U Zʹj$@sdZddlmZmZddlmZddlmZmZddl m Z ddl m Z m Z mZmZmZddlmZddlmZmZdd lmZdd lmZdd lmZdd lmZmZdd lm Z m!Z!m"Z"ddl#m$Z$m%Z%ddl&m'Z'm(Z(m)Z)m*Z*ddl+m,Z,ddl-m.Z.m/Z/m0Z0ddl1m2Z2ddl3m4Z4m5Z5m6Z6m7Z7ddl8m9Z9ddl:m;Z;ddlm?Z?ddl@mAZAmBZBddlCmDZDddlEmFZFmGZGmHZHmIZImJZJddlKmLZLddlMmNZNmOZOddlPmQZQddlRmSZSddlTmUZUGd d!d!eVZWGd"d#d#eZXd$d%ZYd&d'ZZeZd(Z[d)d*Z\e[e\d+fd,d-Z]Gd.d/d/eXZ^d0d1Z_d2d3Z`Gd4d5d5eaZbd6d7ZceZd+dkd8d9Zdd:aeGd;d<dZgeZd+dld?d@ZhGdAdBdBeXZiGdCdDdDeiZjdEdFZkGdGdHdHeiZldIdJZmeZd+dmdKdLZnGdMdNdNeXZoGdOdPdPeoZpdQdRZqGdSdTdTeoZrdUdVZsGdWdXdXeoZtdYdZZuGd[d\d\eoZvd]d^ZweZd+dnd_d`ZxGdadbdbeXZyGdcddddeyZzdedfZ{GdgdhdheyZ|didjZ}dd:l~mmZejZejZejZejZejZejZd:S)oz Integral Transforms )reducewraps)repeat)Spi)Add) AppliedUndef count_opsexpand expand_mulFunction)Mul)igcdilcm)default_sort_key)Dummy)postorder_traversal) factorialrf)reargAbs)exp exp_polar)coshcothsinhtanh)ceiling)MaxMinsqrt)piecewise_fold)coscotsintan)besselj) Heaviside)gamma)meijerg) integrateIntegral)_dummy)to_cnf conjuncts disjunctsOrAnd)roots)factorPoly)CRootOf)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)selfZ transformr<msg __class__Y/var/www/html/bwave/python/venv/lib/python3.8/site-packages/sympy/integrals/transforms.pyr;6s zIntegralTransformError.__init__)__name__ __module__ __qualname____doc__r; __classcell__rArAr?rBr9(s r9c@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. cCs |jdS)z! The function to be transformed. rargsr=rArArBr<SszIntegralTransform.functioncCs |jdS)z; The dependent variable of the function to be transformed. rIrKrArArBfunction_variableXsz#IntegralTransform.function_variablecCs |jdS)z% The independent transform variable. rIrKrArArBtransform_variable]sz$IntegralTransform.transform_variablecCs|jj|jh|jhS)zj This method returns the symbols that will exist when the transform is evaluated. )r< free_symbolsunionrOrMrKrArArBrPbszIntegralTransform.free_symbolscKstdSNNotImplementedErrorr=fxshintsrArArB_compute_transformksz$IntegralTransform._compute_transformcCstdSrRrSr=rVrWrXrArArB _as_integralnszIntegralTransform._as_integralcCs$t|}|dkr t|jjdd|S)NF)r2r9r@name)r=extracondrArArB_collapse_extraqsz!IntegralTransform._collapse_extrac sd}tfddjtD }|rfzjjjjf|}Wn tk rdtdd}YnXj}|j szt |}||fS)Nc3s|]}|jVqdSrR)hasrM).0funcrKrArB ysz2IntegralTransform._try_directly..z6[IT _try ] Caught IntegralTransformError, returns None) anyr<atomsrrZrMrOr9r8is_Addr )r=rYTZ try_directlyfnrArKrB _try_directlyws&   zIntegralTransform._try_directlyc  sdd}dd}|d<jf\}}|dk r<|S|jr<|d<fdd|jD}g}g}|D]\} t| ts| g} || dt| d kr|| d qnt| d krn|| d dg7}qn|dkrt| }nt|}|s|Sz2 |}t |r|ft|WS||fWSWnt k r:YnX|rTt 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)``. needevalFsimplifyTNcs2g|]*}j|gtjddjfqS)rLN)r@listrJdoitrcrWrYr=rArB sz*IntegralTransform.doit..rrNrL)poprkrhrJ isinstancetupleappendlenrrmrar7r9r@_namer< as_coeff_mulrMr