Given an authorative answer, it should not be hard to add that translation 
to the sagemath-fricas interface.  Just let me know.

Martin
On Wednesday 10 April 2024 at 09:16:15 UTC+2 Nasser M. Abbasi wrote:

>
> " Maple page says:
>
> Ei(a, z) = z^(a-1)*GAMMA(1-a, z)
>
> In FriCAS that would be
>
> Ei(a, z) == z^(a-1)*Gamma(1 - a, z)
>
> I am not sure if Maple is right, example above leads to Gamma(0, x)
> which is undefined."
>
> But the help page 
> https://www.maplesoft.com/support/help/maple/view.aspx?path=Ei  says
>
> "with the exception of the point 0 in the case of 
> Ei1(z)."
>
> Ok, so the bottom line is that Fricas does not have a builtin function for 
> the two argument version
> for exponential integral function that sagemath can translate the call to.
>
> In this case, for now, I will remove these problems (135 in total from 
> this particular Rubi test file), because now
> Fricas fails all of them since sagemath is assuming Fricas has the two 
> argument version.
>
> --Nasser
>
>
> On Tuesday, April 9, 2024 at 11:30:13 PM UTC-5 Waldek Hebisch wrote:
>
>> On Tue, Apr 09, 2024 at 08:24:35PM -0700, 'Nasser M. Abbasi' via FriCAS - 
>> computer algebra system wrote: 
>> > I found problem integrating many problems using sagemath calling Fricas 
>> to 
>> > do the integration when using exponential integral function. These are 
>> > problems from Rubi test files. 
>> > 
>> > It works OK with other CAS systems supported by sagemath (Maxima and 
>> GIAC) 
>> > but sagemath 10.3 does not seem to correctly translate the call to 
>> Fricas. 
>> > 
>> > I am having hard time finding what the exponential integral function is 
>> > called before I ask at sagemath forum. I looked at the Fricas book and 
>> do 
>> > not see anything,. I tried Ei but this does not work. (i.e. does not 
>> give 
>> > same answer as other cas systems). 
>>
>> Well, 'Ei' is "true exponential integral". Other systems ofer you 
>> variants, in FriCAS it is just 'Ei'. Some variants are equivalent 
>> to incomplete gamma function, in such case FriCAS gives you 
>> incomplete gamma. 
>>
>> > First here is a link to the special function I am taking about 
>> > 
>> > https://reference.wolfram.com/language/ref/ExpIntegralE.html 
>> > 
>> > https://www.maplesoft.com/support/help/maple/view.aspx?path=Ei 
>> > 
>> > Here is a test to what values it should give for some random input. In 
>> > Mathematica it gives 
>> > 
>> > ExpIntegralE[3, 5.0] 
>> > .000877801 
>> > 
>> > In Maple 
>> > 
>> > Ei(3,5.0) 
>> > 0.0008778008928 
>> > 
>> > IN sagemath 10.3 
>> > 
>> > sage: exp_integral_e(3,5.0) 
>> > 0.000877800892770638 
>> > 
>> > But I tried Ei(3,5.0) in Fricas and it gives error. 
>>
>> 'Ei' takes a single argument. If you need relations between various 
>> functions look into Abramowitz and Stegun, FriCAS Ei is exactly as 
>> defined in Abramowitz and Stegun. 
>>
>> > 
>> > Here is an example, using sagemath trying to integrate. It works OK 
>> with 
>> > maxima and giac but gives error with Fricas. I am using 
>> > 
>> > >fricas --version 
>> > FriCAS 1.3.10 
>> > based on sbcl 2.3.11 
>> > 
>> > with sagemath 
>> > >sage --version 
>> > SageMath version 10.3, Release Date: 2024-03-19 
>> > 
>> > Starting sagemath and typing: 
>> > 
>> > sage: var('x a b') 
>> > (x, a, b) 
>> > 
>> > sage: integrate(exp_integral_e(1,b*x),x,algorithm="giac") 
>> > integrate(exp_integral_e(1, b*x), x) 
>> > 
>> > sage: integrate(exp_integral_e(1,b*x),x,algorithm="maxima") 
>> > -exp_integral_e(2, b*x)/b 
>> > 
>> > sage: integrate(exp_integral_e(1,b*x),x,algorithm="fricas") 
>> > RuntimeError Traceback (most recent call last) 
>> > TypeError: An error occurred when FriCAS evaluated 
>> > 'exp_integral_e(((1)::EXPR INT),(b)*(x))': 
>> > There are no library operations named exp_integral_e 
>> > 
>> > So clearly sagemath did not translate the exp_integral_e to Fricas 
>> > correctly. 
>> > 
>> > What should the translation look like? 
>>
>> Maple page says: 
>>
>> Ei(a, z) = z^(a-1)*GAMMA(1-a, z) 
>>
>> In FriCAS that would be 
>>
>> Ei(a, z) == z^(a-1)*Gamma(1 - a, z) 
>>
>> I am not sure if Maple is right, example above leads to Gamma(0, x) 
>> which is undefined. 
>>
>> -- 
>> Waldek Hebisch 
>>
>

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