"William Stein" <[EMAIL PROTECTED]> writes:
> * Axiom?
Axiom does *elementary* integration. That is, if the Risch algorithm applies,
it will find the result except in a few cases. It does have some pattern
matching abilities, but these are not really worth mentioning.
FriCAS (axiom fork, which is the one used for Sage) fixes quite a few bugs in
the old axiom integrator. We (FriCASers) are lucky to have Waldek as
maintainer, who knows integration fairly well.
I'd like to add that I'm working on a domain (in the FriCAS sense) that
implements "power series that satisfy an ADE", and these are closed under
integration, too. Of course, in most cases, the result will be an ADE, not an
elementary function. Below is a rudimentary example "on foot".
By the way, I think that "axiom.someop" should become "fricas.someop" in SAGE.
"axiom" is (meanwhile) quite misleading, FriCAS and axiom have parted.
Martin
Below, how to integrate "sum n^n/factorial n z^n" on foot:
(1) -> series([n^n/factorial n for n in 0..])$UTS(FRAC INT, x, 0)
(1)
2 9 3 32 4 625 5 324 6 117649 7 131072 8
1 + x + 2x + - x + -- x + --- x + --- x + ------ x + ------ x
2 3 24 5 720 315
+
4782969 9 1562500 10 11
------- x + ------- x + O(x )
4480 567
Type: UnivariateTaylorSeries(Fraction(Integer),x,0)
(2) -> l := [1,1,2,9/2,32/3,625/24,324/5];
Type: List(Fraction(Integer))
(3) -> guessADE l
(3)
n , 3 2
[[function= [[x ]f(x): - xf (x) + f(x) - f(x) = 0,f(0)= 1(0)!],order= 0]]
Type: List(Record(function: Expression(Integer),order: NonNegativeInteger))
(4) -> integrate series([n^n/factorial n for n in 0..])$UTS(FRAC INT, x, 0)
(4)
1 2 2 3 9 4 32 5 625 6 324 7 117649 8 131072 9
x + - x + - x + - x + -- x + --- x + --- x + ------ x + ------ x
2 3 8 15 144 35 5760 2835
+
4782969 10 11
------- x + O(x )
44800
Type: UnivariateTaylorSeries(Fraction(Integer),x,0)
(5) -> guessADE entries complete first(coefficients %, 10)
(5)
[
n ,
[function= [[x ]f(x): (- f(x) - x + 1)f (x) + f(x) - 1= 0,f(0)= 0(0)!],
order= 0]
]
Type: List(Record(function: Expression(Integer),order: NonNegativeInteger))
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