def deltaintegrate(f, x):
"""The idea for integration is the following:
-If we are dealing with a DiracDelta expression, i.e.:
DiracDelta(g(x)), we try to simplify it.
If we could simplify it, then we integrate the resulting expression.
We already know we can integrate a simplified expression, because only
simple DiracDelta expressions are involved.
If we couldn't simplify it, there are two cases:
1) The expression is a simple expression, then we return the integral
Taking care if we are dealing with a Derivative or with a proper DiracDelta
2) The expression is not simple(i.e. DiracDelta(cos(x))), we can do nothing at all
-If the node is a multiplication node having a DiracDelta term
First we expand it.
If the expansion did work, the we try to integrate the expansion
If not, we try to extract a simple DiracDelta term, then we have two cases
1)We have a simple DiracDelta term, so we return the integral
2)We didn't have a simple term, but we do have an expression with simplified
DiracDelta terms, so we integrate this expression
"""
if not f.has(DiracDelta):
return None
# g(x) = DiracDelta(h(x))
if f.func == DiracDelta:
h = f.simplify(x)
if h == f: #can't simplify the expression
#FIXME: the second term tells whether is DeltaDirac or Derivative
#For integrating derivatives of DiracDelta we need the chain rule
if f.is_simple(x):
if (len(f.args) <= 1 or f.args[1] == 0):
return Heaviside(f.args[0])
else:
return (DiracDelta(f.args[0], f.args[1] - 1) /
f.args[0].as_poly().LC())
else: #let's try to integrate the simplified expression
fh = sympy.integrals.integrate(h, x)
return fh
elif f.is_Mul: #g(x)=a*b*c*f(DiracDelta(h(x)))*d*e
g = f.expand()
if f != g: #the expansion worked
fh = sympy.integrals.integrate(g, x)
if fh and not isinstance(fh, sympy.integrals.Integral):
return fh
else: #no expansion performed, try to extract a simple DiracDelta term
dg, rest_mult = change_mul(f, x)
if not dg:
if rest_mult:
fh = sympy.integrals.integrate(rest_mult, x)
return fh
else:
dg = dg.simplify(x)
if dg.is_Mul: # Take out any extracted factors
dg, rest_mult_2 = change_mul(dg, x)
rest_mult = rest_mult * rest_mult_2
point = solve(dg.args[0], x)[0]
return (rest_mult.subs(x, point) * Heaviside(x - point))
return None
def test_Function_change_name():
assert mcode(abs(x)) == "abs(x)"
assert mcode(ceiling(x)) == "ceil(x)"
assert mcode(arg(x)) == "angle(x)"
assert mcode(im(x)) == "imag(x)"
assert mcode(re(x)) == "real(x)"
assert mcode(conjugate(x)) == "conj(x)"
assert mcode(chebyshevt(y, x)) == "chebyshevT(y, x)"
assert mcode(chebyshevu(y, x)) == "chebyshevU(y, x)"
assert mcode(laguerre(x, y)) == "laguerreL(x, y)"
assert mcode(Chi(x)) == "coshint(x)"
assert mcode(Shi(x)) == "sinhint(x)"
assert mcode(Ci(x)) == "cosint(x)"
assert mcode(Si(x)) == "sinint(x)"
assert mcode(li(x)) == "logint(x)"
assert mcode(loggamma(x)) == "gammaln(x)"
assert mcode(polygamma(x, y)) == "psi(x, y)"
assert mcode(RisingFactorial(x, y)) == "pochhammer(x, y)"
assert mcode(DiracDelta(x)) == "dirac(x)"
assert mcode(DiracDelta(x, 3)) == "dirac(3, x)"
assert mcode(Heaviside(x)) == "heaviside(x)"
assert mcode(Heaviside(x, y)) == "heaviside(x, y)"
def test_latex_Heaviside():
assert latex(Heaviside(x)) == r"\theta\left(x\right)"
assert latex(Heaviside(x)**2) == r"\left(\theta\left(x\right)\right)^{2}"
def deltaintegrate(f, x):
"""
deltaintegrate(f, x)
The idea for integration is the following:
- If we are dealing with a DiracDelta expression, i.e. DiracDelta(g(x)),
we try to simplify it.
If we could simplify it, then we integrate the resulting expression.
We already know we can integrate a simplified expression, because only
simple DiracDelta expressions are involved.
If we couldn't simplify it, there are two cases:
1) The expression is a simple expression: we return the integral,
taking care if we are dealing with a Derivative or with a proper
DiracDelta.
2) The expression is not simple (i.e. DiracDelta(cos(x))): we can do
nothing at all.
- If the node is a multiplication node having a DiracDelta term:
First we expand it.
If the expansion did work, then we try to integrate the expansion.
If not, we try to extract a simple DiracDelta term, then we have two
cases:
1) We have a simple DiracDelta term, so we return the integral.
2) We didn't have a simple term, but we do have an expression with
simplified DiracDelta terms, so we integrate this expression.
