaplaceTransform >>> LaplaceTransform._name 'Laplace' Implement ``self._collapse_extra`` if your function returns more than just a number and possibly a convergence condition. cC |jdS)z! The function to be transformed. rargsrbrgrgrhra] zIntegralTransform.functioncCro)z; The dependent variable of the function to be transformed. rprrrgrgrhfunction_variablebrsz#IntegralTransform.function_variablecCro)z% The independent transform variable. rprrrgrgrhtransform_variablegrsz$IntegralTransform.transform_variablecCs|jj|jh|jhS)zj This method returns the symbols that will exist when the transform is evaluated. )ra free_symbolsunionrwrurrrgrgrhrxlszIntegralTransform.free_symbolscKtNNotImplementedErrorrbfxshintsrgrgrh_compute_transformuz$IntegralTransform._compute_transformcCrzr{r|rbrrrrgrgrh _as_integralxrzIntegralTransform._as_integralcCs$t|}|dkrt|jjdd|S)NF)rOr^rfname)rbextracondrgrgrh_collapse_extra{sz!IntegralTransform._collapse_extrac s|d}tfddjtD }|r0zjjjjfi|}Wn ty/d}Ynwj}|js:t |}||fS)Nc3s|] }|jVqdSr{)hasru.0funcrrrgrh sz2IntegralTransform._try_directly..) anyraatomsr rrurwr^is_Addr)rbrT try_directlyfnrgrrrh _try_directlys$    zIntegralTransform._try_directlyc  sdd}dd}|d<jd i\}}|dur |S|jr|d<fdd|jD}g}g}|D].} t| tsB| g} || dt| d krW|| d q8t| d krf|| d dg7}q8|dkrrt| }nt|}|sz|Sz |}t |rt|gt|WS||fWSt yYnw|rt j jjd|j\} } | j t| gtjd dS) a Try to evaluate the transform in closed form. Explanation =========== This general function handles linearity, but apart from that leaves pretty much everything to _compute_transform. Standard hints are the following: - ``simplify``: whether or not to simplify the result - ``noconds``: if True, do not return convergence conditions - ``needeval``: if True, raise IntegralTransformError instead of returning IntegralTransform objects The default values of these hints depend on the concrete transform, usually the default is ``(simplify, noconds, needeval) = (True, False, False)``. needevalFsimplifyTNcs6g|]}j|gtjddjdiqS)rtNrg)rflistrqdoitrrrrbrgrh s.z*IntegralTransform.doit..rrvrtrg)poprrrq isinstancetupleappendlenrrrr\r^rf_namera as_coeff_mulrurr) rbrrrrrresrressrcoeffrestrgrrhrsR          &zIntegralTransform.doitcCs||j|j|jSr{)rrarurwrrrgrgrh as_integrals zIntegralTransform.as_integralcOs|jSr{)r)rbrqkwargsrgrgrh_eval_rewrite_as_Integralz+IntegralTransform._eval_rewrite_as_IntegralN)rirjrkrlpropertyrarurwrxrrrrrrrrgrgrgrhrnFs$    G  rncCs4|rddlm}ddlm}||t|ddS|S)Nrr) powdenestT)polar)sympy.simplifyrsympy.simplify.powsimprr5)exprrrrrgrgrh _simplifys   rcsfdd}|S)aV This is a decorator generator for dropping convergence conditions. Explanation =========== Suppose you define a function ``transform(*args)`` which returns a tuple of the form ``(result, cond1, cond2, ...)``. Decorating it ``@_noconds_(default)`` will add a new keyword argument ``noconds`` to it. If ``noconds=True``, the return value will be altered to be only ``result``, whereas if ``noconds=False`` the return value will not be altered. The default value of the ``noconds`` keyword will be ``default`` (i.e. the argument of this function). cstdfdd }|S)Nnocondscs|i|}|r |dS|SNrrg)rrqrrrrgrhwrappersz0_noconds_..make_wrapper..wrapper)r)rrdefaultrrh make_wrappersz_noconds_..make_wrapperrg)rrrgrrh _noconds_s rFcCst||tjtjfSr{)rHrZeroInfinity)rrrgrgrh_default_integrator rTc stdd|||d||}|ts't|||tjtjftjfS|j s0t d|d|j d\}}|trBt d|dfdd fd d t |D}d d |D}|j d dd|sjt d|d|d\}} } t||||| f| fS)z0 Backend function to compute Mellin transforms. rzmellin-transformrtMellincould not compute integralrintegral in unexpected formcsHddlm}tj}tj}tj}tt|}tddd}|D]}tj}tj} g} t |D]M} | t dd t |} | j rM| jdvsM| sM| |sS| | g7} q+|| |} | j r`| jdvrf| | g7} q+| j|krrt| j| } q+t| j|}q+|tjur||krt||}q| tjur| |krt| |}qt|t| }q|||fS) zN Turn ``cond`` into a strip (a, b), and auxiliary conditions. r_solve_inequalitytTrealcSs |dSr) as_real_imagrrgrgrh3s z:_mellin_transform..process_conds..z==z!=)sympy.solvers.inequalitiesrrNegativeInfinityrtruerLrKrrMreplacer#subs is_Relationalrel_oprltsr1gtsr2rOrN)rrabauxcondsrca_b_aux_dd_solnrrgrh process_conds#sP            z(_mellin_transform..process_condscg|]}|qSrgrgrrrrgrhrJz%_mellin_transform..cSg|] }|ddkr|qS)rvFrgrrgrgrhrKcSs|d|dt|dfS)Nrrtrvr rrgrgrhrLrz#_mellin_transform..keyno convergence found)rJrrIrrrrrr is_Piecewiser^rqrMsort) rrs_ integratorrFrrrrrrg)rrrh_mellin_transforms&  "   ' rc@s,eZdZdZdZddZddZddZd S) MellinTransformz Class representing unevaluated Mellin transforms. For usage of this class, see the :class:`IntegralTransform` docstring. For how to compute Mellin transforms, see the :func:`mellin_transform` docstring. rcKst|||fi|Sr{)rr~rgrgrhrarz"MellinTransform._compute_transformcCs t|||d|tjtjfSNrt)rIrrrrrgrgrhrds zMellinTransform._as_integralc Csg}g}g}|D]\\}}}||g7}||g7}||g7}qt|t|ft|f}|dd|ddkdks>|ddkrDtddd|S)NrrtTFrzno combined convergence.)r1r2rOr^) rbrrrrsasbrrrgrgrhrgs   (zMellinTransform._collapse_extraN)rirjrkrlrrrrrgrgrgrhrUs   rcKt|||jdi|S)a Compute the Mellin transform `F(s)` of `f(x)`, .. math :: F(s) = \int_0^\infty x^{s-1} f(x) \mathrm{d}x. For all "sensible" functions, this converges absolutely in a strip `a < \operatorname{Re}(s) < b`. Explanation =========== The Mellin transform is related via change of variables to