o *ήc@sddlmZddlmZmZmZmZmZm Z m Z ddl m Z ddl mZddlmZmZmZmZmZddlmZmZddlmZmZmZddlmZdd lmZdd l m!Z!dd l"m#Z#Gd d d eZ$GdddeZ%GdddeZ&GdddeZ'GdddeZ(GdddeZ)GdddeZ*GdddeZ+GdddeZ,GdddeZ-d d!Z.Gd"d#d#eZ/d/d%d&Z0d0d(d)Z1d/d*d+Z2d1d-d.Z3d,S)2)Tuple)SAddMulsympifySymbolDummyBasic)Expr) factor_terms)Function DerivativeArgumentIndexError AppliedUndef expand_mul) fuzzy_notfuzzy_or)piIoo)Pow)Eq)sqrt) Piecewisec@peZdZUdZeeed<dZdZdZ e ddZ dddZ dd Z d d Zd d ZddZddZddZdS)rea Returns real part of expression. This function performs only elementary analysis and so it will fail to decompose properly more complicated expressions. If completely simplified result is needed then use ``Basic.as_real_imag()`` or perform complex expansion on instance of this function. Examples ======== >>> from sympy import re, im, I, E, symbols >>> x, y = symbols('x y', real=True) >>> re(2*E) 2*E >>> re(2*I + 17) 17 >>> re(2*I) 0 >>> re(im(x) + x*I + 2) 2 >>> re(5 + I + 2) 7 Parameters ========== arg : Expr Real or complex expression. Returns ======= expr : Expr Real part of expression. See Also ======== im argsTc CsF|tjurtjS|tjurtjS|jr|S|jstj|jr!tjS|jr*|dS|j r9t |t r9t |j dSggg}}}t|}|D]9}|tj}|dur]|js\||qH|tjsl|jrl||qH|j|d}|r|||dqH||qHt|t|krdd|||fD\} } } || t| | SdS)Nrignorecs|]}t|VqdSNr.0xsr%K/tmp/pip-target-vg8gfxp4/lib/python/sympy/functions/elementary/complexes.py izre.eval..)rNaNComplexInfinityis_extended_real is_imaginary ImaginaryUnitZero is_Matrix as_real_imag is_Function isinstance conjugaterrr make_argsas_coefficientappendhaslenim clsargincludedrevertedexcludedrtermcoeff real_imagabcr%r%r&evalDs<         zre.evalcK |tjfS)zF Returns the real number with a zero imaginary part. rr.selfdeephintsr%r%r&r0m zre.as_real_imagcC`|js |jdjrtt|jd|ddS|js|jdjr.tj tt|jd|ddSdSNrTevaluate)r+rrr r,rr-r9rJxr%r%r&_eval_derivativetzre._eval_derivativecKs|jdtjt|jdSNr)rrr-r9rJr<kwargsr%r%r&_eval_rewrite_as_im{szre._eval_rewrite_as_imcC |jdjSrVr is_algebraicrJr%r%r&_eval_is_algebraic~ zre._eval_is_algebraiccCst|jdj|jdjgSrV)rrr,is_zeror]r%r%r& _eval_is_zeroszre._eval_is_zerocC|jdjrdSdSNrTr is_finiter]r%r%r&_eval_is_finite zre._eval_is_finitecCrbrcrdr]r%r%r&_eval_is_complexrgzre._eval_is_complexNT)__name__ __module__ __qualname____doc__tTupler __annotations__r+ unbranched_singularities classmethodrFr0rTrYr^rarfrhr%r%r%r&rs  )  ( rc@r)r9a Returns imaginary part of expression. This function performs only elementary analysis and so it will fail to decompose properly more complicated expressions. If completely simplified result is needed then use ``Basic.as_real_imag()`` or perform complex expansion on instance of this function. Examples ======== >>> from sympy import re, im, E, I >>> from sympy.abc import x, y >>> im(2*E) 0 >>> im(2*I + 17) 2 >>> im(x*I) re(x) >>> im(re(x) + y) im(y) >>> im(2 + 3*I) 3 Parameters ========== arg : Expr Real or complex expression. Returns ======= expr : Expr Imaginary part of expression. See Also ======== re rTc CsP|tjurtjS|tjurtjS|jrtjS|jstj|jr%tj |S|jr.