def integrate(self, x=None, **kwargs):
"""Integrate Formal Power Series.
Examples
========
>>> from sympy import fps, sin, integrate
>>> from sympy.abc import x
>>> f = fps(sin(x))
>>> f.integrate(x).truncate()
-1 + x**2/2 - x**4/24 + O(x**6)
>>> integrate(f, (x, 0, 1))
-cos(1) + 1
"""
from sympy.integrals import integrate
if x is None:
x = self.x
elif iterable(x):
return integrate(self.function, x)
f = integrate(self.function, x)
ind = integrate(self.ind, x)
ind += (f - ind).limit(x, 0) # constant of integration
pow_xk = self._get_pow_x(self.xk.formula)
ak = self.ak
k = ak.variables[0]
if ak.formula.has(x):
form = []
for e, c in ak.formula.args:
temp = S.Zero
for t in Add.make_args(e):
pow_x = self._get_pow_x(t)
temp += t / (pow_xk + pow_x + 1)
form.append((temp, c))
form = Piecewise(*form)
ak = sequence(form.subs(k, k - 1), (k, ak.start + 1, ak.stop))
else:
ak = sequence((ak.formula / (pow_xk + 1)).subs(k, k - 1),
(k, ak.start + 1, ak.stop))
return self.func(f, self.x, self.x0, self.dir, (ak, self.xk, ind))
def integrate(self, x=None, **kwargs):
"""Integrate Formal Power Series.
Examples
========
>>> from sympy import fps, sin, integrate
>>> from sympy.abc import x
>>> f = fps(sin(x))
>>> f.integrate(x).truncate()
-1 + x**2/2 - x**4/24 + O(x**6)
>>> integrate(f, (x, 0, 1))
-cos(1) + 1
"""
from sympy.integrals import integrate
if x is None:
x = self.x
elif iterable(x):
return integrate(self.function, x)
f = integrate(self.function, x)
ind = integrate(self.ind, x)
ind += (f - ind).limit(x, 0) # constant of integration
pow_xk = self._get_pow_x(self.xk.formula)
ak = self.ak
k = ak.variables[0]
if ak.formula.has(x):
form = []
for e, c in ak.formula.args:
temp = S.Zero
for t in Add.make_args(e):
pow_x = self._get_pow_x(t)
temp += t / (pow_xk + pow_x + 1)
form.append((temp, c))
form = Piecewise(*form)
ak = sequence(form.subs(k, k - 1), (k, ak.start + 1, ak.stop))
else:
ak = sequence((ak.formula / (pow_xk + 1)).subs(k, k - 1),
(k, ak.start + 1, ak.stop))
return self.func(f, self.x, self.x0, self.dir, (ak, self.xk, ind))
def _eval_derivative(self, x):
f = self.function.diff(x)
ind = self.ind.diff(x)
pow_xk = self._get_pow_x(self.xk.formula)
ak = self.ak
k = ak.variables[0]
if ak.formula.has(x):
form = []
for e, c in ak.formula.args:
temp = S.Zero
for t in Add.make_args(e):
pow_x = self._get_pow_x(t)
temp += t * (pow_xk + pow_x)
form.append((temp, c))
form = Piecewise(*form)
ak = sequence(form.subs(k, k + 1), (k, ak.start - 1, ak.stop))
else:
ak = sequence((ak.formula * pow_xk).subs(k, k + 1),
(k, ak.start - 1, ak.stop))
return self.func(f, self.x, self.x0, self.dir, (ak, self.xk, ind))
def _eval_derivative(self, x):
f = self.function.diff(x)
ind = self.ind.diff(x)
pow_xk = self._get_pow_x(self.xk.formula)
ak = self.ak
k = ak.variables[0]
if ak.formula.has(x):
form = []
for e, c in ak.formula.args:
temp = S.Zero
for t in Add.make_args(e):
pow_x = self._get_pow_x(t)
temp += t * (pow_xk + pow_x)
form.append((temp, c))
form = Piecewise(*form)
ak = sequence(form.subs(k, k + 1), (k, ak.start - 1, ak.stop))
else:
ak = sequence((ak.formula * pow_xk).subs(k, k + 1),
(k, ak.start - 1, ak.stop))
return self.func(f, self.x, self.x0, self.dir, (ak, self.xk, ind))
def test_Function_subs():
f, g, h, i = symbols('f g h i', cls=Function)
p = Piecewise((g(f(x, y)), x < -1), (g(x), x <= 1))
assert p.subs(g, h) == Piecewise((h(f(x, y)), x < -1), (h(x), x <= 1))
assert (f(y) + g(x)).subs({f: h, g: i}) == i(x) + h(y)
