# 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, sign
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

========

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

========

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)
result = 0
valid = True
darg = abs(diff(self.args[0], x))
for r, m in argroots.items():
if r.is_real is not False and m == 1:
result += self.func(x - r)/darg.subs(x, r)
else:
# don't handle non-real and if m != 1 then
# a polynomial will have a zero in the derivative (darg)
# at r
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

========

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

========

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))

def _eval_rewrite_as_sign(self, arg):
if arg.is_real:
return (sign(arg)+1)/2