rn) r=rYrlrmrjriresr_ressrWcoeffrestrArqrBrosR        zIntegralTransform.doitcCs||j|j|jSrR)r\r<rMrOrKrArArB as_integrals zIntegralTransform.as_integralcOs|jSrR)r~)r=rJkwargsrArArB_eval_rewrite_as_Integralsz+IntegralTransform._eval_rewrite_as_IntegralN)rCrDrErFpropertyr<rMrOrPrZr\rarkror~rrArArArBrH<s"    G rHcCs4|r0ddlm}ddlm}||t|ddS|S)Nr)rm) powdenestT)Zpolar)sympy.simplifyrmZsympy.simplify.powsimprr")exprrormrrArArB _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||}|r|dS|SNrrA)rrJrrzrdrArBwrappers z0_noconds_..make_wrapper..wrapper)r)rdrdefaultrrB make_wrappersz_noconds_..make_wrapperrA)rrrArrB _noconds_s rFcCst||tjtjfSrR)r+rZeroInfinity)rVrWrArArB_default_integratorsrTc stdd|||d||}|tsNt|||tjtjftjfS|j s`t d|d|j d\}}|trt d|dfdd fd d t |D}d d |D}|j d dd|st d|d|d\}} } t||||| f| fS)z0 Backend function to compute Mellin transforms. rXzmellin-transformrLMellincould not compute integralrintegral in unexpected formcsRddlm}tj}tj}tj}tt|}tddd}|D] }tj}tj} g} t |D]} | t dd t |} | j r| jdks| s| |s| | g7} qX|| |} | j r| jdkr| | g7} qX| j|krt| j| } qXt| j|}qX|tjk r||krt||}q:| tjk r8| |kr8t| |}q:t|t| }q:|||fS) zN Turn ``cond`` into a strip (a, b), and auxiliary conditions. r)_solve_inequalitytT)realcSs |dSr) as_real_imagrWrArArB)z:_mellin_transform..process_conds..)z==z!=)Zsympy.solvers.inequalitiesrrNegativeInfinityrtruer/r.rr0replacersubsZ is_RelationalZrel_oprbZltsrZgtsr r2r1)r`rabauxcondsrca_b_Zaux_dZd_Zsoln)rXrArB process_condssT           z(_mellin_transform..process_condscsg|] }|qSrArArcr)rrArBrr@sz%_mellin_transform..cSsg|]}|ddkr|qS)rNFrArprArArBrrAs cSs|d|dt|dfS)NrrLrN)r rrArArBrBrz#_mellin_transform..keyzno convergence found)r-rbr,rrrrrr is_Piecewiser9rJr0sort) rVrWZs_Z integratorrmFr`rrrrrA)rrXrB_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|||f|SrR)rrUrArArBrZWsz"MellinTransform._compute_transformcCs t|||d|tjtjfSNrL)r,rrrr[rArArBr\ZszMellinTransform._as_integralc Csg}g}g}|D]*\\}}}||g7}||g7}||g7}qt|t|ft|f}|dd|ddkdks||ddkrtddd|S)NrrLTFrzno combined convergence.)rr r2r9) r=r_rrr`sasbrrzrArArBra]s   (zMellinTransform._collapse_extraN)rCrDrErFrxrZr\rarArArArBrKs  rcKst|||jf|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 )rro)rVrWrXrYrArArBmellin_transformls)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) rrL)r rrrr))Zm_nrXrrmnrrArArB _rewrite_sins %  rc@seZdZdZdS)MellinTransformStripErrorzF Exception raised by _rewrite_gamma. Mainly for internal use. N)rCrDrErFrArArArBrsrc-st||g\fdd}g}tD]F}|s>> 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|}dkr"tjkr"dSdkr2|kSdkrB|kS|kdkrRdS|kdkrbdS|rjdSjs|js||jrdStddS)zU Decide whether pole at c lies to the left of the fundamental strip. NTFzPole inside critical strip?)r rrrrPr)ris_numer)rrrArBleft s    z_rewrite_gamma..leftrrLcSsg|]}|jrt|n|qSrA)is_extended_realrrprArArBrr8sz"_rewrite_gamma..csg|] }|qSrArArp)common_coefficientrArBrr>scss|] }|jVqdSrR) is_RationalrprArArBre?sz!