the Fourier transform, and also to the (bilateral) Laplace transform. This function returns ``(F, (a, b), cond)`` where ``F`` is the Mellin transform of ``f``, ``(a, b)`` is the fundamental strip (as above), and ``cond`` are auxiliary convergence conditions. If the integral cannot be computed in closed form, this function returns an unevaluated :class:`MellinTransform` object. For a description of possible hints, refer to the docstring of :func:`sympy.integrals.transforms.IntegralTransform.doit`. If ``noconds=False``, then only `F` will be returned (i.e. not ``cond``, and also not the strip ``(a, b)``). Examples ======== >>> from sympy import mellin_transform, exp >>> from sympy.abc import x, s >>> mellin_transform(exp(-x), x, s) (gamma(s), (0, oo), True) See Also ======== inverse_mellin_transform, laplace_transform, fourier_transform hankel_transform, inverse_hankel_transform Nrg)rr)rrrrrgrgrhmellin_transformvs)rcCsp|\}}t|t}t|t}t| ||d}t||||td||||d|tfS)a Re-write the sine function ``sin(m*s + n)`` as gamma functions, compatible with the strip (a, b). Return ``(gamma1, gamma2, fac)`` so that ``f == fac/(gamma1 * gamma2)``. Examples ======== >>> from sympy.integrals.transforms import _rewrite_sin >>> from sympy import pi, S >>> from sympy.abc import s >>> _rewrite_sin((pi, 0), s, 0, 1) (gamma(s), gamma(1 - s), pi) >>> _rewrite_sin((pi, 0), s, 1, 0) (gamma(s - 1), gamma(2 - s), -pi) >>> _rewrite_sin((pi, 0), s, -1, 0) (gamma(s + 1), gamma(-s), -pi) >>> _rewrite_sin((pi, pi/2), s, S(1)/2, S(3)/2) (gamma(s - 1/2), gamma(3/2 - s), -pi) >>> _rewrite_sin((pi, pi), s, 0, 1) (gamma(s), gamma(1 - s), -pi) >>> _rewrite_sin((2*pi, 0), s, 0, S(1)/2) (gamma(2*s), gamma(1 - 2*s), pi) >>> _rewrite_sin((2*pi, 0), s, S(1)/2, 1) (gamma(2*s - 1), gamma(2 - 2*s), -pi) rrt)rrr0rrE)m_nrrrmnrrgrgrh _rewrite_sins %  6rc@seZdZdZdS)MellinTransformStripErrorzF Exception raised by _rewrite_gamma. Mainly for internal use. N)rirjrkrlrgrgrgrhrsrc-st||g\fdd}g}tD]#}|sq|jd}|jr-|d}|\}} ||g7}qtt t t D]%}|sJqB|jd}|jrY|d}|\}} ||t g7}qBdd|D}tj |D] } | js}| nqtfdd|D}tdd |Drjstd d d ttd d|Dtj } | krt|dkr} n ttdd|D} | tj | } tj | } d urڈ| 9d ur| 9\}}t|}t|}tt|tdtt|td}g}g}g}g}g}fdd|r|\r(||}}|}n||}}|}fdd}sD|g7}njsNtt rjrYj!}j }nt"d}j }|j#r{}|dkrn| }|||fgt$|7}q|s||\}}sd|}|||g7}|||g7}ng%r#t&}|'dkr|(d}t)|}t||'krt*+|}||g7}|fdd|D7}q|,\}}||g7}|| }||r |tj | dfg7}|tj | fg7}n|dg7}|tj-|dfg7}|tj-|fg7}nttra|jd\}}rY|dkrE|| |dksU|dkrY|| |dkrYt.d|||fg7}nttrjd}rt|t td|t t } }!}"n t/||\} }!}"||  f|! fg7}||"g7}nett rÈjd}|t|ddftt d|dd fg7}nBtt rވjd}|tt d|ddfg7}n'tt rjd}|tt d|ddft|dd fg7}n|s| t|t|9} ggggf\}#}$}%}&||#|%df||&|$dffD]\}'}(})|'r|'\}}|dkr|dkrt$t|}||}*||}+|j#sWt0dt1|D]},|'|*|+|,|fg7}'q[r| dt d|d||tj29} |||g7}n| dt d|d||tj2} ||| g7}q.|dkr|(3d|n|)3||'s1q(t|}|#j4t5d|$j4t5d|%j4t5d|&j4t5d|#|$f|%|&f|| | fS)a Try to rewrite the product f(s) as a product of gamma functions, so that the inverse Mellin transform of f can be expressed as a meijer G function. Explanation =========== Return (an, ap), (bm, bq), arg, exp, fac such that G((an, ap), (bm, bq), arg/z**exp)*fac is the inverse Mellin transform of f(s). Raises IntegralTransformError or MellinTransformStripError on failure. It is asserted that f has no poles in the fundamental strip designated by (a, b). One of a and b is allowed to be None. The fundamental strip is important, because it determines the inversion contour. This function can handle exponentials, linear factors, trigonometric functions. This is a helper function for inverse_mellin_transform that will not attempt any transformations on f. Examples ======== >>> from sympy.integrals.transforms import _rewrite_gamma >>> from sympy.abc import s >>> from sympy import oo >>> _rewrite_gamma(s*(s+3)*(s-1), s, -oo, oo) (([], [-3, 0, 1]), ([-2, 1, 2], []), 1, 1, -1) >>> _rewrite_gamma((s-1)**2, s, -oo, oo) (([], [1, 1]), ([2, 2], []), 1, 1, 1) Importance of the fundamental strip: >>> _rewrite_gamma(1/s, s, 0, oo) (([1], []), ([], [0]), 1, 1, 1) >>> _rewrite_gamma(1/s, s, None, oo) (([1], []), ([], [0]), 1, 1, 1) >>> _rewrite_gamma(1/s, s, 0, None) (([1], []), ([], [0]), 1, 1, 1) >>> _rewrite_gamma(1/s, s, -oo, 0) (([], [1]), ([0], []), 1, 1, -1) >>> _rewrite_gamma(1/s, s, None, 0) (([], [1]), ([0], []), 1, 1, -1) >>> _rewrite_gamma(1/s, s, -oo, None) (([], [1]), ([0], []), 1, 1, -1) >>> _rewrite_gamma(2**(-s+3), s, -oo, oo) (([], []), ([], []), 1/2, 1, 8) cstt|}durtjurdSdur|kSdur!|kS|kdkr)dS|kdkr1dS|r5dSjs>js>|jr@dStd)zU Decide whether pole at c lies to the left of the fundamental strip. NTFzPole inside critical strip?)r r#rrrxr)ris_numer)rrrgrhlefts   z_rewrite_gamma..leftrrtcSsg|] }|jr t|n|qSrg)is_extended_realr%rrgrgrhrBsz"_rewrite_gamma..csg|]}|qSrgrgr)common_coefficientrgrhrHrcss|]}|jVqdSr{) is_RationalrrgrgrhrIsz!_rewrite_gamma..GammaNzNonrational multipliercSg|]}t|jqSrg)rqrrgrgrhrLscSrrg)rprrgrgrhrSTFcstdd|S)NInverse MellinzUnrecognised form '%s'.)r^)factrrgrh exceptionkz!