|dS|j r>t |t r>t |j d Sggg}}}t|}|D]9}|tj}|durh|jsb||qM||qM|tjsq|js|j|d}|r||dqM||qMt|t|krdd|||fD\} } } || t| | SdS)Nrrcsrr r!r"r%r%r&r'r(zim.eval..)rr)r*r+r.r,r-r/r0r1r2r3r9rrr4r5r6r7r8rr:r%r%r&rFs<          zim.evalcKrG)zC Return the imaginary part with a zero real part. rHrIr%r%r&r0rMzim.as_real_imagcCrNrO)r+rr9r r,rr-rrRr%r%r&rTrUzim._eval_derivativecKs tj |jdt|jdSrV)rr-rrrWr%r%r&_eval_rewrite_as_re zim._eval_rewrite_as_recCrZrVr[r]r%r%r&r^r_zim._eval_is_algebraiccCrZrVrr+r]r%r%r&rar_zim._eval_is_zerocCrbrcrdr]r%r%r&rfrgzim._eval_is_finitecCrbrcrdr]r%r%r&rhrgzim._eval_is_complexNri)rjrkrlrmrnr ror+rprqrrrFr0rTrtr^rarfrhr%r%r%r&r9s  )  ' r9cseZdZdZdZdZfddZeddZddZ d d Z d d Z d dZ ddZ ddZddZddZddZd$ddZddZddZd d!Zd"d#ZZS)%signa Returns the complex sign of an expression: Explanation =========== If the expression is real the sign will be: * $1$ if expression is positive * $0$ if expression is equal to zero * $-1$ if expression is negative If the expression is imaginary the sign will be: * $I$ if im(expression) is positive * $-I$ if im(expression) is negative Otherwise an unevaluated expression will be returned. When evaluated, the result (in general) will be ``cos(arg(expr)) + I*sin(arg(expr))``. Examples ======== >>> from sympy import sign, I >>> sign(-1) -1 >>> sign(0) 0 >>> sign(-3*I) -I >>> sign(1 + I) sign(1 + I) >>> _.evalf() 0.707106781186548 + 0.707106781186548*I Parameters ========== arg : Expr Real or imaginary expression. Returns ======= expr : Expr Complex sign of expression. See Also ======== Abs, conjugate Tc s>t}||kr|jdjdur|jdt|jdS|S)NrF)superdoitrr`Abs)rJrLs __class__r%r&ryCs z sign.doitc CsD|jrX|\}}g}t|}|D].}|jr| }q|jrq|jr:t|}|jr4|tj 9}|jr3| }q| |q| |q|tj urOt |t |krOdS|||j |S|tjur`tjS|jrftjS|jrltj S|jrrtjS|jr|t|tr||S|jr|jr|jtjurtj Stj |}|jrtj S|jrtj SdSdSr )is_Mul as_coeff_mulrwis_extended_negativeis_extended_positiver,r9 is_comparablerr-r6Oner8 _new_rawargsr)r`r. NegativeOner1r2is_PowexpHalf) r;r<rErunkr{rCaiarg2r%r%r&rFIsT       z sign.evalcCst|jdjr tjSdSrV)rrr`rrr]r%r%r& _eval_Abs{szsign._eval_AbscCstt|jdSrV)rwr3rr]r%r%r&_eval_conjugatezsign._eval_conjugatecCs|jdjrddlm}dt|jd|dd||jdS|jdjrBddlm}dt|jd|dd|tj |jdSdS)Nr) DiracDeltaTrP)rr+'sympy.functions.special.delta_functionsrr