_rewrite_gamma..GammaNzNonrational multipliercSsg|]}t|jqSrA)rqrprArArBrrBscSsg|]}t|jqSrA)rprprArArBrrIsTFcstdd|S)NInverse MellinzUnrecognised form '%s'.)r9)fact)rVrArB exceptionasz!_rewrite_gamma..exceptioncs8|st|}|dkr0|S)z7 Test if arg is of form a*s+b, raise exception if not. rL) is_polynomialr5degree all_coeffs)rr)rrrXrArB linear_argls    z"_rewrite_gamma..linear_argcsg|]}|fqSrArAr)rrXrArBrrsrz Gammas partially over the strip.)evaluaterNza is not an integerr)6rrgr)rbrJrhZas_independentryr%r#r&r$rOnerallrr9rrrwrrZas_numer_denomr Z make_argsrnziprrsZis_PowrtrbaserZ is_Integerrrr5rLTr3r6Z all_rootsrZ NegativeOnerTr TypeErrorrangeZHalfrvrr)-rVrXrrrZ s_multipliersgrr|_rWZ s_multiplierfacexponentZnumerdenomrJZfacsZdfacsZ numer_gammasZ denom_gammasZ exponentialsZugammasZlgammasZufacsrrZexp_r`rrsrZgamma1Zgamma2Zfac_ZanapZbmbqZgammasplusminusZnewaZnewckrA)rrrrrVrrrXrB_rewrite_gammasR?               $                        &              &&     rc sbtdd|dd|t}t|t|t|fD]}|jrfdd|jD}dd|D}dd|D}t|}st|| t d }| |t |fSz$t |d d \} } } } } Wntk rYq0YnXzt| | | | }Wntk rYq0YnXr&|}nzd d lm}||}Wn"tk r`td |dYnX|jrt|jdkrt t| |jd jd t t| |jd jd }tt|j|jtkg}|t tt|jt|jkd t|jd ktt|j|jtkg7}t|}|dkr:td |d||  ||fStd |ddS)zs A helper for the real inverse_mellin_transform function, this one here assumes x to be real and positive. rzinverse-mellin-transformT)Zpositivec s g|]}t|ddqS)Fr)_inverse_mellin_transform)rcG as_meijergrXstriprWrArBrrs z-_inverse_mellin_transform..cSsg|] }|dqS)rLrArcrrArArBrrscSsg|] }|dqS)rrArrArArBrrs)ZgensrrL) hyperexpandrzCould not calculate integralFzdoes not converger]N) r-Zrewriter)r4r r rhrJrrgr(rr2rr9r* ValueErrorrrrTrrwrrargumentdeltarr1rrrnu)rrXZx_rrrr{rrzrrCerrhrr`rArrBrsb $     * rNc@sHeZdZdZdZedZedZddZe 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. rNonercKs4|dkrtj}|dkrtj}tj||||||f|SrR)r_none_sentinelrH__new__)clsrrXrWrroptsrArArBrCs zInverseMellinTransform.__new__cCs:|jd|jd}}|tjkr$d}|tjkr2d}||fS)Nr)rJrr)r=rrrArArBfundamental_stripJs   z(InverseMellinTransform.fundamental_stripc Ks|ddtdkr0tttttttt t t t t h at|D].}|jr8||r8|jtkr8td|d|q8|j}t||||f|S)NrmTrzComponent %s not recognised.)rs_allowedrr)r%r#r&r$rrrrrrrZ is_Functionrbrdr9rr)r=rrXrWrYrVrrArArBrZSs,  z)InverseMellinTransform._compute_transformcCsJ|jj}t||| ||tjtj|tjtjfdtjtjSNrN)r@_cr,r ImaginaryUnitrPi)r=rrXrWrrArArBr\cs   z#InverseMellinTransform._as_integralN) rCrDrErFrxrrrrrrrZr\rArArArBr5s  rcKs t||||d|djf|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 rrL)rro)rrXrWrrYrArArBinverse_mellin_transformis5rc Cst||t|tj|||tjtjf}|tsHt||tj fSt||tjtjf}|tjtjtj fksz|trt ||d|j st ||d|j d\}} |trt ||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$function not integrable on real axisrrr)r+rrrrrrbr,rrNaNr9rrJ) rVrWrrrr^rmrZ integral_fr`rArArB_fourier_transforms .     