I guess the problem is limit(Ei(-x),x=%plusInfinity) --> failed
https://en.wikipedia.org/wiki/Exponential_integral
Even with c:=1:
I:=integrate(exp(-x)*log(x),x=eps..1/eps,"noPole")
1
- ---
- eps 2 eps 1 1
%e log(eps ) - %e log(----) + 2 Ei(- ---) - 2 Ei(- eps)
2 eps
eps
(1) ---------------------------------------------------------------
2
Type: Union(f1: OrderedCompletion(Expression(Integer)),...)
limit(%,eps,0)
limit(%,eps=0)
(42) [leftHandLimit = - infinity, rightHandLimit = "failed"]
Type: Union(Record(leftHandLimit:
Union(OrderedCompletion(Expression(Integer)),"failed"),rightHandLimit:
Union(OrderedCompletion(Expression(Integer)),"failed")),...)
So the MM/M-results are not better.
On 05.09.2022 14:48, Ralf Hemmecke wrote:
> Consider c>0.
>
> (1) -> integrate(exp(-c*x)*log(x),x=0..%plusInfinity)
>
> (1) "faileed"
>
> Can FriCAS specify "c>0" somehow?
>
> In Mathematica I get.
>
> In[1]:= Integrate[Exp[-c*x]*Log[x], {x, 0, Infinity}]
>
> Out[1]= ConditionalExpression[-((EulerGamma + Log[c])/c), Re[c] > 0]
>
> In Maple I get at least...
>
>> integrate(exp(-c*x)*log(x),x=0..infinity);
>
> exp(-c x) ln(x) + ln(c) + Ei(1, c x) + gamma
> lim - --------------------------------------------
> x -> infinity c
>
> Since we do not have "lim" in OutputForm, it would be hard to return a result
> similar to Maple.
>
> Ralf
>
I:=integrate(exp(-c*x)*log(x),x=eps..1/eps,"noPole")
c
- ---
- c eps 2 eps 1 c
%e log(eps ) - %e log(----) + 2 Ei(- ---) - 2 Ei(- c eps)
2 eps
eps
(1) -------------------------------------------------------------------
2 c
Type: Union(f1: OrderedCompletion(Expression(Integer)),...)
I.f1
c
- ---
- c eps 2 eps 1 c
%e log(eps ) - %e log(----) + 2 Ei(- ---) - 2 Ei(- c eps)
2 eps
eps
(2) -------------------------------------------------------------------
2 c
Type: OrderedCompletion(Expression(Integer))
limit(I.f1,eps=0)
(3) "failed"
Type: Union("failed",...)
Using c^2 instead of c =>
limit(I.f1,eps=0)
(22) [leftHandLimit = - infinity, rightHandLimit = "failed"]
Type: Union(Record(leftHandLimit:
Union(OrderedCompletion(Expression(Integer)),"failed"),rightHandLimit:
Union(OrderedCompletion(Expression(Integer)),"failed")),...)
J:=integrate(exp(-c^2*x)*log(x),x)
2
- c x 2
- %e log(x) + Ei(- c x)
(24) ---------------------------
2
c
Type: Union(Expression(Integer),...)
limit(J,x=0)
(27)
2
log(c ) - log(- 1) - digamma(1)
[leftHandLimit = -------------------------------, rightHandLimit = "failed"]
2
c
Type: Union(Record(leftHandLimit:
Union(OrderedCompletion(Expression(Integer)),"failed"),rightHandLimit:
Union(OrderedCompletion(Expression(Integer)),"failed")),...)
limit(J,x=%plusInfinity)
(34) "failed"
Type: Union("failed",...)
limit(Ei(-x),x=%plusInfinity)
(37) "failed"
Type: Union("failed",...)
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