Source code for sympy.functions.special.delta_functions
from __future__ import print_function, division
from sympy.core import S, sympify, diff
from sympy.core.function import Function, ArgumentIndexError
from sympy.core.relational import Eq
from sympy.polys.polyerrors import PolynomialError
from sympy.functions.elementary.complexes import im
from sympy.functions.elementary.piecewise import Piecewise
###############################################################################
################################ DELTA FUNCTION ###############################
###############################################################################
[docs]class DiracDelta(Function):
"""
The DiracDelta function and its derivatives.
DiracDelta function has the following properties:
1) ``diff(Heaviside(x),x) = DiracDelta(x)``
2) ``integrate(DiracDelta(x-a)*f(x),(x,-oo,oo)) = f(a)`` and
``integrate(DiracDelta(x-a)*f(x),(x,a-e,a+e)) = f(a)``
3) ``DiracDelta(x) = 0`` for all ``x != 0``
4) ``DiracDelta(g(x)) = Sum_i(DiracDelta(x-x_i)/abs(g'(x_i)))``
Where ``x_i``-s are the roots of ``g``
Derivatives of ``k``-th order of DiracDelta have the following property:
5) ``DiracDelta(x,k) = 0``, for all ``x != 0``
See Also
========
Heaviside
simplify, is_simple
sympy.functions.special.tensor_functions.KroneckerDelta
References
==========
.. [1] http://mathworld.wolfram.com/DeltaFunction.html
"""
is_real = True
def fdiff(self, argindex=1):
if argindex == 1:
#I didn't know if there is a better way to handle default arguments
k = 0
if len(self.args) > 1:
k = self.args[1]
return self.func(self.args[0], k + 1)
else:
raise ArgumentIndexError(self, argindex)
@classmethod
def eval(cls, arg, k=0):
k = sympify(k)
if not k.is_Integer or k.is_negative:
raise ValueError("Error: the second argument of DiracDelta must be \
a non-negative integer, %s given instead." % (k,))
arg = sympify(arg)
if arg is S.NaN:
return S.NaN
if arg.is_positive or arg.is_negative:
return S.Zero
[docs] def simplify(self, x):
"""simplify(self, x)
Compute a simplified representation of the function using
property number 4.
x can be:
- a symbol
Examples
========
>>> from sympy import DiracDelta
>>> from sympy.abc import x, y
>>> DiracDelta(x*y).simplify(x)
DiracDelta(x)/Abs(y)
>>> DiracDelta(x*y).simplify(y)
DiracDelta(y)/Abs(x)
>>> DiracDelta(x**2 + x - 2).simplify(x)
DiracDelta(x - 1)/3 + DiracDelta(x + 2)/3
See Also
========
is_simple, Directdelta
"""
from sympy.polys.polyroots import roots
if not self.args[0].has(x) or (len(self.args) > 1 and self.args[1] != 0 ):
return self
try:
argroots = roots(self.args[0], x, multiple=True)
result = 0
valid = True
darg = diff(self.args[0], x)
for r in argroots:
#should I care about multiplicities of roots?
if r.is_real is not False and not darg.subs(x, r).is_zero:
result += self.func(x - r)/abs(darg.subs(x, r))
else:
valid = False
break
if valid:
return result
except PolynomialError:
pass
return self
[docs] def is_simple(self, x):
"""is_simple(self, x)
Tells whether the argument(args[0]) of DiracDelta is a linear
expression in x.
x can be:
- a symbol
Examples
========
>>> from sympy import DiracDelta, cos
>>> from sympy.abc import x, y
>>> DiracDelta(x*y).is_simple(x)
True
>>> DiracDelta(x*y).is_simple(y)
True
>>> DiracDelta(x**2+x-2).is_simple(x)
False
>>> DiracDelta(cos(x)).is_simple(x)
False
See Also
========
simplify, Directdelta
"""
p = self.args[0].as_poly(x)
if p:
return p.degree() == 1
return False
@staticmethod
def _latex_no_arg(printer):
return r'\delta'
###############################################################################
############################## HEAVISIDE FUNCTION #############################
###############################################################################
[docs]class Heaviside(Function):
"""Heaviside Piecewise function
Heaviside function has the following properties [*]_:
1) ``diff(Heaviside(x),x) = DiracDelta(x)``
``( 0, if x < 0``
2) ``Heaviside(x) = < ( 1/2 if x==0 [*]``
``( 1, if x > 0``
.. [*] Regarding to the value at 0, Mathematica defines ``H(0) = 1``,
but Maple uses ``H(0) = undefined``
I think is better to have H(0) = 1/2, due to the following::
integrate(DiracDelta(x), x) = Heaviside(x)
integrate(DiracDelta(x), (x, -oo, oo)) = 1
and since DiracDelta is a symmetric function,
``integrate(DiracDelta(x), (x, 0, oo))`` should be 1/2 (which is what
Maple returns).
If we take ``Heaviside(0) = 1/2``, we would have
``integrate(DiracDelta(x), (x, 0, oo)) = ``
``Heaviside(oo) - Heaviside(0) = 1 - 1/2 = 1/2``
and
``integrate(DiracDelta(x), (x, -oo, 0)) = ``
``Heaviside(0) - Heaviside(-oo) = 1/2 - 0 = 1/2``
If we consider, instead ``Heaviside(0) = 1``, we would have
``integrate(DiracDelta(x), (x, 0, oo)) = Heaviside(oo) - Heaviside(0) = 0``
and
``integrate(DiracDelta(x), (x, -oo, 0)) = Heaviside(0) - Heaviside(-oo) = 1``
See Also
========
DiracDelta
References
==========
.. [1] http://mathworld.wolfram.com/HeavisideStepFunction.html
"""
is_real = True
def fdiff(self, argindex=1):
if argindex == 1:
# property number 1
return DiracDelta(self.args[0])
else:
raise ArgumentIndexError(self, argindex)
@classmethod
def eval(cls, arg):
arg = sympify(arg)
if arg is S.NaN:
return S.NaN
elif im(arg).is_nonzero:
raise ValueError("Function defined only for Real Values. Complex part: %s found in %s ." % (repr(im(arg)), repr(arg)) )
elif arg.is_negative:
return S.Zero
elif arg.is_zero:
return S.Half
elif arg.is_positive:
return S.One
def _eval_rewrite_as_Piecewise(self, arg):
if arg.is_real:
return Piecewise((1, arg > 0), (S(1)/2, Eq(arg, 0)), (0, True))