rc@s0eZdZdZddZddZddZdd Zd S) FourierTypeTransformz# Base class for Fourier transforms.cCstd|jdSNz,Class %s must implement a(self) but does notrTr@rKrArArBrszFourierTypeTransform.acCstd|jdSNz,Class %s must implement b(self) but does notrrKrArArBrszFourierTypeTransform.bcKs"t||||||jjf|SrR)rrrr@rxr=rVrWrrYrArArBrZsz'FourierTypeTransform._compute_transformcCs>|}|}t||t|tj|||tjtjfSrR)rrr,rrrrr)r=rVrWrrrrArArBr\sz!FourierTypeTransform._as_integralNrCrDrErFrrrZr\rArArArBrs rc@s$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. ZFouriercCsdSrrArKrArArBrszFourierTransform.acCs dtjS)NrrrKrArArBrszFourierTransform.bNrCrDrErFrxrrrArArArBrs rcKst|||jf|S)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 )rrorVrWrrYrArArBfourier_transforms'rc@s$eZdZdZdZddZddZdS)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 FouriercCsdSrrArKrArArBr#szInverseFourierTransform.acCs dtjSrrrKrArArBr&szInverseFourierTransform.bNrrArArArBr s r cKst|||jf|S)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 )r rorrrWrYrArArBinverse_fourier_transform*s'r c Cst|||||||tjtjf}|tsBt||tjfS|jsTt ||d|j d\}} |trxt ||dt||| fS)a  Compute a general sine or cosine-type transform F(k) = a int_0^oo b*sin(x*k) f(x) dx. F(k) = a int_0^oo b*cos(x*k) f(x) dx. For suitable choice of a and b, this reduces to the standard sine/cosine and inverse sine/cosine transforms. rrr) r+rrrrbr,rrrr9rJ) rVrWrrrKr^rmrr`rArArB_sine_cosine_transformXs (    r c@s0eZdZdZddZddZddZdd Zd S) SineCosineTypeTransformzK Base class for sine and cosine transforms. Specify cls._kern. cCstd|jdSrrrKrArArBrwszSineCosineTypeTransform.acCstd|jdSrrrKrArArBr{szSineCosineTypeTransform.bcKs(t||||||jj|jjf|SrR)r rrr@_kernrxrrArArBrZsz*SineCosineTypeTransform._compute_transformcCs@|}|}|jj}t|||||||tjtjfSrR)rrr@rr,rrr)r=rVrWrrrr rArArBr\sz$SineCosineTypeTransform._as_integralNrrArArArBrqs rc@s(eZdZdZdZeZddZddZdS) SineTransformz Class representing unevaluated sine transforms. For usage of this class, see the :class:`IntegralTransform` docstring. For how to compute sine transforms, see the :func:`sine_transform` docstring. ZSinecCstdttSrr!rrKrArArBrszSineTransform.acCstjSrRrrrKrArArBrszSineTransform.bN rCrDrErFrxr%rrrrArArArBrs  rcKst|||jf|S)a1 Compute the unitary, ordinary-frequency sine transform of `f`, defined as .. math:: F(k) = \sqrt{\frac{2}{\pi}} \int_{0}^\infty f(x) \sin(2\pi x k) \mathrm{d} x. Explanation =========== If the transform cannot be computed in closed form, this function returns an unevaluated :class:`SineTransform` 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 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 )rrorrArArBsine_transforms$rc@s(eZdZdZdZeZddZddZdS)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 SinecCstdttSrrrKrArArBrszInverseSineTransform.acCstjSrRrrKrArArBrszInverseSineTransform.bNrrArArArBrs  rcKst|||jf|S)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 )rror rArArBinverse_sine_transforms%rc@s(eZdZdZdZeZddZddZdS)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. ZCosinecCstdttSrrrKrArArBrszCosineTransform.acCstjSrRrrKrArArBrszCosineTransform.bN rCrDrErFrxr#rrrrArArArBrs  rcKst|||jf|S)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 )rrorrArArBcosine_transforms$rc@s(eZdZdZdZeZddZddZdS)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 CosinecCstdttSrrrKrArArBrLszInverseCosineTransform.acCstjSrRrrKrArArBrOszInverseCosineTransform.bNrrArArArBr?s  rcKst|||jf|S)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 )rror rArArBinverse_cosine_transformSs$rcCst|t|||||tjtjf}|ts@t||tjfS|j sRt ||d|j d\}}|trvt ||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) r+r'rrrrbr,rrrr9rJ)rVrrrr^rmrr`rArArB_hankel_transform~s&    rc@s4eZdZdZddZddZddZedd Zd S) HankelTypeTransformz+ Base class for Hankel transforms. cKs |j|j|j|j|jdf|SNr)rZr<rMrOrJ)r=rYrArArBroszHankelTypeTransform.doitcKst|||||jf|SrR)rrx)r=rVrrrrYrArArBrZsz&HankelTypeTransform._compute_transformcCs&t|t|||||tjtjfSrR)r,r'rrr)r=rVrrrrArArBr\sz HankelTypeTransform._as_integralcCs||j|j|j|jdSr)r\r<rMrOrJrKrArArBr~s zHankelTypeTransform.as_integralN) rCrDrErFrorZr\rr~rArArArBrs rc@seZdZdZdZdS)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. ZHankelNrCrDrErFrxrArArArBrs rcKst||||jf|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 )rro)rVrrrrYrArArBhankel_transforms.r!c@seZdZdZdZdS)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 HankelNr rArArArBr"s r"cKst||||jf|S)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 )r"ro)rrrrrYrArArBinverse_hankel_transforms.r#)F)T)T)T)rF functoolsrr itertoolsrZ sympy.corerrZsympy.core.addrZsympy.core.functionrr r r r Zsympy.core.mulr Zsympy.core.intfuncrrZsympy.core.sortingrZsympy.core.symbolrZsympy.core.traversalrZ(sympy.functions.combinatorial.factorialsrrZ$sympy.functions.elementary.complexesrrrZ&sympy.functions.elementary.exponentialrrZ%sympy.functions.elementary.hyperbolicrrrrZ#sympy.functions.elementary.integersrZ(sympy.functions.elementary.miscellaneousrr r!Z$sympy.functions.elementary.piecewiser"Z(sympy.functions.elementary.trigonometricr#r$r%r&Zsympy.functions.special.besselr'Z'sympy.functions.special.delta_functionsr(Z'sympy.functions.special.gamma_functionsr)Zsympy.functions.special.hyperr*Zsympy.integralsr+r,Zsympy.integrals.meijerintr-Zsympy.logic.boolalgr.r/r0r1r2Zsympy.polys.polyrootsr3Zsympy.polys.polytoolsr4r5Zsympy.polys.rootoftoolsr6Zsympy.utilities.iterablesr7Zsympy.utilities.miscr8rTr9rHrrZ_nocondsrrrrrrrrrrrrrrrrr r r rrrrrrrrrrrrr!r"r#Zsympy.integrals.laplaceZ integralsZlaplaceZ_laplaceZLaplaceTransformZlaplace_transformZlaplace_correspondenceZlaplace_initial_condsZInverseLaplaceTransformZinverse_laplace_transformrArArArBs                 D!,-- :4< *. 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