_rewrite_gamma..exceptioncs8|s t|}|dkr|S)z7 Test if arg is of form a*s+b, raise exception if not. rt) is_polynomialrUdegree all_coeffs)r$r )rr rrgrh linear_argvs   z"_rewrite_gamma..linear_argcsg|]}|fqSrgrgr)rrrgrhrsrz Gammas partially over the strip.)evaluatervza is not an integerr)6rrrErrqras_independentrr8r6r9r7rOnerallrr^rrrrras_numer_denomr make_argsrziprris_Powrr(baser* is_Integerr%rrUrLTrSrW all_rootsr NegativeOner}r TypeErrorrangeHalfrrr)-rrrrr s_multipliersgr$r_r s_multiplierfacexponentnumerdenomrqfacsdfacs numer_gammas denom_gammas exponentialsugammaslgammasufacsrrexp_rr rsrgamma1gamma2fac_anapbmbqgammasplusminusnewanewckrg)rrrrrr rrrh_rewrite_gammasZ?             $                       &        h     &&      rDc sBtdd|dd|t}t|t|t|fD]}|jrTfdd|jD}dd|D}dd|D}t|}sHt|| t d }| |t |fSzt |d d \} } } } } Wn tyoYqwz t| | | | }Wn tyYqwr|}nDz d d lm}||}Wn tytd |dw|jrt|jdkrt t| |jd jd t t| |jd jd }tt|j|jtkg}|t tt|jt|jkd t|jd ktt|j|jtkg7}t|}|dkrtd |d||  ||fStd |d)zs A helper for the real inverse_mellin_transform function, this one here assumes x to be real and positive. rzinverse-mellin-transformT)positivec s g|] }t|ddqS)Fr)_inverse_mellin_transform)rG as_meijergrstriprrgrhr s   z-_inverse_mellin_transform..cSg|]}|dqS)rtrgrr rgrgrhrrcSrK)rrgrLrgrgrhrr)gensrrt) hyperexpandr zCould not calculate integralFzdoes not converger) rJrewriterErTrr rrqrrr@rrOrDr^rG ValueErrorrrNr}rrr%r$argumentdeltarrNr;r=r#nu)rrx_rJrIr&rrrrrCer)rGhrNrrgrHrhrFs` $     *  rFNc@HeZdZdZdZedZedZddZe ddZ d d Z d d Z d S)InverseMellinTransformz Class representing unevaluated inverse Mellin transforms. For usage of this class, see the :class:`IntegralTransform` docstring. For how to compute inverse Mellin transforms, see the :func:`inverse_mellin_transform` docstring. r NonercKs8|durtj}|durtj}tj||||||fi|Sr{)rZ_none_sentinelrn__new__)clsrrrrroptsrgrgrhr]Ms zInverseMellinTransform.__new__cCs:|jd|jd}}|tjurd}|tjurd}||fS)NrO)rqrZr\)rbrrrgrgrhfundamental_stripTs   z(InverseMellinTransform.fundamental_stripc Ks|ddtdurtttttttt t t t t h at|D]}|jr3||r3|jtvr3td|d|q|j}t||||fi|S)NrTr zComponent %s not recognised.)r_allowedr(rEr8r6r9r7r+r-r.r,r!r"r is_Functionrrr^rarF)rbrrrrrrJrgrgrhr]s  z)InverseMellinTransform._compute_transformcCsJ|jj}t||| ||tjtj|tjtjfdtjtjSNrv)rf_crIr ImaginaryUnitrPi)rbrrrrrgrgrhrms   z#InverseMellinTransform._as_integralN) rirjrkrlrrr\rer]rrarrrgrgrgrhrZ?s   rZcKs$t||||d|djdi|S)a" Compute the inverse Mellin transform of `F(s)` over the fundamental strip given by ``strip=(a, b)``. Explanation =========== This can be defined as .. math:: f(x) = \frac{1}{2\pi i} \int_{c - i\infty}^{c + i\infty} x^{-s} F(s) \mathrm{d}s, for any `c` in the fundamental strip. Under certain regularity conditions on `F` and/or `f`, this recovers `f` from its Mellin transform `F` (and vice versa), for positive real `x`. One of `a` or `b` may be passed as ``None``; a suitable `c` will be inferred. If the integral cannot be computed in closed form, this function returns an unevaluated :class:`InverseMellinTransform` object. Note that this function will assume x to be positive and real, regardless of the SymPy assumptions! For a description of possible hints, refer to the docstring of :func:`sympy.integrals.transforms.IntegralTransform.doit`. Examples ======== >>> from sympy import inverse_mellin_transform, oo, gamma >>> from sympy.abc import x, s >>> inverse_mellin_transform(gamma(s), s, x, (0, oo)) exp(-x) The fundamental strip matters: >>> f = 1/(s**2 - 1) >>> inverse_mellin_transform(f, s, x, (-oo, -1)) x*(1 - 1/x**2)*Heaviside(x - 1)/2 >>> inverse_mellin_transform(f, s, x, (-1, 1)) -x*Heaviside(1 - x)/2 - Heaviside(x - 1)/(2*x) >>> inverse_mellin_transform(f, s, x, (1, oo)) (1/2 - x**2/2)*Heaviside(1 - x)/x See Also ======== mellin_transform hankel_transform, inverse_hankel_transform rrtNrg)rZr)rrrrJrrgrgrhinverse_mellin_transformss$5rhcsfddfddfddfdd}d d }d d lm}||}||t}||tfd d}||t|}t|S)a Naively simplify some conditions occurring in ``expr``, given that `\operatorname{Re}(s) > a`. Examples ======== >>> from sympy.integrals.transforms import _simplifyconds as simp >>> from sympy.abc import x >>> from sympy import sympify as S >>> simp(abs(x**2) < 1, x, 1) False >>> simp(abs(x**2) < 1, x, 2) False >>> simp(abs(x**2) < 1, x, 0) Abs(x**2) < 1 >>> simp(abs(1/x**2) < 1, x, 1) True >>> simp(S(1) < abs(x), x, 1) True >>> simp(S(1) < abs(1/x), x, 1) False >>> from sympy import Ne >>> simp(Ne(1, x**3), x, 1) True >>> simp(Ne(1, x**3), x, 2) True >>> simp(Ne(1, x**3), x, 0) Ne(1, x**3) cs&|krdS|jr|jkr|jSdSr)rrr()exrrgrhpowers z_simplifyconds..powercs|r |r dSt|tr|jd}t|tr |jd}|r.d|d|S|}|dur8dSz,|dkrLt|t|kdkrLWdS|dkr_t|t|kdkrbWdSWdSWdStynYdSw)z_ Return True only if |ex1| > |ex2|, False only if |ex1| < |ex2|. Else return None. NrrtTF)rrr%rqr")ex1ex2r)rbiggerrjrrgrhrms(        z_simplifyconds..biggercsH|jst|tr|jst|ts||kS||}|dur | S||kS)z simplify x < y N) is_positiverr%)ryrrmrgrhreplies z_simplifyconds..repliecs ||}|dvr dSt||S)NTFT)r)rrorrprgrhreplues  z_simplifyconds..repluecWs|dvrt|S|j|S)Nrr)boolr)rirqrgrgrhrepls z_simplifyconds..replr) collect_abscs ||Sr{rg)rro)rqrgrhr z _simplifyconds..)sympy.simplify.radsimprvrrrr)rrrrsrurvrg)rrmrjrqrrh_simplifycondss      rycCst||tS)zs Expand an expression involving DiractDelta to get it as a linear combination of DiracDelta functions. )rQrr?rrgrgrhexpand_dirac_deltasr{cs*tdtdgd}g}g}zt\}}Wn ty$tddw|D]4\} } | t|} | rF|| d| |q)| j d dj rVtdd|| | q)t t t t |g|Rtjtjft |} | tst| |tjtjfS| jstdd| j d\} } | trtdd fd d fd d t| D}dd |D}|sdd |D}tt|}dd|jfddd|stdd|d\}}fdd}|rt| |} t||}t| |||t||fS)z The backend function for Laplace transforms. This backend assumes that the frontend has already split sums such that `f` is to an addition anymore. rrexcludeLaplacez'could not expand DiracDelta expressionsrznot implemented yet.rrc s|ddlm}tj}tj}tt|}tdtgd\}}}}}} } |t t |||k|t t |||kt t |||||kt t |||||kt t t |||||kt t t |||||kf} |D]5} 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!krt"d d t#|j!