Re: [Maxima-discuss] simplify_sum leads to a wrong value

[andre m. <and...@gm...>]

FYI. simplify_sum gives for me a sum of 6 gamma_incomplete_lower terms

with the help of
https://mathworld.wolfram.com/IncompleteGammaFunction.html
shouldn't e.g. gamma_incomplete_lower(4,-1) be computable exactly?

I wonder what's the preferred way of exactly evaluating such integrals with Maxima?

- Andre

-----
$ rmaxima
Maxima 5.45.1 https://maxima.sourceforge.io
using Lisp SBCL 2.0.1-8.fc36
Distributed under the GNU Public License. See the file COPYING.
Dedicated to the memory of William Schelter.
The function bug_report() provides bug reporting information.
(%i1) display2d : false;

(%o1) false
(%i2) load("simplify_sum");

(%o2) "/usr/share/maxima/5.45.1/share/solve_rec/simplify_sum.mac"
(%i3) S : sum(sin(n*%pi/3)^3*cos(n*%pi/3)/n,n,1,inf);

(%o3) 'sum((cos((%pi*n)/3)*sin((%pi*n)/3)^3)/n,n,1,inf)
(%i4) h : ratsimp(simplify_sum(S));

(%o4) (3^(3/2)*'realpart(gamma_incomplete_lower(4,-1))
  +3^(3/2)*%i*'imagpart(gamma_incomplete_lower(4,-1))
  -3^(5/2)*'realpart(gamma_incomplete_lower(3,-1))
  -3^(5/2)*%i*'imagpart(gamma_incomplete_lower(3,-1))
  +3^(3/2)*'realpart(gamma_incomplete_lower(2,-1))
  +3^(3/2)*%i*'imagpart(gamma_incomplete_lower(2,-1)))
  /8
(%i5) a:4; x:-1; hgfred([a],[1+a],-x);

(%o5) 4
(%o6) -1
(%o7) 4*gamma_incomplete_lower(4,-1)
-----

On 5/14/23 14:28, philippe wrote:
> Hi,
>
> 3 years after our discussion about simplify_sum errors, I've found another example where maxima (and sagemath) fails to simplify a summation. Wolframalpha.com also fails to find the simplest expression (pi/16) but gives different closed forms and a numerical value compatible with it. Here is a minimal example showing the problem  in maxima :
>
> 'S=S:sum(sin(n*%pi/3)^3*cos(n*%pi/3)/n,n,1,inf);
> 'S=S:ratsimp(simplify_sum(S2)); /* false!*/
> S_exact:%pi/16;
> S_num:float(sum(sin(n*%pi/3)^3*cos(n*%pi/3)/n,n,1,10000));
> S_exact=float(S_exact);
>
> you can check that the value given by simplify_sum is far from the numerical value, and this one is compatible with pi/16 (which has been obtained by Dirichlet Theorem for the Fourier series of some  piecewise constant function):
>
> S= 3/8*sqrt(3)*e + 3/8*sqrt(3) = 2.4150948914066888
> S_exact= 1/16*pi = 0.19634954084936207
> S_num= 0.1965659389111564
>
> I've added this example in comment to bug report 3630 :
>
> https://sourceforge.net/p/maxima/bugs/3630/
>
> thanks for reading,
> Philippe
>
> Le 07/04/2020 à 22:52, philippe a écrit :
>> Hi,
>>
>> I want to check the limit of the following serie
>>
>> sum((k*sin((k*%pi)/3))/(4*k^2-1),k,1,inf)
>>
>> I know it converge to (sqrt(3)*%pi)/16=0.3400873807939158...
>>
>> simplify_sum find the limit in terms of erf(%i) but floating point
>> approximation is far from the expected value : 2.564242426927912...
>>
>> I don't know if the problem comes from simplify_sum or floating point
>> approximations of erf function, but there is no error for the similar serie
>>
>> sum(k*sin(k*%pi/2)/(4*k^2-1),k,1,inf)=sqrt(2)*%pi/16
>>
>> if some one is interested in this problem here is a short piece of code
>> to test it :
>>
>> load(simplify_sum)$
>> S2:sum(k*sin(k*%pi/3)/(4*k^2-1),k,1,inf)$
>> S2s:simplify_sum(S2)$
>> S2c:sqrt(3)*%pi/16$
>> print(S2,"=",S2s,"=",float(S2s));
>> print(S2,"=",S2c,"=",float(S2c));
>>
>> thanks for reading,
>> Philippe
>
>
>
>
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