ó ¡¼™\c@ sdZddlmZmZddlmZmZmZmZddl m Z ddl m Z m Z ddlmZmZmZddlmZe d„ƒZe d „ƒZe d „ƒZe d „ƒZe d „ƒZe d „ƒZe d„ƒZe ed„ƒZdS(sŸ This module implements sums and products containing the Kronecker Delta function. References ========== - http://mathworld.wolfram.com/KroneckerDelta.html iÿÿÿÿ(tprint_functiontdivision(tAddtMultStDummy(tcacheit(tdefault_sort_keytrange(tKroneckerDeltat Piecewisetpiecewise_fold(tIntervalcC s½|js |Sd}t}tdƒg}xˆ|jD]}}|dkr’|jr’t||ƒr’t}|j}g|jD]}|d|^qu}q2g|D]}||^q™}q2W||ŒS(sB Expand the first Add containing a simple KroneckerDelta. iiN( tis_MultNoneRRtargstis_Addt_has_simple_deltatTruetfunc(texprtindextdeltaRttermsthtt((s3lib/python2.7/site-packages/sympy/concrete/delta.pyt _expand_deltas $ '!cC s´t||ƒsd|fSt|tƒr8|tdƒfS|jsPtdƒ‚nd}g}xB|jD]7}|dkrt||ƒr|}qf|j |ƒqfW||j |ŒfS(ss Extract a simple KroneckerDelta from the expression. Returns the tuple ``(delta, newexpr)`` where: - ``delta`` is a simple KroneckerDelta expression if one was found, or ``None`` if no simple KroneckerDelta expression was found. - ``newexpr`` is a Mul containing the remaining terms; ``expr`` is returned unchanged if no simple KroneckerDelta expression was found. Examples ======== >>> from sympy import KroneckerDelta >>> from sympy.concrete.delta import _extract_delta >>> from sympy.abc import x, y, i, j, k >>> _extract_delta(4*x*y*KroneckerDelta(i, j), i) (KroneckerDelta(i, j), 4*x*y) >>> _extract_delta(4*x*y*KroneckerDelta(i, j), k) (None, 4*x*y*KroneckerDelta(i, j)) See Also ======== sympy.functions.special.tensor_functions.KroneckerDelta deltaproduct deltasummation isIncorrect exprN( RRt isinstanceR RR t ValueErrorRt_is_simple_deltatappendR(RRRRtarg((s3lib/python2.7/site-packages/sympy/concrete/delta.pyt_extract_delta&s   cC se|jtƒrat||ƒr"tS|js4|jrax'|jD]}t||ƒr>tSq>WqantS(sØ Returns True if ``expr`` is an expression that contains a KroneckerDelta that is simple in the index ``index``, meaning that this KroneckerDelta is nonzero for a single value of the index ``index``. ( thasR RRRR RRtFalse(RRR((s3lib/python2.7/site-packages/sympy/concrete/delta.pyRVscC s\t|tƒrX|j|ƒrX|jd|jdj|ƒ}|rX|jƒdkSntS(su Returns True if ``delta`` is a KroneckerDelta and is nonzero for a single value of the index ``index``. ii(RR R!Rtas_polytdegreeR"(RRtp((s3lib/python2.7/site-packages/sympy/concrete/delta.pyRgs !cC sOddlm}|jr8|jttt|jƒƒŒS|jsE|Sg}g}xO|jD]D}t |t ƒr’|j |jd|jdƒq[|j |ƒq[W|s­|S||dt ƒ}t |ƒdkrØtjSt |ƒdkrKx6|djƒD]$}|j t ||d|ƒƒqûW|j|Œ}||krKt|ƒSn|S(s0 Evaluate products of KroneckerDelta's. iÿÿÿÿ(tsolveiitdict(t sympy.solversR&RRtlisttmapt_remove_multiple_deltaRR RR RRtlenRtZerotkeys(RR&teqstnewargsRtsolnstkeytexpr2((s3lib/python2.7/site-packages/sympy/concrete/delta.pyR+ts.  ""  cC s³ddlm}t|tƒr¯yy||jd|jddtƒ}|r—t|ƒdkr—tg|djƒD]\}}t||fŒ^qrŒSWq¯t k r«q¯Xn|S(sB Rewrite a KroneckerDelta's indices in its simplest form. iÿÿÿÿ(R&iiR'( R(R&RR RRR,RtitemstNotImplementedError(RR&tslnsR2tvalue((s3lib/python2.7/site-packages/sympy/concrete/delta.pyt_simplify_delta“s$7 c C s[ddlm}|d|ddktkr5tjS|jtƒsQ|||ƒS|jrDd }g}xRt |j dt ƒD];}|d krªt ||dƒrª|}q||j |ƒq|W|j|Œ}tddtƒ}t|dtƒr¦t|dtƒr¦t||ƒtgtt|dƒt|ddƒƒD]`}t||d|d|dfƒ|j|d|ƒt||d|d|dfƒ^q9ƒ} n”t||ƒtt||d|d|dfƒ|j|d|ƒt||d|d|dfƒ||d|dfd t ||dƒƒ} t| ƒSt||dƒ\}} |s×t||dƒ} || krÊdd lm} y| t| |ƒƒSWqÊtk rÆt| |ƒSXn|||ƒSdd lm} | |d|ddƒ}t|j|d|dƒt|d|dƒƒtjtt|d|ddƒƒS( sÅ Handle products containing a KroneckerDelta. See Also ======== deltasummation sympy.functions.special.tensor_functions.KroneckerDelta sympy.concrete.products.product iÿÿÿÿ(tproductiiiR2tkprimetintegert no_piecewise(tfactor(tEqN(tsympy.concrete.productsR9RRtOneR!R RRtsortedRRRRRRRtintt deltaproducttsumRtsubstdeltasummationR+R RtsympyR=tAssertionErrorR>R8(tftlimitR9RRRtnewexprtktiktresultt_tgR=R>tc((s3lib/python2.7/site-packages/sympy/concrete/delta.pyRC¤sL    &’5"    2c C s§ddlm}ddlm}|d|ddktkrEtjS|jtƒsa|||ƒS|d}t ||ƒ}|j r¸t |j g|j D]}t|||ƒ^q–ŒƒSt||ƒ\}} |sà|||ƒS||j d|j d|ƒ} t| ƒdkrtjSt| ƒdkrIddlm} | ||ƒS| d} |ri| j|| ƒSt| j|| ƒt|dd!Œj| ƒftjtfƒS( s_ Handle summations containing a KroneckerDelta. The idea for summation is the following: - If we are dealing with a KroneckerDelta expression, i.e. KroneckerDelta(g(x), j), we try to simplify it. If we could simplify it, then we sum the resulting expression. We already know we can sum a simplified expression, because only simple KroneckerDelta expressions are involved. If we couldn't simplify it, there are two cases: 1) The expression is a simple expression: we return the summation, taking care if we are dealing with a Derivative or with a proper KroneckerDelta. 2) The expression is not simple (i.e. KroneckerDelta(cos(x))): we can do nothing at all. - If the expr is a multiplication expr having a KroneckerDelta term: First we expand it. If the expansion did work, then we try to sum the expansion. If not, we try to extract a simple KroneckerDelta term, then we have two cases: 1) We have a simple KroneckerDelta term, so we return the summation. 2) We didn't have a simple term, but we do have an expression with simplified KroneckerDelta terms, so we sum this expression. Examples ======== >>> from sympy import oo, symbols >>> from sympy.abc import k >>> i, j = symbols('i, j', integer=True, finite=True) >>> from sympy.concrete.delta import deltasummation >>> from sympy import KroneckerDelta, Piecewise >>> deltasummation(KroneckerDelta(i, k), (k, -oo, oo)) 1 >>> deltasummation(KroneckerDelta(i, k), (k, 0, oo)) Piecewise((1, i >= 0), (0, True)) >>> deltasummation(KroneckerDelta(i, k), (k, 1, 3)) Piecewise((1, (i >= 1) & (i <= 3)), (0, True)) >>> deltasummation(k*KroneckerDelta(i, j)*KroneckerDelta(j, k), (k, -oo, oo)) j*KroneckerDelta(i, j) >>> deltasummation(j*KroneckerDelta(i, j), (j, -oo, oo)) i >>> deltasummation(i*KroneckerDelta(i, j), (i, -oo, oo)) j See Also ======== deltaproduct sympy.functions.special.tensor_functions.KroneckerDelta sympy.concrete.sums.summation iÿÿÿÿ(t summation(R&iii(tSumi(tsympy.concrete.summationsRRR(R&RRR-R!R RRR RRRFR R,RSRER R t as_relational( RIRJR<RRR&txRPRRRR1RSR7((s3lib/python2.7/site-packages/sympy/concrete/delta.pyRFås4A   2 !  +N(t__doc__t __future__RRt sympy.coreRRRRtsympy.core.cacheRtsympy.core.compatibilityRRtsympy.functionsR R R t sympy.setsR RR RRR+R8RCR"RF(((s3lib/python2.7/site-packages/sympy/concrete/delta.pyt s"0 A