r,rr-)rJrSrr%r%r&rTs     zsign._eval_derivativecCrbrc)ris_nonnegativer]r%r%r&_eval_is_nonnegativergzsign._eval_is_nonnegativecCrbrc)ris_nonpositiver]r%r%r&_eval_is_nonpositivergzsign._eval_is_nonpositivecCrZrV)rr,r]r%r%r&_eval_is_imaginaryr_zsign._eval_is_imaginarycCrZrVrvr]r%r%r&_eval_is_integerr_zsign._eval_is_integercCrZrV)rr`r]r%r%r&rar_zsign._eval_is_zerocCs.t|jdjr|jr|jrtjSdSdSdSrV)rrr` is_integeris_evenrr)rJotherr%r%r& _eval_powers zsign._eval_powerrcCsV|jd}||d}|dkr||S|dkr|||}t|dkr(tj StjSrV)rsubsfuncdirrrr)rJrSnlogxcdirarg0x0r%r%r& _eval_nseriess    zsign._eval_nseriescKs&|jrtd|dkfd|dkfdSdS)Nrsr)rT)r+rrWr%r%r&_eval_rewrite_as_Piecewiseszsign._eval_rewrite_as_PiecewisecKs&ddlm}|jr||ddSdS)Nr Heavisiderrsrrr+rJr<rXrr%r%r&_eval_rewrite_as_Heavisides zsign._eval_rewrite_as_HeavisidecKs tdt|df|t|dfSrc)rrrzrWr%r%r&_eval_rewrite_as_Absruzsign._eval_rewrite_as_AbscKs|t|jdSrV)rr r)rJrXr%r%r&_eval_simplifyszsign._eval_simplifyr)rjrkrlrm is_complexrqryrrrFrrrTrrrrrarrrrrr __classcell__r%r%r|r&rw s*6  1   rwc@seZdZUdZeeed<dZdZdZ dZ dZ d,ddZ e dd Zd d Zd d ZddZddZddZddZddZddZddZddZd-dd Zd!d"Zd#d$Zd%d&Zd'd(Zd)d*Zd+S).rzab Return the absolute value of the argument. Explanation =========== This is an extension of the built-in function ``abs()`` to accept symbolic values. If you pass a SymPy expression to the built-in ``abs()``, it will pass it automatically to ``Abs()``. Examples ======== >>> from sympy import Abs, Symbol, S, I >>> Abs(-1) 1 >>> x = Symbol('x', real=True) >>> Abs(-x) Abs(x) >>> Abs(x**2) x**2 >>> abs(-x) # The Python built-in Abs(x) >>> Abs(3*x + 2*I) sqrt(9*x**2 + 4) >>> Abs(8*I) 8 Note that the Python built-in will return either an Expr or int depending on the argument:: >>> type(abs(-1)) <... 'int'> >>> type(abs(S.NegativeOne)) Abs will always return a SymPy object. Parameters ========== arg : Expr Real or complex expression. Returns ======= expr : Expr Absolute value returned can be an expression or integer depending on input arg. See Also ======== sign, conjugate rTFrscCs |dkr t|jdSt||)zE Get the first derivative of the argument to Abs(). rsr)rwrr)rJargindexr%r%r&fdiffs z Abs.fdiffcsddlm}tdr}|dur|Stts"tdt|dd\}}|j r<|j s<||||Sj rg}g}j D];}|j rm|j jrm|j jrm||j} t| |rc||qF|t| |j qF||} t| |r|||qF|| qFt|}|r|t|ddntj}||StjurtjStjurtjSddlm } m} j r \} }| jr|jr|jrĈS| tjurtjSt| |S| j r| t!|S| j"r| t!|| tj# t$|SdS| %t&s | | '\}}|t(|}| t!