Source code for sympy.functions.special.tensor_functions

from __future__ import print_function, division

from sympy.core.function import Function
from sympy.core import S, Integer
from sympy.core.mul import prod
from sympy.core.logic import fuzzy_not
from sympy.utilities.iterables import (has_dups, default_sort_key)
from sympy.core.compatibility import range

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###################### Kronecker Delta, Levi-Civita etc. ######################
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[docs]def Eijk(*args, **kwargs): """ Represent the Levi-Civita symbol. This is just compatibility wrapper to ``LeviCivita()``. See Also ======== LeviCivita """ return LeviCivita(*args, **kwargs)
[docs]def eval_levicivita(*args): """Evaluate Levi-Civita symbol.""" from sympy import factorial n = len(args) return prod( prod(args[j] - args[i] for j in range(i + 1, n)) / factorial(i) for i in range(n)) # converting factorial(i) to int is slightly faster
[docs]class LeviCivita(Function): """Represent the Levi-Civita symbol. For even permutations of indices it returns 1, for odd permutations -1, and for everything else (a repeated index) it returns 0. Thus it represents an alternating pseudotensor. Examples ======== >>> 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 """ is_integer = True @classmethod def eval(cls, *args): if all(isinstance(a, (int, Integer)) for a in args): return eval_levicivita(*args) if has_dups(args): return S.Zero def doit(self): return eval_levicivita(*self.args)
[docs]class KroneckerDelta(Function): """The discrete, or Kronecker, delta function. 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. Parameters ========== i : Number, Symbol The first index of the delta function. j : Number, Symbol The second index of the delta function. Examples ======== A simple example with integer indices:: >>> from sympy.functions.special.tensor_functions 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) See Also ======== eval sympy.functions.special.delta_functions.DiracDelta References ========== .. [1] http://en.wikipedia.org/wiki/Kronecker_delta """ is_integer = True @classmethod
[docs] def eval(cls, i, j): """ Evaluates the discrete delta function. Examples ======== >>> from sympy.functions.special.tensor_functions 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 """ diff = i - j if diff.is_zero: return S.One elif fuzzy_not(diff.is_zero): return S.Zero if i.assumptions0.get("below_fermi") and \ j.assumptions0.get("above_fermi"): return S.Zero if j.assumptions0.get("below_fermi") and \ i.assumptions0.get("above_fermi"): return S.Zero # to make KroneckerDelta canonical # following lines will check if inputs are in order # if not, will return KroneckerDelta with correct order if i is not min(i, j, key=default_sort_key): return cls(j, i)
def _eval_power(self, expt): if expt.is_positive: return self if expt.is_negative and not -expt is S.One: return 1/self @property def is_above_fermi(self): """ True if Delta can be non-zero above fermi Examples ======== >>> from sympy.functions.special.tensor_functions import KroneckerDelta >>> from sympy import 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 """ if self.args[0].assumptions0.get("below_fermi"): return False if self.args[1].assumptions0.get("below_fermi"): return False return True @property def is_below_fermi(self): """ True if Delta can be non-zero below fermi Examples ======== >>> from sympy.functions.special.tensor_functions import KroneckerDelta >>> from sympy import 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 """ if self.args[0].assumptions0.get("above_fermi"): return False if self.args[1].assumptions0.get("above_fermi"): return False return True @property def is_only_above_fermi(self): """ True if Delta is restricted to above fermi Examples ======== >>> from sympy.functions.special.tensor_functions import KroneckerDelta >>> from sympy import 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 """ return ( self.args[0].assumptions0.get("above_fermi") or self.args[1].assumptions0.get("above_fermi") ) or False @property def is_only_below_fermi(self): """ True if Delta is restricted to below fermi Examples ======== >>> from sympy.functions.special.tensor_functions import KroneckerDelta >>> from sympy import 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 """ return ( self.args[0].assumptions0.get("below_fermi") or self.args[1].assumptions0.get("below_fermi") ) or False @property def indices_contain_equal_information(self): """ Returns True if indices are either both above or below fermi. Examples ======== >>> from sympy.functions.special.tensor_functions import KroneckerDelta >>> from sympy import 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 """ if (self.args[0].assumptions0.get("below_fermi") and self.args[1].assumptions0.get("below_fermi")): return True if (self.args[0].assumptions0.get("above_fermi") and self.args[1].assumptions0.get("above_fermi")): return True # if both indices are general we are True, else false return self.is_below_fermi and self.is_above_fermi @property def preferred_index(self): """ 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 same information, 'a' is preferred before 'b'. Examples ======== >>> from sympy.functions.special.tensor_functions import KroneckerDelta >>> from sympy import 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 """ if self._get_preferred_index(): return self.args[1] else: return self.args[0] @property def killable_index(self): """ Returns the index which is preferred to substitute in the final expression. The index to substitute is the index with less information regarding fermi level. If indices contain same information, 'a' is preferred before 'b'. Examples ======== >>> from sympy.functions.special.tensor_functions import KroneckerDelta >>> from sympy import 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 """ if self._get_preferred_index(): return self.args[0] else: return self.args[1] def _get_preferred_index(self): """ 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 same information, index 0 is returned. """ if not self.is_above_fermi: if self.args[0].assumptions0.get("below_fermi"): return 0 else: return 1 elif not self.is_below_fermi: if self.args[0].assumptions0.get("above_fermi"): return 0 else: return 1 else: return 0 @staticmethod def _latex_no_arg(printer): return r'\delta' @property def indices(self): return self.args[0:2] def _sage_(self): import sage.all as sage return sage.kronecker_delta(self.args[0]._sage_(), self.args[1]._sage_())