\h0SSKJr SSKJrJr SSKJr SSKJr SSK J r SSK J r SSK Jr SSKJr SS KJr SS KJr S rS r"S S\5r"SS\5rg))prod)SInteger)DefinedFunction) fuzzy_not)Ne)default_sort_key) SYMPY_INTS) factorial) Piecewise)has_dupsc[U0UD6$)zy Represent the Levi-Civita symbol. This is a compatibility wrapper to ``LeviCivita()``. See Also ======== LeviCivita ) LeviCivita)argskwargss `/var/www/auris/envauris/lib/python3.13/site-packages/sympy/functions/special/tensor_functions.pyEijkrs t &v &&cZ^^[T5m[UU4Sj[T555$)zEvaluate Levi-Civita symbol.c3>^# UH4m[UU4Sj[TS-T555[T5- v M6 g7f)c3:># UHnTUTT- v M g7fN).0jris r ,eval_levicivita...%s 81T!WtAw sN)rranger )rrrns @rr"eval_levicivita..$s9* (1 8a!eQ 88 A,  (s>> from sympy import LeviCivita >>> from sympy.abc import i, j, k >>> LeviCivita(1, 2, 3) 1 >>> LeviCivita(1, 3, 2) -1 >>> LeviCivita(1, 2, 2) 0 >>> LeviCivita(i, j, k) LeviCivita(i, j, k) >>> LeviCivita(i, j, i) 0 See Also ======== Eijk Tc[SU55(a[U6$[U5(a[R$g)Nc3N# UHn[U[[45v M g7fr) isinstancer r)ras rr"LeviCivita.eval..QsBTz!j'233Ts#%)allr$r rZero)clsrs revalLeviCivita.evalOs5 BTB B B"D) ) D>>66M rc &[UR6$r)r$r)selfhintss rdoitLeviCivita.doitVs **rrN) __name__ __module__ __qualname____firstlineno____doc__ is_integer classmethodr/r4__static_attributes__rrrrr*s& DJ +rrc\rSrSrSrSr\SSj5r\S5r Sr \S5r \S 5r \S 5r \S 5r\S 5r\S 5r\S5rSr\S5rSrSrg)KroneckerDeltaZa The discrete, or Kronecker, delta function. Explanation =========== A function that takes in two integers $i$ and $j$. It returns $0$ if $i$ and $j$ are not equal, or it returns $1$ if $i$ and $j$ are equal. Examples ======== An example with integer indices: >>> from sympy import KroneckerDelta >>> KroneckerDelta(1, 2) 0 >>> KroneckerDelta(3, 3) 1 Symbolic indices: >>> from sympy.abc import i, j, k >>> KroneckerDelta(i, j) KroneckerDelta(i, j) >>> KroneckerDelta(i, i) 1 >>> KroneckerDelta(i, i + 1) 0 >>> KroneckerDelta(i, i + 1 + k) KroneckerDelta(i, i + k + 1) Parameters ========== i : Number, Symbol The first index of the delta function. j : Number, Symbol The second index of the delta function. See Also ======== eval DiracDelta References ========== .. [1] https://en.wikipedia.org/wiki/Kronecker_delta TNc(UbpUupEXA- S:S:Xa[R$XB- S:S:Xa[R$XQ- S:S:Xa[R$XR- S:S:Xa[R$X- nUR(a[R$[ UR5(a[R$UR R S5(a0UR R S5(a[R$UR R S5(a0UR R S5(a[R$[U5[U5:aU(a U"X!U5$U"X!5$g)a? Evaluates the discrete delta function. Examples ======== >>> from sympy import KroneckerDelta >>> from sympy.abc import i, j, k >>> KroneckerDelta(i, j) KroneckerDelta(i, j) >>> KroneckerDelta(i, i) 1 >>> KroneckerDelta(i, i + 1) 0 >>> KroneckerDelta(i, i + 1 + k) KroneckerDelta(i, i + k + 1) # indirect doctest NrT below_fermi above_fermi)rr-is_zeroOner assumptions0getr )r.rr delta_rangedinfdsupdiffs rr/KroneckerDelta.evals60  "$JD1 %vv 1 %vv 1 %vv 1 %vv u <<55L t|| $ $66M >>  m , ,""=1166M >>  m , ,""=1166M A !1!!4 41--1y 5rcT[UR5S:aURS$g)N)r#rr2s rrHKroneckerDelta.delta_ranges% tyy>A 99Q<  rc~UR(aU$UR(aU[RLaSU- $gg)Nr) is_positive is_negativer NegativeOne)r2expts r _eval_powerKroneckerDelta._eval_powers5   K   AMM 9T6M!: rcURSRRS5(agURSRRS5(agg)a True if Delta can be non-zero above fermi. Examples ======== >>> from sympy import KroneckerDelta, Symbol >>> a = Symbol('a', above_fermi=True) >>> i = Symbol('i', below_fermi=True) >>> p = Symbol('p') >>> q = Symbol('q') >>> KroneckerDelta(p, a).is_above_fermi True >>> KroneckerDelta(p, i).is_above_fermi False >>> KroneckerDelta(p, q).is_above_fermi True See Also ======== is_below_fermi, is_only_below_fermi, is_only_above_fermi rrBFrTrrFrGrOs ris_above_fermiKroneckerDelta.is_above_fermiK4 99Q< $ $ ( ( 7 7 99Q< $ $ ( ( 7 7rcURSRRS5(agURSRRS5(agg)a True if Delta can be non-zero below fermi. Examples ======== >>> from sympy import KroneckerDelta, Symbol >>> a = Symbol('a', above_fermi=True) >>> i = Symbol('i', below_fermi=True) >>> p = Symbol('p') >>> q = Symbol('q') >>> KroneckerDelta(p, a).is_below_fermi False >>> KroneckerDelta(p, i).is_below_fermi True >>> KroneckerDelta(p, q).is_below_fermi True See Also ======== is_above_fermi, is_only_above_fermi, is_only_below_fermi rrCFrTrYrOs ris_below_fermiKroneckerDelta.is_below_fermir\rcURSRRS5=(d( URSRRS5=(d S$)a True if Delta is restricted to above fermi. Examples ======== >>> from sympy import KroneckerDelta, Symbol >>> a = Symbol('a', above_fermi=True) >>> i = Symbol('i', below_fermi=True) >>> p = Symbol('p') >>> q = Symbol('q') >>> KroneckerDelta(p, a).is_only_above_fermi True >>> KroneckerDelta(p, q).is_only_above_fermi False >>> KroneckerDelta(p, i).is_only_above_fermi False See Also ======== is_above_fermi, is_below_fermi, is_only_below_fermi rrCrFrYrOs ris_only_above_fermi"KroneckerDelta.is_only_above_fermiP41**..