||St| r| t!j dStt)r1j*r(Sjr/ SdSj+rM%tjtj,rMt-dd 'DrMtjSj.rTtj/Sj rZSj0ra Sj1rqtj2 }|j rq|SjrwdS|3dd4t34t3}|rt5fd d |DrdSkrӈ krՈ4t}6d d |D}d d|j D}|rt5fdd |Dst7t8SdSdSdS)Nr)signsimprzBad argument type for Abs(): %sFrP)rlogcss|]}|jVqdSr ) is_infiniter#rCr%r%r&r'PszAbs.eval..c3s |] }|jdVqdS)rN)r7rr#ir<r%r&r'bscSsi|]}|tddqS)T)real)rrr%r%r& fzAbs.eval..cSsg|] }|jdur|qSr )r+rr%r%r& gzAbs.eval..c3s|] }t|VqdSr )r7r3)r#u)conjr%r&r'hs)9sympy.simplify.simplifyrhasattrrr2r TypeErrortypeas_numer_denom free_symbolsr~rrrr is_negativebaser6rrrrr)r*Infinity&sympy.functions.elementary.exponentialr as_base_expr+rrrzis_extended_nonnegativerrPir9r7rr0rr is_positiveis_AddNegativeInfinityanyr`r.is_extended_nonpositiver,r-r3atomsallxreplacerr)r;r<robjrdknownrtbnewtnewrrrexponentrCrDzrnew_conjr abs_free_argr%)r<rr&rF s                   "      zAbs.evalcCrbrcrdr]r%r%r& _eval_is_realkrgzAbs._eval_is_realcC|jdjr |jdjSdSrV)rr+rr]r%r%r&ro  zAbs._eval_is_integercCt|jdjSrVr_argsr`r]r%r%r&_eval_is_extended_nonzeroszAbs._eval_is_extended_nonzerocCrZrV)rr`r]r%r%r&ravr_zAbs._eval_is_zerocCrrVrr]r%r%r&_eval_is_extended_positiveyrzAbs._eval_is_extended_positivecCrrV)rr+ is_rationalr]r%r%r&_eval_is_rational|rzAbs._eval_is_rationalcCrrV)rr+rr]r%r%r& _eval_is_evenrzAbs._eval_is_evencCrrV)rr+is_oddr]r%r%r& _eval_is_oddrzAbs._eval_is_oddcCrZrVr[r]r%r%r&r^r_zAbs._eval_is_algebraiccCsP|jdjr&|jr&|jr|jd|S|tjur&|jr&|jd|d|SdS)Nrrs)rr+rrrr is_Integer)rJrr%r%r&rs zAbs._eval_powerrcCsdddlm}|jd|d}|||r||||}|jdj|||d}t||S)Nr)r)rr) rrrleadtermr7rrrwexpand)rJrSrrrr directionr{r%r%r&rs zAbs._eval_nseriescCs|jdjs |jdjrt|jd|ddtt|jdSt|jdtt|jd|ddt|jdtt|jd|ddt|jd}| tSrO) rr+r,r rwr3rr9rzrewrite)rJrSrvr%r%r&rTs  zAbs._eval_derivativecKs,ddlm}|jr||||| SdS)Nrrrrr%r%r&rs zAbs._eval_rewrite_as_HeavisidecKsL|jrt||dkf| dfS|jr$tt|t|dkft |dfSdSrc)r+rr,rrWr%r%r&rs $zAbs._eval_rewrite_as_PiecewisecKs |t|Sr rwrWr%r%r&_eval_rewrite_as_signr_zAbs._eval_rewrite_as_signcKst|t|Sr )rr3rWr%r%r&_eval_rewrite_as_conjugaterzAbs._eval_rewrite_as_conjugateN)rsr)rjrkrlrmrnr ror+rrrprqrrrrFrrrrarrrrr^rrrTrrrrr%r%r%r&rzs6  9  `   rzc@s<eZdZdZdZdZdZdZeddZ ddZ ddZ d S) r<a Returns the argument (in radians) of a complex number. The argument is evaluated in consistent convention with ``atan2`` where the branch-cut is taken along the negative real axis and ``arg(z)`` is in the interval $(-\pi,\pi]$. For a positive number, the argument is always 0; the argument of a negative number is $\pi$; and the argument of 0 is undefined and returns ``nan``. So the ``arg`` function will never nest greater than 3 levels since at the 4th application, the result must be nan; for a real number, nan is returned on the 3rd application. Examples ======== >>> from sympy import arg, I, sqrt, Dummy >>> from sympy.abc import x >>> arg(2.0) 0 >>> arg(I) pi/2 >>> arg(sqrt(2) + I*sqrt(2)) pi/4 >>> arg(sqrt(3)/2 + I/2) pi/6 >>> arg(4 + 3*I) atan(3/4) >>> arg(0.8 + 0.6*I) 0.643501108793284 >>> arg(arg(arg(arg(x)))) nan >>> real = Dummy(real=True) >>> arg(arg(arg(real))) nan Parameters ========== arg : Expr Real or complex expression. Returns ======= value : Expr Returns arc tangent of arg measured in radians. Tc Cs |}tdD]}t||r|jd}q|dkr|jrtjSntjSddlm}t||r4t|t S|j sSt | \}}|j rLtdd|jD}t||}n|}tdd|tDrcdSdd lm}|\}} || |} | jry| S||kr||d d SdS) Nrr exp_polarcSs$g|]}t|dvr |nt|qS))rrsrrr%r%r&rs zarg.eval..css|]}|jduVqdSr )rrr%r%r&r'szarg.eval..atan2FrP)ranger2rr+rr)rrperiodic_argumentris_Atomr as_coeff_Mulr~rrwrrr(sympy.functions.elementary.trigonometricrr0 is_number) r;r<rCrrrEarg_rrSyrr%r%r&rFs:           zarg.evalcCsF|jd\}}|t||dd|t||dd|d|dS)NrTrPr)rr0r )rJrrSrr%r%r&rTs  zarg._eval_derivativecKs(ddlm}|jd\}}|||S)Nrr)rrrr0)rJr<rXrrSrr%r%r&_eval_rewrite_as_atan2s  zarg._eval_rewrite_as_atan2N) rjrkrlrmr+is_realrerqrrrFrTrr%r%r%r&r<s/  r<c@sXeZdZdZdZeddZddZddZd d Z d d Z d dZ ddZ ddZ dS)r3a> Returns the *complex conjugate* [1]_ of an argument. In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. Thus, the conjugate of the complex number :math:`a + ib` (where $a$ and $b$ are real numbers) is :math:`a - ib` Examples ======== >>> from sympy import conjugate, I >>> conjugate(2) 2 >>> conjugate(I) -I >>> conjugate(3 + 2*I) 3 - 2*I >>> conjugate(5 - I) 5 + I Parameters ========== arg : Expr Real or complex expression. Returns ======= arg : Expr Complex conjugate of arg as real, imaginary or mixed expression. See Also ======== sign, Abs References ========== .. [1] https://en.wikipedia.org/wiki/Complex_conjugation TcC|}|dur |SdSr )rr;r<rr%r%r&rFGzconjugate.evalcCstSr )r3r]r%r%r&inverseMszconjugate.inversecCt|jdddSrOrzrr]r%r%r&rPrzconjugate._eval_AbscCt|jdSrV transposerr]r%r%r& _eval_adjointSzconjugate._eval_adjointcC |jdSrVrr]r%r%r&rV zconjugate._eval_conjugatecCsB|jrtt|jd|ddS|jrtt|jd|dd SdSrO)rr3r rr,rRr%r%r&rTYs zconjugate._eval_derivativecCr rVadjointrr]r%r%r&_eval_transpose_rzconjugate._eval_transposecCrZrVr[r]r%r%r&r^br_zconjugate._eval_is_algebraicN)rjrkrlrmrqrrrFr rrrrTrr^r%r%r%r&r3s+  r3c@s4eZdZdZeddZddZddZdd Zd S) ra Linear map transposition. Examples ======== >>> from sympy import transpose, Matrix, MatrixSymbol >>> A = MatrixSymbol('A', 25, 9) >>> transpose(A) A.T >>> B = MatrixSymbol('B', 9, 22) >>> transpose(B) B.T >>> transpose(A*B) B.T*A.T >>> M = Matrix([[4, 5], [2, 1], [90, 12]]) >>> M Matrix([ [ 4, 5], [ 2, 1], [90, 12]]) >>> transpose(M) Matrix([ [4, 2, 90], [5, 1, 12]]) Parameters ========== arg : Matrix Matrix or