| } q| tj urt$| |}q{t%|t&|}q{||jr|j'fS|fS) z7 Turn ``conds`` into a strip and auxiliary conditions. rrzp q w1 w2 w3 w4 w5r^r}rvc3s|]}|jVqdSr{)rn)rwildrrgrhr_z<_laplace_transform..process_conds..cSs|dSr)r rrrgrgrhrbsz;_laplace_transform..process_conds..rr~zconvergence not in half-plane?)(rrrrrrLrKrrr%r$r'r&rrMrrhsrxreversedrrr reversedsignmatchrnrr#r6rrrrrrr^r2r1rOrN canonical)rrrrr r w1w2w3w4w5patternsrrrrpatrr)rrrrrhr9s      ":4"(       z)_laplace_transform..process_condscrrgrgrrrgrhrxrz&_laplace_transform..cSs*g|]}|ddkr|dtjur|qS)rtFr)rrrrgrgrhrys*cSr)rtFrgrrgrgrhr{rcSs|dvrdS|S)Nrrrrrzrgrgrhcnt~sz_laplace_transform..cntcs|d |dfSNrrtrgr)rrgrhrr z$_laplace_transform..rrcs |Sr{)rrz)rrrgrhsbs z_laplace_transform..sbs)rrr{rRr^itemsrr?rrrqis_zerorrHr(rrrrrIrrrrrMrrrryr)rrrrr deltazero deltanonzero integratable deltadict dirac_func dirac_coeffr rrrconds2rrrg)rrrrrrrh_laplace_transformsh     ?     $rcsP|j}t|j}t|jdkr|Sfdd|D}|jr$||S||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)`. rcsg|]}t|qSrg_laplace_deep_collect)rr$rrgrhrr z)_laplace_deep_collect..)rrrqrrcollect)rrrrqrgrrhrs rc stdgd}tdgd}tdgd}tdgd}tdgd}fdd}g|||tjtj|ft||t| ||t|tt|d k|d kt|d k|d ktj|ft||td tt|d k|d kt|d k|d ktj|fd d |tjtj|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 |d tjtj|fd ||t| || t | ||||d ktj|fd t ||t |t |t|||t t |||||d ktj|ft |t t |t t |t||t t ||tjtj|f||td  d d |td  d d t ||td d t|||t t |||||d ktj|ftd d |d t |td d t |td d t||t t ||tjtj|fd |t dt |td d t||t t ||tjtj|f|t|d ||d |d ktj|f|||t|d |||t| ||||d |t|d k|d ktj|f||||t|d t| |||d ktj|ft||t| ||tj||ft||t| ||d tj||f|t|t|d |||d |d k||ft| d t t d|t|d d|t |t d||d ktj|ft| d d d |d t t d|td d |t |t d|tjtj|ft| d t ||td d t |||d ktj|ft t| td d t t |d d d t ||tdt |||d ktj|ft| t t t |tdt |||d ktj|ft| t t t |tdt |||d ktj|f|t| d |||d d t|d d t |||d ktj|ftdt ||d t t ||td  d t||t t ||tjtj|ftdt |t t |td d t||t t ||tjtj|ft|||d |d tjt||ft|d d |d |d d|d |d tjtd ||ft|t||||d tj||f|t|t|d d ||| d ||| d |dkt||ftd t |t t ||t |t||tjtj|ft td t |t td d |td d |d |t||tt |||td d |dtjtj|ftd t |t t td d |td  d t||tt ||tjtj|ftt |d t t td d d |td  d t||d tjtj|ft|||d |d tjt||ft|d |d d |d |d d|d |d tjtd ||f|t|t|d d ||| d ||| d |d kt||ftd t |d |t t ||t |t||tt ||tjtj|ft td t |t td d |td d |d |t||tjtj|ftd t |t t td d |td  d t||tjtj|ftt |d t t td d d |td  d t||d tjtj|ft|t||tj ||d ktj|ftd |t|| |t | |tjtj|ft||t|t|||||t | ||||d ktj|ftt t t | td|tjtjtj|f|tt|d || d t|d t||d ktj|ft|d t||tjd t d d||d ktj|ft| tt||tj ||tj| |ft|||d |d tjtj|ftt|||d |d tt |d ||d ktj|ft|t||tjtj|ft|d td d|d |d dtjtj|ft|d d |td |||td d|d |d dtjtj|ftd t |t t ||t |t| ||d ktj|ftd t |t tt |||d ktj|ft|||d |d tjtj|ft|d |d d |d |d d|d |tjtj|ft td t |t t d |td d |d |t| ||d ktj|ftd t |t t t |t| ||d ktj|ft|t|d ||||d ||d |d ||d tjtj|ft|t|||d |d |d |d ||d |d ||d tjtj|ft|t|||d |d |d |d ||d |d ||d tjtj|ft|t||||d |d tj||ft|t|||||d |d tj||ft|t|d d |d t |d |||d ktj|ftt |t |t ||||d ktj|ft|tt |t |t ||||d k||ftt |d d tt || ||d ktj|ft t |t ||t |t ||||d ktj|ft|t t |d |t |||d ktj|ft t |d tt || ||d ktj|ft||||t |d |d |t |d |d |t|d k|d ktj|f|t||d |t t t|tj|||d |d | tjtt|d k|tj kt||tj|f|t||d |d t t t|td d ||||d |d | td d tt|d k|d kt||d tj|ftd d t |t| |||d ktj|f|t|d t |||d || d t| |tt|d k|d kt||tjtj|ftd |t d |t|||t |d |d t |d |d |d ktj|ft||||t |d |d |t |d |d |t|d k|d kt||f|t||d |t t t|tj|||d |d | tjtt|d k|tj kt||t||f|t||d |d t t t|td d ||||d |d | td d tt|d k|d kt||d t||f|t|d t |||d || d t||tt|d k|d kt||tjtj|ftd |dt t ||t |d |d |d ktj|ftd |t|t |d |d t |d |d |d kt||f}|S)z This is an internal helper function that returns the table of Laplace transfrom rules in terms of the time variable `t` and the frequency variable `s`. It is used by `_laplace_apply_rules`. rr|rrtauomegacs t|Sr{rrrrgrhrrwz&_laplace_build_rules..rrtrvrOrg?r`)!rrrrr?r(r%rNrOr@rCr3rrBrErFr=r-r)rAr+ EulerGammarDr8r,r:r6r<r$rr;r>r/) rrrrrrrdcolaplace_transform_rulesrgrrh_laplace_build_ruless * * !  %( *6 .2 2X 6B :. > B6F" JNR X6 \B  `$ dB h l p0 t@ x.  | & 2 " ` < 4   !.%2): -8 1, 54  9 = A4 F K, O$ S  W [& a  e i6 m$ q  u y* 6   6  B B  * ! % ) -" 1 5  96=@&AT&E I&(M6  Q6U@(YT(]$( a&  e.ircKs |dd }|r|||fS|S)z Internal helper function that will return `(f, a, c)` unless `**hints` contains `noconds=True`, in which case it will only return `f`. rF)get)rrrrrrgrgrh _laplace_crs rc s|dd}tdgd}tddd}|jdd \}} | |} | r| |jd } | |} | r| |d krtd td || | ftd | |dkr|dkrvtfdd| | t Dsv|t | | ||dS|t | | |fi|St| | || |fd|i|} z| \}}}|| ||||fWSty|| || YSwdS)a This internal helper function tries to apply the time-scaling rule of the Laplace transform and returns `None` if it cannot do it. Time-scaling means the following: if $F(s)$ is the Laplace transform of, $f(t)$, then, for any $a>0$, the Laplace transform of $f(at)$ will be $\frac1a F(\frac{s}{a})$. This scaling will