}== ! ))--m< rcURSRRS5=(d( URSRRS5=(d S$)a True if Delta is restricted to below fermi. Examples ======== >>> from sympy import KroneckerDelta, Symbol >>> a = Symbol('a', above_fermi=True) >>> i = Symbol('i', below_fermi=True) >>> p = Symbol('p') >>> q = Symbol('q') >>> KroneckerDelta(p, i).is_only_below_fermi True >>> KroneckerDelta(p, q).is_only_below_fermi False >>> KroneckerDelta(p, a).is_only_below_fermi False See Also ======== is_above_fermi, is_below_fermi, is_only_above_fermi rrBrFrYrOs ris_only_below_fermi"KroneckerDelta.is_only_below_fermi4rcrcURSRRS5(a.URSRRS5(agURSRRS5(a.URSRRS5(agUR=(a UR$)a Returns True if indices are either both above or below fermi. Examples ======== >>> from sympy import KroneckerDelta, Symbol >>> a = Symbol('a', above_fermi=True) >>> i = Symbol('i', below_fermi=True) >>> p = Symbol('p') >>> q = Symbol('q') >>> KroneckerDelta(p, q).indices_contain_equal_information True >>> KroneckerDelta(p, q+1).indices_contain_equal_information True >>> KroneckerDelta(i, p).indices_contain_equal_information False rrBrTrC)rrFrGr^rZrOs r!indices_contain_equal_information0KroneckerDelta.indices_contain_equal_informationSs* IIaL % % ) )- 8 8 ! ))--m<< IIaL % % ) )- 8 8IIaL--11-@@"":t':'::rchUR5(aURS$URS$)aa Returns the index which is preferred to keep in the final expression. Explanation =========== The preferred index is the index with more information regarding fermi level. If indices contain the same information, 'a' is preferred before 'b'. Examples ======== >>> from sympy import KroneckerDelta, Symbol >>> a = Symbol('a', above_fermi=True) >>> i = Symbol('i', below_fermi=True) >>> j = Symbol('j', below_fermi=True) >>> p = Symbol('p') >>> KroneckerDelta(p, i).preferred_index i >>> KroneckerDelta(p, a).preferred_index a >>> KroneckerDelta(i, j).preferred_index i See Also ======== killable_index rr_get_preferred_indexrrOs rpreferred_indexKroneckerDelta.preferred_indexrs0B  $ $ & &99Q< 99Q< rchUR5(aURS$URS$)ai Returns the index which is preferred to substitute in the final expression. Explanation =========== The index to substitute is the index with less information regarding fermi level. If indices contain the same information, 'a' is preferred before 'b'. Examples ======== >>> from sympy import KroneckerDelta, Symbol >>> a = Symbol('a', above_fermi=True) >>> i = Symbol('i', below_fermi=True) >>> j = Symbol('j', below_fermi=True) >>> p = Symbol('p') >>> KroneckerDelta(p, i).killable_index p >>> KroneckerDelta(p, a).killable_index p >>> KroneckerDelta(i, j).killable_index j See Also ======== preferred_index rrrkrOs rkillable_indexKroneckerDelta.killable_indexs0D  $ $ & &99Q< 99Q< rcUR(d/URSRRS5(aggUR(d/URSRRS5(aggg)z Returns the index which is preferred to keep in the final expression. The preferred index is the index with more information regarding fermi level. If indices contain the same information, index 0 is returned. rrBrrC)rZrrFrGr^rOs rrl#KroneckerDelta._get_preferred_indexsc""yy|((,,];;$$yy|((,,];;rc URSS$)NrrN)rrOs rindicesKroneckerDelta.indicessyy1~rc8Uup4[S[X454S5$)Nr)rT)r r)r2rrrrs r_eval_rewrite_as_Piecewise)KroneckerDelta._eval_rewrite_as_Piecewises!RX 22rrr)r6r7r8r9r:r;r<r/propertyrHrVrZr^rarerhrmrprlrurxr=rrrr?r?Zs3jJ5!5!n   >><<;;<# # J$ $ L*3rr?N)mathr sympy.corerrsympy.core.functionrsympy.core.logicrsympy.core.relationalrsympy.core.sortingr sympy.external.gmpyr (sympy.functions.combinatorial.factorialsr $sympy.functions.elementary.piecewiser sympy.utilities.iterablesr rr$rr?rrrrsH!/&$/*>:. '*-+-+`@3_@3r