matrix expression to take the transpose of. Returns ======= value : Matrix Transpose of arg. cCrr )rrr%r%r&rFrztranspose.evalcCr rVr3rr]r%r%r&rrztranspose._eval_adjointcCr rVrr]r%r%r&rrztranspose._eval_conjugatecCrrVrr]r%r%r&rrztranspose._eval_transposeN) rjrkrlrmrrrFrrrr%r%r%r&rfs(  rc@sFeZdZdZeddZddZddZdd Zdd d Z d dZ d S)ra Conjugate transpose or Hermite conjugation. Examples ======== >>> from sympy import adjoint, MatrixSymbol >>> A = MatrixSymbol('A', 10, 5) >>> adjoint(A) Adjoint(A) Parameters ========== arg : Matrix Matrix or matrix expression to take the adjoint of. Returns ======= value : Matrix Represents the conjugate transpose or Hermite conjugation of arg. cCs0|}|dur |S|}|durt|SdSr )rrr3rr%r%r&rFsz adjoint.evalcCrrVrr]r%r%r&rrzadjoint._eval_adjointcCr rVr r]r%r%r&rrzadjoint._eval_conjugatecCr rVrr]r%r%r&rrzadjoint._eval_transposeNcGs,||jd}d|}|rd||f}|S)Nrz %s^{\dagger}z\left(%s\right)^{%s})_printr)rJprinterrrr<texr%r%r&_latexs  zadjoint._latexcGsJddlm}|j|jdg|R}|jr||d}|S||d}|S)Nr) prettyFormu†+) sympy.printing.pretty.stringpictrrr _use_unicode)rJrrrpformr%r%r&_prettys   zadjoint._prettyr ) rjrkrlrmrrrFrrrrr!r%r%r%r&rs   rc@s4eZdZdZdZdZeddZddZdd Z d S) polar_lifta Lift argument to the Riemann surface of the logarithm, using the standard branch. Examples ======== >>> from sympy import Symbol, polar_lift, I >>> p = Symbol('p', polar=True) >>> x = Symbol('x') >>> polar_lift(4) 4*exp_polar(0) >>> polar_lift(-4) 4*exp_polar(I*pi) >>> polar_lift(-I) exp_polar(-I*pi/2) >>> polar_lift(I + 2) polar_lift(2 + I) >>> polar_lift(4*x) 4*polar_lift(x) >>> polar_lift(4*p) 4*p Parameters ========== arg : Expr Real or complex expression. See Also ======== sympy.functions.elementary.exponential.exp_polar periodic_argument TFc Csddlm}|jr*||}|dtdt dtfvr*ddlm}|t|t|S|jr1|j }n|g}g}g}g}|D]}|j rG||g7}q<|j rP||g7}q<||g7}q>> from sympy import exp_polar, periodic_argument >>> from sympy import I, pi >>> periodic_argument(exp_polar(10*I*pi), 2*pi) 0 >>> periodic_argument(exp_polar(5*I*pi), 4*pi) pi >>> from sympy import exp_polar, periodic_argument >>> from sympy import I, pi >>> periodic_argument(exp_polar(5*I*pi), 2*pi) pi >>> periodic_argument(exp_polar(5*I*pi), 3*pi) -pi >>> periodic_argument(exp_polar(5*I*pi), pi) 0 Parameters ========== ar : Expr A polar number. period : Expr The period $P$. See Also ======== sympy.functions.elementary.exponential.exp_polar polar_lift : Lift argument to the Riemann surface of the logarithm principal_branch c Csddlm}m}|jr|j}n|g}d}|D]I}|js"|t|7}qt||r1||j d7}q|j rN|j \}}||t |j ||t |j 