also affect the transform's convergence plane. rTrr|r&rtnargsFas_Addr_laplace_apply_rules match: f: %s ( %s, %s )z1 rule: amplitude and time scaling (1.1, 1.2)c3s|]}|VqdSr{)rrrrgrhr8rz*_laplace_rule_timescale..rrN)rrrrrrqrr]rrr rrLaplaceTransform_laplace_apply_rulesr")rrrrrrrr&rCrma1r$ma2Lrr rrgrrh_laplace_rule_timescale!s@     "  rcKs\|ddtd|gd}td|gd}td}tddd }|j|d d \} } | t||} | r| |||} | |jd |||} | r| |d kr| r| || |krtd td|| | | ftdt | | |||fd|i|}z|\}}}| t | | ||||fWSt y| t | | ||YSwdS)z This internal helper function tries to transform a product containing the `Heaviside` function and returns `None` if it cannot do it. rTrr|rror&rtrFrrrz f: %s ( %s, %s, %s )z rule: time shift (1.3)rN) rrrrrr@rqrr]rrr(r")rrrrrrrror&rCrrrma3rrr rrgrgrh_laplace_rule_heavisideHs,  $" " rcKs|ddtd|gd}td}td}|j|dd\}} | t||} | ru| ||||} | rutd td || | ftd t| |||| |fd |i|} z| \} }}| || ||fWStyt| YSwd S)z This internal helper function tries to transform a product containing the `exp` function and returns `None` if it cannot do it. rTrr|rozFrrrz# rule: multiply with exp (1.5)rN) rrrrr(rr]rr")rrrrrrrorrCrrrrrr rrgrgrh_laplace_rule_expds( $  rcKs:|dd}td|gd}td}td}|j|dd\} } t|d d d d ft|d d d d ft|d t d tft|dd d tfg} | D]} | \} }}}}| | |}|r|| |||}|rt dt d|||ft d| j |ft ||||fd|i|}z:|\}}}|d krt ||}nd}||||||||||||||d|||fWSty|dkr|dkr||||||||||||||dYS||||||||||||||dYSwqCdS)z This internal helper function tries to transform a product containing a trigonometric function (`sin`, `cos`, `sinh`, `cosh`, ) and returns `None` if it cannot do it. rTrr|rorFrz1.6rtrz1.7z1.81.9rrz! rule: multiply with %s (%s)rrrvN)rrrr-r+r8rr6rrr]rrr%rrr")rrrrrrrrorrCr trigrulestrigrulefmrTs1s2sdrrrrr rcp_shiftrgrgrh_laplace_rule_trig~sZ  "    rc KsL|ddtd|gd}td}td|gd}tddd }||t|||f} | r| |jd |kr| |jrtd td |ftd g} t| |D]1} | d krc| | | |d }nt| | ||| f |d }| || || d|qQ|| |t | | |||fd|i|} | || t | SdS)z This internal helper function tries to transform an expression containing a derivative of an undefined function and returns `None` if it cannot do it. rTrr|rorr&rtrrr f: %sz( rule: time derivative (1.11, 1.12)rN)rrrrr rq is_integerr]r#rrrrr) rrrrrrrorr&rrrCrrgrgrh_laplace_rule_diffs*     $rc Ks6|j|dd\}}t||}|D]a\}} } } } | ||} | rqtdtd|ftd|| fz0td| f| | }td| |f|dkrbt|| | | | tjfi|WSWqtyptd Yqwq| t ryd St t t ttg}|D]}||||fd |i|}|d ur|Sqd S) aM Helper function for the class LaplaceTransform. This function does a Laplace transform based on rules and, after applying the rules, hands the rest over to `_laplace_transform`, which will attempt to integrate. If it is called with `doit=False`, then it will instead return `LaplaceTransform` objects. Frrrz rule: %s o---o %sz try %sz check %s -> %sTz#_laplace_apply_rules did not match.Nr)rrrr]xreplacerrr Exceptionrr?rrrrr)rrrrrrCr simple_rulest_doms_domcheckplaneprepmar prog_rulesp_rulerrgrgrhrsD       rc@s4eZdZdZdZddZddZddZd d Zd S) rz 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. r~cKs`t|||fi|}|dur.|dd}td|ftd|ft|||fd|i|S|S)NrTz0_laplace_apply_rules could not match function %sz hints: %s)rrr]r)rbrrrrrrrgrgrhrs z#LaplaceTransform._compute_transformcCs"t|t| ||tjtjfSr{)rIr(rrr)rbrrrrgrgrhrs"zLaplaceTransform._as_integralcCsVg}g}|D]\}}||||qt|}t|}|dkr'tddd||fS)NFr~zNo combined convergence.)rrOr1r^)rbrrplanesrrrgrgrhrs   z LaplaceTransform._collapse_extracKsx|j}td|f|j}|j}d}|js8t|}z|j|||fi|}W||fSty7d}Y||fSw||fS)Nz----> _try_directly: %s)rar]rurwrrrr^)rbrrt_rrrgrgrhrs zLaplaceTransform._try_directlyN) rirjrkrlrrrrrrgrgrgrhrs   rc  s td|ft|trzt|drzdd }|rE|rEtddddtt|fd d Wd S1s?wYn5fd d |D}|rnt |\}}} t |g|j |R} | t |t | fSt |g|j |RSt|jdiS)ak 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 attemped, you can switch debug information on by setting ``sympy.SYMPY_DEBUG=True``. 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 integral cannot be fully computed in closed form, this function returns an unevaluated :class:`LaplaceTransform` object. 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) (s/(a + s), Max(0, -a), True) See Also ======== inverse_laplace_transform, mellin_transform, fourier_transform hankel_transform, inverse_hankel_transform z$ ***** laplace_transform(%s, %s, %s) applyfuncrFz 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. rz#deprecated-laplace-transform-matrix)deprecated_since_versionactive_deprecations_targetcst|fiSr{laplace_transform)fijrrrrgrhrrz#laplace_transform..Ncs g|] }t|fiqSrgr)rrrrgrhrs z%laplace_transform..rg)r]rrPhasattrrrYr[rZrrtypeshaper1rOrr) rrr legacy_matrixrrelements_transelementsavals conditions f_laplacergrrhr.s&B "rc sddlm}mtdddfdd}|r|}|jr.pw_simpcsg|] }t|qSrg)_inverse_laplace_transform)rX)rrrrrgrhrsz._inverse_laplace_transform..NF)rrInverse Laplacerz(inversion integral of unrecognised form.ucs||t }|rt||Sddlm}||dk}|jkr1t|j}t||St|j}t| |S)Nrr) rr(rr@rrrr)r)r$H0rrrelrC)rrrgrhsimp_heavisides      z2_inverse_laplace_transform..simp_heavisidecSs tt|Sr{)r r()r$rgrgrhsimp_exprz,_inverse_laplace_transform..simp_exp)sympy.integrals.meijerintrrris_rational_functionapartrrrqrrrhr(rrr^rrrIrrr4r$r@) rrrrrrrrrrrrg)rrrrrrrhrsL         rc@rY)InverseLaplaceTransformz 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. rr[rcKs(|durtj}tj|||||fi|Sr{)rr\rnr])r^rrrrr_rgrgrhr]szInverseLaplaceTransform.