from __future__ import print_function, division
from sympy.core import S, Integer
from sympy.core.compatibility import range, SYMPY_INTS
from sympy.core.function import Function
from sympy.core.logic import fuzzy_not
from sympy.core.mul import prod
from sympy.utilities.iterables import (has_dups, default_sort_key)
###############################################################################
###################### Kronecker Delta, Levi-Civita etc. ######################
###############################################################################
[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, (SYMPY_INTS, 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] https://en.wikipedia.org/wiki/Kronecker_delta
"""
is_integer = True
[docs] @classmethod
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
@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_())