7}qt|tr]|t|jd7}qdS|S)Nr)rrrs)rrrr~rr%r<r2rr0runbranched_argumentrr$r") r;r'rrrrprCrr9r%r%r&_getunbranched_s*  z periodic_argument._getunbranchedc Cs |jsdS|tkrt|trt|jSt|tr&|dtkr&t|jd|S|jrAdd|jD}t |t |jkrAtt ||S| |}|durLdSddl m }m}|t||r]dS|tkrc|S|tkrddlm}|||tj|}||s||SdSdS)NrrcSsg|]}|js|qSr%)rr#rSr%r%r&rz*periodic_argument.eval..)atanrceiling)rrr2principal_branchrrr"rr~r8rr,rr/rr7#sympy.functions.elementary.integersr1rr) r;r'periodnewargsrpr/rr1rr%r%r&rFvs2    zperiodic_argument.evalcCsn|j\}}|tkrt|}|dur|S||St|t|}ddlm}||||tj||S)Nrr0) rrrr,r)r3r1rr)rJr*rr4rpubr1r%r%r&r)s     zperiodic_argument._eval_evalfN)rjrkrlrmrrr,rFr)r%r%r%r&r6s(   rcCs t|tS)a\ Returns periodic argument of arg with period as infinity. Examples ======== >>> from sympy import exp_polar, unbranched_argument >>> from sympy import I, pi >>> unbranched_argument(exp_polar(15*I*pi)) 15*pi >>> unbranched_argument(exp_polar(7*I*pi)) 7*pi See also ======== periodic_argument )rrrr%r%r&r+s r+c@s,eZdZdZdZdZeddZddZdS) r2a Represent a polar number reduced to its principal branch on a quotient of the Riemann surface of the logarithm. Explanation =========== This is a function of two arguments. The first argument is a polar number `z`, and the second one a positive real number or infinity, `p`. The result is ``z mod exp_polar(I*p)``. Examples ======== >>> from sympy import exp_polar, principal_branch, oo, I, pi >>> from sympy.abc import z >>> principal_branch(z, oo) z >>> principal_branch(exp_polar(2*pi*I)*3, 2*pi) 3*exp_polar(0) >>> principal_branch(exp_polar(2*pi*I)*3*z, 2*pi) 3*principal_branch(z, 2*pi) Parameters ========== x : Expr A polar number. period : Expr Positive real number or infinity. See Also ======== sympy.functions.elementary.exponential.exp_polar polar_lift : Lift argument to the Riemann surface of the logarithm periodic_argument TFcCsddlm}t|trt|jd|S|tkr|St|t}t||}||krj|tsj|tsjt|}dd}| t|}t|t}|tsj||krX|t |||}n|}|j sh||sh||d9}|S|j ss|d} } n|j |j \} } g} | D]} | jr| | 9} q| | g7} qt| } t| |} | trdS| jrt| | ks| dkr| dkr| dkr| dkrt| tt| |St|t | t| |t| S| jrt| |dkdks| |dkr| dkr|| t t| SdSdSdS) NrrcSst|ts t|S|Sr )r2rr")exprr%r%r&mrs z!principal_branch.eval..mrr%rsrT)rrr2r"r2rrrr7replacerr%rrrtuplerr+r$r)rJrSr4rr6bargplr8resrEmothersrr<r%r%r&rFs`             "& zprincipal_branch.evalcCsZ|j\}}t|||}t|tks|t kr|Sddlm}t||t||S)Nr)r)rrr)r$rrrr)rJr*rr4prr%r%r&r)s  zprincipal_branch._eval_evalfN) rjrkrlrmr%rrrrFr)r%r%r%r&r2s(  3r2Fc s`ddlm}|jr |S|jrst|St|tr!s!r!t|S|jr&|S|jr>|j fdd|j D}r.FrCcrB)FrCrErGrHr%r&r1rrsrIrDcs(g|]}t|trt|dn|qS)rC)r2r rFrGrJr%r&r<s  )sympy.integrals.integralsrAr%rr"r2rrrrrrrrExp1rFrr1functionr6r:) eqrIrDrArrlimitslimitvarrestr%rJr&rFs:   