__new__cCs|jd}|tjur d}|SNrO)rqrr\)rbrrgrgrhfundamental_planes  z)InverseLaplaceTransform.fundamental_planecKst||||jfi|Sr{)rr)rbrrrrrgrgrhrsz*InverseLaplaceTransform._compute_transformcCsL|jj}tt|||||tjtj|tjtjfdtjtjSrd)rfrerIr(rrfrrg)rbrrrrrgrgrhrs  z$InverseLaplaceTransform._as_integralN) rirjrkrlrrr\rer]rrrrrgrgrgrhrs   rc sFt|trt|dr|fddSt|jdiS)a Compute the inverse Laplace transform of `F(s)`, defined as .. math :: f(t) = \frac{1}{2\pi i} \int_{c-i\infty}^{c+i\infty} e^{st} F(s) \mathrm{d}s, for `c` so large that `F(s)` has no singularites in the half-plane `\operatorname{Re}(s) > c-\epsilon`. Explanation =========== The plane can be specified by argument ``plane``, but will be inferred if passed as None. Under certain regularity conditions, this recovers `f(t)` from its Laplace Transform `F(s)`, for non-negative `t`, and vice versa. If the integral cannot be computed in closed form, this function returns an unevaluated :class:`InverseLaplaceTransform` object. Note that this function will always assume `t` to be real, regardless of the SymPy assumption on `t`. For a description of possible hints, refer to the docstring of :func:`sympy.integrals.transforms.IntegralTransform.doit`. Examples ======== >>> from sympy import inverse_laplace_transform, exp, Symbol >>> from sympy.abc import s, t >>> a = Symbol('a', positive=True) >>> inverse_laplace_transform(exp(-a*s)/s, s, t) Heaviside(-a + t) See Also ======== laplace_transform, _fast_inverse_laplace hankel_transform, inverse_hankel_transform rcst|fiSr{)inverse_laplace_transform)Fijrrrrrgrhr0r z+inverse_laplace_transform..Nrg)rrPrrrr)rrrrrrgrrhrs+rcsltdtgd\fddfddfdd fd d fd d |S) zEFast inverse Laplace transform of rational function including RootSumza, b, nrcsN|s|S|jr|S|jr|S|jr|St|tr%|Str{)rris_MulrrrXr}rW)_ilt_add_ilt_mul_ilt_pow _ilt_rootsumrrgrh_ilt8s  z#_fast_inverse_laplace.._iltcs|jt|jSr{)rmaprqrrrgrhrFsz'_fast_inverse_laplace.._ilt_addcs$|\}}|jr t||Sr{)rrr})rWrr)rrrgrhrIs z'_fast_inverse_laplace.._ilt_mulcs|}|durM|||}}}|jr>|dkr>| dt|| || t| S|dkrMt|| |Str)rrr(rEr})rWrnmamr<)rrrrrrgrhrOs4z'_fast_inverse_laplace.._ilt_powcs,|jj}|jj\}t|jt|t|Sr{)funr variablesrXpolyrrV)rWrvariablerrgrhrYs z+_fast_inverse_laplace.._ilt_rootsum)rr)rWrrrg) rrrrrrrrrrrh_fast_inverse_laplace4s  r c Cst||t|tj|||tjtjf}|ts$t||tj fSt||tjtjf}|tjtjtj fvs=|trCt ||d|j sLt ||d|j d\}} |tr^t ||dt||| fS)z Compute a general Fourier-type transform .. math:: F(k) = a \int_{-\infty}^{\infty} e^{bixk} f(x)\, dx. For suitable choice of *a* and *b*, this reduces to the standard Fourier and inverse Fourier transforms. z$function not integrable on real axisrrr)rHr(rrfrrrrIrrNaNr^rrq) rrrCrrrrr integral_frrgrgrh_fourier_transformes.     r c@0eZdZdZddZddZddZdd Zd S) FourierTypeTransformz# Base class for Fourier transforms.cCtd|jNz,Class %s must implement a(self) but does notr}rfrrrgrgrhrzFourierTypeTransform.acCrNz,Class %s must implement b(self) but does notrrrrgrgrhrrzFourierTypeTransform.bcKs&t||||||jjfi|Sr{)r rrrfrrbrrrCrrgrgrhrs  z'FourierTypeTransform._compute_transformcCs>|}|}t||t|tj|||tjtjfSr{)rrrIr(rrfrr)rbrrrCrrrgrgrhrs.z!FourierTypeTransform._as_integralNrirjrkrlrrrrrgrgrgrhrs  rc@$eZdZdZdZddZddZdS)FourierTransformz Class representing unevaluated Fourier transforms. For usage of this class, see the :class:`IntegralTransform` docstring. For how to compute Fourier transforms, see the :func:`fourier_transform` docstring. FouriercCdSrrgrrrgrgrhrrzFourierTransform.acC dtjS)Nrrrgrrrgrgrhr zFourierTransform.bNrirjrkrlrrrrgrgrgrhr   rcKr)a Compute the unitary, ordinary-frequency Fourier transform of ``f``, defined as .. math:: F(k) = \int_{-\infty}^\infty f(x) e^{-2\pi i x k} \mathrm{d} x. Explanation =========== If the transform cannot be computed in closed form, this function returns an unevaluated :class:`FourierTransform` object. For other Fourier transform conventions, see the function :func:`sympy.integrals.transforms._fourier_transform`. For a description of possible hints, refer to the docstring of :func:`sympy.integrals.transforms.IntegralTransform.doit`. Note that for this transform, by default ``noconds=True``. Examples ======== >>> from sympy import fourier_transform, exp >>> from sympy.abc import x, k >>> fourier_transform(exp(-x**2), x, k) sqrt(pi)*exp(-pi**2*k**2) >>> fourier_transform(exp(-x**2), x, k, noconds=False) (sqrt(pi)*exp(-pi**2*k**2), True) See Also ======== inverse_fourier_transform sine_transform, inverse_sine_transform cosine_transform, inverse_cosine_transform hankel_transform, inverse_hankel_transform mellin_transform, laplace_transform Nrg)rrrrrCrrgrgrhfourier_transform'r!c@r)InverseFourierTransformz Class representing unevaluated inverse Fourier transforms. For usage of this class, see the :class:`IntegralTransform` docstring. For how to compute inverse Fourier transforms, see the :func:`inverse_fourier_transform` docstring. zInverse FouriercCrrrgrrrgrgrhrrzInverseFourierTransform.acCrrdrrrrgrgrhrrzInverseFourierTransform.bNrrgrgrgrhr#rr#cKr)a Compute the unitary, ordinary-frequency inverse Fourier transform of `F`, defined as .. math:: f(x) = \int_{-\infty}^\infty F(k) e^{2\pi i x k} \mathrm{d} k. Explanation =========== If the transform cannot be computed in