Examples
========
>>> from sympy.abc import x, y, z
>>> from sympy.integrals.deltafunctions import deltaintegrate
>>> from sympy import sin, cos, DiracDelta, Heaviside
>>> deltaintegrate(x*sin(x)*cos(x)*DiracDelta(x - 1), x)
sin(1)*cos(1)*Heaviside(x - 1)
>>> deltaintegrate(y**2*DiracDelta(x - z)*DiracDelta(y - z), y)
z**2*DiracDelta(x - z)*Heaviside(y - z)
See Also
========
sympy.functions.special.delta_functions.DiracDelta
sympy.integrals.integrals.Integral
"""
if not f.has(DiracDelta):
return None
from sympy.integrals import Integral, integrate
from sympy.solvers import solve
# g(x) = DiracDelta(h(x))
if f.func == DiracDelta:
h = f.simplify(x)
if h == f: # can't simplify the expression
#FIXME: the second term tells whether is DeltaDirac or Derivative
#For integrating derivatives of DiracDelta we need the chain rule
if f.is_simple(x):
if (len(f.args) <= 1 or f.args[1] == 0):
return Heaviside(f.args[0])
else:
return (DiracDelta(f.args[0], f.args[1] - 1) /
f.args[0].as_poly().LC())
else: # let's try to integrate the simplified expression
fh = integrate(h, x)
return fh
elif f.is_Mul or f.is_Pow: # g(x) = a*b*c*f(DiracDelta(h(x)))*d*e
g = f.expand()
if f != g: # the expansion worked
fh = integrate(g, x)
if fh is not None and not isinstance(fh, Integral):
return fh
else:
# no expansion performed, try to extract a simple DiracDelta term
dg, rest_mult = change_mul(f, x)
if not dg:
if rest_mult:
fh = integrate(rest_mult, x)
return fh
else:
dg = dg.simplify(x)
if dg.is_Mul: # Take out any extracted factors
dg, rest_mult_2 = change_mul(dg, x)
rest_mult = rest_mult*rest_mult_2
point = solve(dg.args[0], x)[0]
return (rest_mult.subs(x, point)*Heaviside(x - point))
return None
def deltaintegrate(f, x):
"""
deltaintegrate(f, x)
The idea for integration is the following:
- If we are dealing with a DiracDelta expression, i.e. DiracDelta(g(x)),
we try to simplify it.
If we could simplify it, then we integrate the resulting expression.
We already know we can integrate a simplified expression, because only
simple DiracDelta expressions are involved.
If we couldn't simplify it, there are two cases:
1) The expression is a simple expression: we return the integral,
taking care if we are dealing with a Derivative or with a proper
DiracDelta.
2) The expression is not simple (i.e. DiracDelta(cos(x))): we can do
nothing at all.
- If the node is a multiplication node having a DiracDelta term:
First we expand it.
If the expansion did work, then we try to integrate the expansion.
If not, we try to extract a simple DiracDelta term, then we have two
cases:
1) We have a simple DiracDelta term, so we return the integral.
2) We didn't have a simple term, but we do have an expression with
simplified DiracDelta terms, so we integrate this expression.
Examples
========
>>> from sympy.abc import x, y, z
>>> from sympy.integrals.deltafunctions import deltaintegrate
>>> from sympy import sin, cos, DiracDelta, Heaviside
>>> deltaintegrate(x*sin(x)*cos(x)*DiracDelta(x - 1), x)
sin(1)*cos(1)*Heaviside(x - 1)
>>> deltaintegrate(y**2*DiracDelta(x - z)*DiracDelta(y - z), y)
z**2*DiracDelta(x - z)*Heaviside(y - z)
See Also
========
sympy.functions.special.delta_functions.DiracDelta
sympy.integrals.integrals.Integral
"""
if not f.has(DiracDelta):
return None
from sympy.integrals import Integral, integrate
from sympy.solvers import solve
# g(x) = DiracDelta(h(x))
if f.func == DiracDelta:
h = f.expand(diracdelta=True, wrt=x)
if h == f: # can't simplify the expression
#FIXME: the second term tells whether is DeltaDirac or Derivative
#For integrating derivatives of DiracDelta we need the chain rule
if f.is_simple(x):
if (len(f.args) <= 1 or f.args[1] == 0):
return Heaviside(f.args[0])
else:
return (DiracDelta(f.args[0], f.args[1] - 1) /
f.args[0].as_poly().LC())
else: # let's try to integrate the simplified expression
fh = integrate(h, x)
return fh
elif f.is_Mul or f.is_Pow: # g(x) = a*b*c*f(DiracDelta(h(x)))*d*e
g = f.expand()
if f != g: # the expansion worked
fh = integrate(g, x)
if fh is not None and not isinstance(fh, Integral):
return fh
else:
# no expansion performed, try to extract a simple DiracDelta term
deltaterm, rest_mult = change_mul(f, x)
if not deltaterm:
if rest_mult:
fh = integrate(rest_mult, x)
return fh
else:
deltaterm = deltaterm.expand(diracdelta=True, wrt=x)
if deltaterm.is_Mul: # Take out any extracted factors
deltaterm, rest_mult_2 = change_mul(deltaterm, x)
rest_mult = rest_mult*rest_mult_2
point = solve(deltaterm.args[0], x)[0]
# Return the largest hyperreal term left after
# repeated integration by parts. For example,
#
# integrate(y*DiracDelta(x, 1),x) == y*DiracDelta(x,0), not 0
#
# This is so Integral(y*DiracDelta(x).diff(x),x).doit()
# will return y*DiracDelta(x) instead of 0 or DiracDelta(x),
# both of which are correct everywhere the value is defined
# but give wrong answers for nested integration.
n = (0 if len(deltaterm.args)==1 else deltaterm.args[1])
m = 0
while n >= 0:
r = (-1)**n*rest_mult.diff(x, n).subs(x, point)
if r.is_zero:
n -= 1
m += 1
else:
if m == 0:
return r*Heaviside(x - point)
else:
return r*DiracDelta(x,m-1)
# In some very weak sense, x=0 is still a singularity,
# but we hope will not be of any practical consequence.
return S.Zero
return None