rFTcCsN|rd}tt||}|s|Sdd|jD}||}|dd|DfS)a Turn all numbers in eq into their polar equivalents (under the standard choice of argument). Note that no attempt is made to guess a formal convention of adding polar numbers, expressions like $1 + x$ will generally not be altered. Note also that this function does not promote ``exp(x)`` to ``exp_polar(x)``. If ``subs`` is ``True``, all symbols which are not already polar will be substituted for polar dummies; in this case the function behaves much like :func:`~.posify`. If ``lift`` is ``True``, both addition statements and non-polar symbols are changed to their ``polar_lift()``ed versions. Note that ``lift=True`` implies ``subs=False``. Examples ======== >>> from sympy import polarify, sin, I >>> from sympy.abc import x, y >>> expr = (-x)**y >>> expr.expand() (-x)**y >>> polarify(expr) ((_x*exp_polar(I*pi))**_y, {_x: x, _y: y}) >>> polarify(expr)[0].expand() _x**_y*exp_polar(_y*I*pi) >>> polarify(x, lift=True) polar_lift(x) >>> polarify(x*(1+y), lift=True) polar_lift(x)*polar_lift(y + 1) Adds are treated carefully: >>> polarify(1 + sin((1 + I)*x)) (sin(_x*polar_lift(1 + I)) + 1, {_x: x}) FcSsi|] }|t|jddqS)T)polar)rname)r#r{r%r%r&rmszpolarify..cSsi|]\}}||qSr%r%)r#r{rOr%r%r&ror.)rFrrritems)rNrrIrepsr%r%r&polarify@s( rXcsRt|tr|jr |S|slddlm}m}t||r!|t|jSt|tr7|jddt kr7t|jdS|j sR|j sR|j sR|j r_|jdvrMd|jvsR|jdvr_|jfdd|jDSt|trlt|jdS|jrt|j}t|j|jo~| }||S|jrt|jdd r|jfd d|jDS|jfd d|jDS) Nr)rrrsr)z==z!=csg|]}t|qSr% _unpolarifyr-exponents_onlyr%r&rr.z_unpolarify..rpFcsg|]}t|qSr%rYr-r[r%r&rscsg|]}t|dqSrirYr-r[r%r&rr)r2r rrrrrZr2rrrr~ is_Boolean is_Relationalrel_oprr"rrrr1getattr)rNr\rDrrexporr%r[r&rZrsH    rZNcCst|tr|St|}|durt||Sd}d}|rd}|r9d}t|||}||kr0d}|}t|tr7|S|s ddlm}||ddtddiS)a If `p` denotes the projection from the Riemann surface of the logarithm to the complex line, return a simplified version `eq'` of `eq` such that `p(eq') = p(eq)`. Also apply the substitution subs in the end. (This is a convenience, since ``unpolarify``, in a certain sense, undoes :func:`polarify`.) Examples ======== >>> from sympy import unpolarify, polar_lift, sin, I >>> unpolarify(polar_lift(I + 2)) 2 + I >>> unpolarify(sin(polar_lift(I + 7))) sin(7 + I) NTFrrrs) r2boolr unpolarifyrrZrrr")rNrr\changedrDr=rr%r%r&rcs(    rc)F)TF)NF)4typingrrn sympy.corerrrrrrr sympy.core.exprr sympy.core.exprtoolsr sympy.core.functionr r rrrsympy.core.logicrrsympy.core.numbersrrrsympy.core.powerrsympy.core.relationalr(sympy.functions.elementary.miscellaneousr$sympy.functions.elementary.piecewiserrr9rwrzr<r3rrr"rr+r2rFrXrZrcr%r%r%r&s: $      z{6{aM9BUj i ! 2!