closed form, this function returns an unevaluated :class:`InverseFourierTransform` object. For other Fourier transform conventions, see the function :func:`sympy.integrals.transforms._fourier_transform`. For a description of possible hints, refer to the docstring of :func:`sympy.integrals.transforms.IntegralTransform.doit`. Note that for this transform, by default ``noconds=True``. Examples ======== >>> from sympy import inverse_fourier_transform, exp, sqrt, pi >>> from sympy.abc import x, k >>> inverse_fourier_transform(sqrt(pi)*exp(-(pi*k)**2), k, x) exp(-x**2) >>> inverse_fourier_transform(sqrt(pi)*exp(-(pi*k)**2), k, x, noconds=False) (exp(-x**2), True) See Also ======== fourier_transform sine_transform, inverse_sine_transform cosine_transform, inverse_cosine_transform hankel_transform, inverse_hankel_transform mellin_transform, laplace_transform Nrg)r#rrrCrrrgrgrhinverse_fourier_transformr"r%c Cst|||||||tjtjf}|ts!t||tjfS|js*t ||d|j d\}} |tr>> from sympy import sine_transform, exp >>> from sympy.abc import x, k, a >>> sine_transform(x*exp(-a*x**2), x, k) sqrt(2)*k*exp(-k**2/(4*a))/(4*a**(3/2)) >>> sine_transform(x**(-a), x, k) 2**(1/2 - a)*k**(a - 1)*gamma(1 - a/2)/gamma(a/2 + 1/2) See Also ======== fourier_transform, inverse_fourier_transform inverse_sine_transform cosine_transform, inverse_cosine_transform hankel_transform, inverse_hankel_transform mellin_transform, laplace_transform Nrg)r+rr rgrgrhsine_transforma $r3c@r*)InverseSineTransformz Class representing unevaluated inverse sine transforms. For usage of this class, see the :class:`IntegralTransform` docstring. For how to compute inverse sine transforms, see the :func:`inverse_sine_transform` docstring. z Inverse SinecCr-rdr.rrrgrgrhr rzInverseSineTransform.acCr/r{r0rrrgrgrhr rzInverseSineTransform.bNr1rgrgrgrhr5 r2r5cKr)am Compute the unitary, ordinary-frequency inverse sine transform of `F`, defined as .. math:: f(x) = \sqrt{\frac{2}{\pi}} \int_{0}^\infty F(k) \sin(2\pi x k) \mathrm{d} k. Explanation =========== If the transform cannot be computed in closed form, this function returns an unevaluated :class:`InverseSineTransform` object. For a description of possible hints, refer to the docstring of :func:`sympy.integrals.transforms.IntegralTransform.doit`. Note that for this transform, by default ``noconds=True``. Examples ======== >>> from sympy import inverse_sine_transform, exp, sqrt, gamma >>> from sympy.abc import x, k, a >>> inverse_sine_transform(2**((1-2*a)/2)*k**(a - 1)* ... gamma(-a/2 + 1)/gamma((a+1)/2), k, x) x**(-a) >>> inverse_sine_transform(sqrt(2)*k*exp(-k**2/(4*a))/(4*sqrt(a)**3), k, x) x*exp(-a*x**2) See Also ======== fourier_transform, inverse_fourier_transform sine_transform cosine_transform, inverse_cosine_transform hankel_transform, inverse_hankel_transform mellin_transform, laplace_transform Nrg)r5rr$rgrgrhinverse_sine_transform s%r6c@r*)CosineTransformz Class representing unevaluated cosine transforms. For usage of this class, see the :class:`IntegralTransform` docstring. For how to compute cosine transforms, see the :func:`cosine_transform` docstring. CosinecCr-rdr.rrrgrgrhr rzCosineTransform.acCr/r{r0rrrgrgrhr rzCosineTransform.bN rirjrkrlrr6r)rrrgrgrgrhr7 r2r7cKr)a: Compute the unitary, ordinary-frequency cosine transform of `f`, defined as .. math:: F(k) = \sqrt{\frac{2}{\pi}} \int_{0}^\infty f(x) \cos(2\pi x k) \mathrm{d} x. Explanation =========== If the transform cannot be computed in closed form, this function returns an unevaluated :class:`CosineTransform` object. For a description of possible hints, refer to the docstring of :func:`sympy.integrals.transforms.IntegralTransform.doit`. Note that for this transform, by default ``noconds=True``. Examples ======== >>> from sympy import cosine_transform, exp, sqrt, cos >>> from sympy.abc import x, k, a >>> cosine_transform(exp(-a*x), x, k) sqrt(2)*a/(sqrt(pi)*(a**2 + k**2)) >>> cosine_transform(exp(-a*sqrt(x))*cos(a*sqrt(x)), x, k) a*exp(-a**2/(2*k))/(2*k**(3/2)) See Also ======== fourier_transform, inverse_fourier_transform, sine_transform, inverse_sine_transform inverse_cosine_transform hankel_transform, inverse_hankel_transform mellin_transform, laplace_transform Nrg)r7rr rgrgrhcosine_transform r4r:c@r*)InverseCosineTransformz Class representing unevaluated inverse cosine transforms. For usage of this class, see the :class:`IntegralTransform` docstring. For how to compute inverse cosine transforms, see the :func:`inverse_cosine_transform` docstring. zInverse CosinecCr-rdr.rrrgrgrhr rzInverseCosineTransform.acCr/r{r0rrrgrgrhr rzInverseCosineTransform.bNr9rgrgrgrhr; r2r;cKr)a( Compute the unitary, ordinary-frequency inverse cosine transform of `F`, defined as .. math:: f(x) = \sqrt{\frac{2}{\pi}} \int_{0}^\infty F(k) \cos(2\pi x k) \mathrm{d} k. Explanation =========== If the transform cannot be computed in closed form, this function returns an unevaluated :class:`InverseCosineTransform` object. For a description of possible hints, refer to the docstring of :func:`sympy.integrals.transforms.IntegralTransform.doit`. Note that for this transform, by default ``noconds=True``. Examples ======== >>> from sympy import inverse_cosine_transform, sqrt, pi >>> from sympy.abc