def _tderivative_(self, f, x, a, b, diff_param=None):
"""
Return the derivative of symbolic integration
EXAMPLES::
sage: from sage.symbolic.integration.integral import definite_integral
sage: f = function('f'); a,b=var('a,b')
sage: h = definite_integral(f(x), x,a,b)
sage: h.diff(x) # indirect doctest
0
sage: h.diff(a)
-f(a)
sage: h.diff(b)
f(b)
"""
if not x.has(diff_param):
# integration variable != differentiation variable
ans = definite_integral(f.diff(diff_param), x, a, b)
else:
ans = SR.zero()
return (ans + f.subs(x == b) * b.diff(diff_param) -
f.subs(x == a) * a.diff(diff_param))
def _tderivative_(self, f, x, a, b, diff_param=None):
"""
Return the derivative of symbolic integration.
EXAMPLES::
sage: from sage.symbolic.integration.integral import definite_integral
sage: f = function('f'); a,b=var('a,b')
sage: h = definite_integral(f(x), x,a,b)
...
sage: h.diff(x) # indirect doctest
0
sage: h.diff(a)
-f(a)
sage: h.diff(b)
f(b)
TESTS:
Check for :trac:`28656`::
sage: t = var("t")
sage: f = function("f")
sage: F(x) = integrate(f(t),t,0,x)
sage: F(x).diff(x)
f(x)
"""
if not x.has(diff_param):
# integration variable != differentiation variable
ans = definite_integral(f.diff(diff_param), x, a, b)
else:
ans = SR.zero()
if hasattr(b, 'diff'):
ans += f.subs(x == b) * b.diff(diff_param)
if hasattr(a, 'diff'):
ans -= f.subs(x == a) * a.diff(diff_param)
return ans
def closed_form(self, n='n'):
r"""
Return a symbolic expression in ``n``, which equals the n-th term of
the sequence.
It is a well-known property of C-finite sequences ``a_n`` that they
have a closed form of the type:
.. MATH::
a_n = \sum_{i=1}^d c_i(n) \cdot r_i^n,
where ``r_i`` are the roots of the characteristic equation and
``c_i(n)`` is a polynomial (whose degree equals the multiplicity of
``r_i`` minus one). This is a natural generalization of Binet's
formula for Fibonacci numbers. See, for instance, [KP2011, Theorem 4.1].
Note that if the o.g.f. has a polynomial part, that is, if the
numerator degree is not strictly less than the denominator degree,
then this closed form holds only when ``n`` exceeds the degree of that
polynomial part. In that case, the returned expression will differ
from the sequence for small ``n``.
EXAMPLES::
sage: CFiniteSequence(1/(1-x)).closed_form()
1
sage: CFiniteSequence(x^2/(1-x)).closed_form()
1
sage: CFiniteSequence(1/(1-x^2)).closed_form()
1/2*(-1)^n + 1/2
sage: CFiniteSequence(1/(1+x^3)).closed_form()
1/3*(-1)^n + 1/3*(1/2*I*sqrt(3) + 1/2)^n + 1/3*(-1/2*I*sqrt(3) + 1/2)^n
sage: CFiniteSequence(1/(1-x)/(1-2*x)/(1-3*x)).closed_form()
9/2*3^n - 4*2^n + 1/2
Binet's formula for the Fibonacci numbers::
sage: CFiniteSequence(x/(1-x-x^2)).closed_form()
sqrt(1/5)*(1/2*sqrt(5) + 1/2)^n - sqrt(1/5)*(-1/2*sqrt(5) + 1/2)^n
sage: [_.subs(n=k).full_simplify() for k in range(6)]
[0, 1, 1, 2, 3, 5]
sage: CFiniteSequence((4*x+3)/(1-2*x-5*x^2)).closed_form()
1/2*(sqrt(6) + 1)^n*(7*sqrt(1/6) + 3) - 1/2*(-sqrt(6) + 1)^n*(7*sqrt(1/6) - 3)
Examples with multiple roots::
sage: CFiniteSequence(x*(x^2+4*x+1)/(1-x)^5).closed_form()
1/4*n^4 + 1/2*n^3 + 1/4*n^2
sage: CFiniteSequence((1+2*x-x^2)/(1-x)^4/(1+x)^2).closed_form()
1/12*n^3 - 1/8*(-1)^n*(n + 1) + 3/4*n^2 + 43/24*n + 9/8
sage: CFiniteSequence(1/(1-x)^3/(1-2*x)^4).closed_form()
4/3*(n^3 - 3*n^2 + 20*n - 36)*2^n + 1/2*n^2 + 19/2*n + 49
sage: CFiniteSequence((x/(1-x-x^2))^2).closed_form()
1/5*(n - sqrt(1/5))*(1/2*sqrt(5) + 1/2)^n + 1/5*(n + sqrt(1/5))*(-1/2*sqrt(5) + 1/2)^n
"""
from sage.arith.all import binomial
from sage.rings.qqbar import QQbar
from sage.symbolic.ring import SR
n = SR(n)
expr = SR.zero()
R = FractionField(PolynomialRing(QQbar, self.parent().variable_name()))
ogf = R(self.ogf())
__, parts = ogf.partial_fraction_decomposition(decompose_powers=False)
for part in parts:
denom = part.denominator().factor()
denom_base, denom_exp = denom[0]
# denominator is of the form (x+b)^{m+1}
m = denom_exp - 1
b = denom_base.constant_coefficient()
# check that the partial fraction decomposition was indeed done correctly
# (that is, there is only one factor, of degree 1, and monic)
assert len(denom) == 1 and len(denom_base.list(
)) == 2 and denom_base[1] == 1 and denom.unit() == 1
r = SR((-1 / b).radical_expression())
c = SR.zero()
for k, a in enumerate(part.numerator()):
a = -QQbar(a) if k % 2 else QQbar(a)
bino = binomial(n + m - k, m)
c += bino * SR((a * b**(k - m - 1)).radical_expression())
expr += c.expand() * r**n
return expr