import x, k, a >>> inverse_cosine_transform(sqrt(2)*a/(sqrt(pi)*(a**2 + k**2)), k, x) exp(-a*x) >>> inverse_cosine_transform(1/sqrt(k), k, x) 1/sqrt(x) See Also ======== fourier_transform, inverse_fourier_transform, sine_transform, inverse_sine_transform cosine_transform hankel_transform, inverse_hankel_transform mellin_transform, laplace_transform Nrg)r;rr$rgrgrhinverse_cosine_transform r4r<cCst|t|||||tjtjf}|ts t||tjfS|j s)t ||d|j d\}}|tr;t ||dt|||fS)zv Compute a general Hankel transform .. math:: F_\nu(k) = \int_{0}^\infty f(r) J_\nu(k r) r \mathrm{d} r. rrr) rHr<rrrrrIrrrr^rq)rrrCrTrrrrrgrgrh_hankel_transform> s&    r=c@s4eZdZdZddZddZddZedd Zd S) HankelTypeTransformz+ Base class for Hankel transforms. cKs$|j|j|j|j|jdfi|Sr)rrarurwrq)rbrrgrgrhrY szHankelTypeTransform.doitcKst|||||jfi|Sr{)r=r)rbrrrCrTrrgrgrhr` sz&HankelTypeTransform._compute_transformcCs&t|t|||||tjtjfSr{)rIr<rrr)rbrrrCrTrgrgrhrc s&z HankelTypeTransform._as_integralcCs||j|j|j|jdSr)rrarurwrqrrrgrgrhrf s zHankelTypeTransform.as_integralN) rirjrkrlrrrrrrgrgrgrhr>T sr>c@eZdZdZdZdS)HankelTransformz Class representing unevaluated Hankel transforms. For usage of this class, see the :class:`IntegralTransform` docstring. For how to compute Hankel transforms, see the :func:`hankel_transform` docstring. HankelNrirjrkrlrrgrgrgrhr@n  r@cKt||||jdi|S)a Compute the Hankel transform of `f`, defined as .. math:: F_\nu(k) = \int_{0}^\infty f(r) J_\nu(k r) r \mathrm{d} r. Explanation =========== If the transform cannot be computed in closed form, this function returns an unevaluated :class:`HankelTransform` object. For a description of possible hints, refer to the docstring of :func:`sympy.integrals.transforms.IntegralTransform.doit`. Note that for this transform, by default ``noconds=True``. Examples ======== >>> from sympy import hankel_transform, inverse_hankel_transform >>> from sympy import exp >>> from sympy.abc import r, k, m, nu, a >>> ht = hankel_transform(1/r**m, r, k, nu) >>> ht 2*k**(m - 2)*gamma(-m/2 + nu/2 + 1)/(2**m*gamma(m/2 + nu/2)) >>> inverse_hankel_transform(ht, k, r, nu) r**(-m) >>> ht = hankel_transform(exp(-a*r), r, k, 0) >>> ht a/(k**3*(a**2/k**2 + 1)**(3/2)) >>> inverse_hankel_transform(ht, k, r, 0) exp(-a*r) See Also ======== fourier_transform, inverse_fourier_transform sine_transform, inverse_sine_transform cosine_transform, inverse_cosine_transform inverse_hankel_transform mellin_transform, laplace_transform Nrg)r@r)rrrCrTrrgrgrhhankel_transform{ .rEc@r?)InverseHankelTransformz Class representing unevaluated inverse Hankel transforms. For usage of this class, see the :class:`IntegralTransform` docstring. For how to compute inverse Hankel transforms, see the :func:`inverse_hankel_transform` docstring. zInverse HankelNrBrgrgrgrhrG rCrGcKrD)a Compute the inverse Hankel transform of `F` defined as .. math:: f(r) = \int_{0}^\infty F_\nu(k) J_\nu(k r) k \mathrm{d} k. Explanation =========== If the transform cannot be computed in closed form, this function returns an unevaluated :class:`InverseHankelTransform` object. For a description of possible hints, refer to the docstring of :func:`sympy.integrals.transforms.IntegralTransform.doit`. Note that for this transform, by default ``noconds=True``. Examples ======== >>> from sympy import hankel_transform, inverse_hankel_transform >>> from sympy import exp >>> from sympy.abc import r, k, m, nu, a >>> ht = hankel_transform(1/r**m, r, k, nu) >>> ht 2*k**(m - 2)*gamma(-m/2 + nu/2 + 1)/(2**m*gamma(m/2 + nu/2)) >>> inverse_hankel_transform(ht, k, r, nu) r**(-m) >>> ht = hankel_transform(exp(-a*r), r, k, 0) >>> ht a/(k**3*(a**2/k**2 + 1)**(3/2)) >>> inverse_hankel_transform(ht, k, r, 0) exp(-a*r) See Also ======== fourier_transform, inverse_fourier_transform sine_transform, inverse_sine_transform cosine_transform, inverse_cosine_transform hankel_transform mellin_transform, laplace_transform Nrg)rGr)rrCrrTrrgrgrhinverse_hankel_transform rFrH)F)Tr{)rl functoolsrr itertoolsr sympy.corerrrsympy.core.addrsympy.core.functionr r r r r rrrrsympy.core.mulrsympy.core.numbersrrsympy.core.relationalrrrrrrsympy.core.sortingrrsympy.core.symbolrrrsympy.core.traversalr (sympy.functions.combinatorial.factorialsr!r"$sympy.functions.elementary.complexesr#r$r%r&r'&sympy.functions.elementary.exponentialr(r)r*%sympy.functions.elementary.hyperbolicr+r,r-r.r/#sympy.functions.elementary.integersr0(sympy.functions.elementary.miscellaneousr1r2r3$sympy.functions.elementary.piecewiser4r5(sympy.functions.elementary.trigonometricr6r7r8r9r:sympy.functions.special.besselr;r<r=r>'sympy.functions.special.delta_functionsr?r@'sympy.functions.special.error_functionsrArBrC'sympy.functions.special.gamma_functionsrDrErFsympy.functions.special.hyperrGsympy.integralsrHrIrrJsympy.logic.boolalgrKrLrMrNrOsympy.matrices.matricesrPsympy.polys.matrices.linsolverQrRsympy.polys.polyrootsrSsympy.polys.polytoolsrTrUsympy.polys.rationaltoolsrVsympy.polys.rootoftoolsrWrXsympy.utilities.exceptionsrYrZr[sympy.utilities.iterablesr\sympy.utilities.miscr]r}r^rnrr_nocondsrrrrrrQrrDrFrbrZrhryr{rrrrrrrrrrrrrrrr r rrr!r#r%r'r(r+r3r5r6r7r:r;r<r=r>r@rErGrHrgrgrgrhs  ,           D!,-- :4<X w '   5 ) 5b P #01 *. 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