Post AVZHBbEQL7qJpglyV6 by uxor@mastodon.xyz
(DIR) More posts by uxor@mastodon.xyz
(DIR) Post #AVZHBTXGxAANxpWulk by uxor@mastodon.xyz
2020-04-08T06:39:38Z
2 likes, 3 repeats
#mathematics #maths I've found these two identities while working on an examination about Fourier Series. Do you think the same formula is true when replacing the number in red by 4? If not what should be the result ?
(DIR) Post #AVZHBUY1BsMD6Qt1ZQ by uxor@mastodon.xyz
2020-04-08T09:12:08Z
0 likes, 0 repeats
#mathematics #maths when I want to check such identities I try some computer algebra system like #wxmaxima . the software often give a closed form more complicated than mine, so I compare numerical values and strange things can happend ...
(DIR) Post #AVZHBVHOTAeZNA7GIS by uxor@mastodon.xyz
2020-04-28T12:27:48Z
1 likes, 0 repeats
#mathematics #maths #maxima #wxmaxima I made a #bugreport on the simplify_sum bug I discovered 3 weeks ago. I don't know if the problem comes from the Zeilberger/Gosper algorithm or a lack of knowledge on some special functions ....https://sourceforge.net/p/maxima/bugs/3630/
(DIR) Post #AVZHBbEQL7qJpglyV6 by uxor@mastodon.xyz
2020-04-08T09:25:39Z
0 likes, 0 repeats
#mathematics #maths for those who think that the problem comes from #wxmaxima only you can check that #sage do the same computations . But the question is why ? is the simplification false or is the numerical value of erf function at complex value i wrong?
(DIR) Post #AVZHBfqrAAxiA2mixc by uxor@mastodon.xyz
2020-04-08T09:32:27Z
0 likes, 0 repeats
#mathematics #maths I also tried #wolframalpha webapp , the closed form for the series is even more complicated , but the numerical value looks compatible with my result !
(DIR) Post #AVZHBid8pcm2mNOJV2 by uxor@mastodon.xyz
2020-04-08T09:42:41Z
0 likes, 0 repeats
#mathematics #maths there was a last way to check where does the problem comes from : compute the numerical value of closed form given by #wxmaxima and #sage using #wolframalpha ... so there is a bug in simplify_sum used by #maxima and #sage !
(DIR) Post #AVZMNe0eKxrnh4tPe4 by ai@cawfee.club
2023-05-12T05:43:31.359257Z
0 likes, 0 repeats
@uxor Looks like you can break into two series by rewriting k/(4k^2-1) by factoring the denominator as a difference of squares and then using partial fractions. Also rewrite the sin using exp. The two series should be something like x^k / (2k + 1) summed over k. Then substitute x^2 for x and the series looks like every other term of the Taylor series for log(1+x). So you can get it as log(1+x) + log(1-x). Not sure if these ideas work. I’m sleeping now, but I’ll come back to this tomorrow.
(DIR) Post #AVaKGSpvTS86GRHkVU by uxor@mastodon.xyz
2023-05-12T10:47:13Z
2 likes, 0 repeats
@ai I don't think so. I proved the formulas above using the Dirichlet theorem applied to the Fourier series expansion of the 2*pi periodic function :f(x)= cos(x) for x in [0,pi[ f(x)=0 for x in [-pi,0[my post was about the bug appearing when I want to verify the formula with in maxima (and also with #sagemath) see http://rouxph.blogspot.com/2020/04/calcul-de-series-et-bug-dans-wxmaxima.html
(DIR) Post #AVaKGToBrOKrHLTsRM by ai@cawfee.club
2023-05-12T16:54:32.575838Z
1 likes, 0 repeats
@uxor What you said is a lot nicer, but my way works too (see pic). I get your point about the bug. It's really quite disturbing that maxima can spit out a closed form that is so far off numerically. Three years later, it looks like your bug report has not been addressed at all. Do you know if there's been any work done on this, since your original discussion from 2020-04-07 on the maxima mailing list? Also, I was tired last night and didn't realize I was reviving a 3-year-old post. My apologies. For this, I blame my wonderful mutuals @jeffcliff and @Hyolobrika for not showing me this post sooner. ;)
(DIR) Post #AVaKGUqhzVwaVRfP0K by uxor@mastodon.xyz
2023-05-12T11:04:05Z
2 likes, 0 repeats
@ai so for the sum of k/(4k^2-1)* sin(k*pi/4) you should take x=pi/8 and apply th Dirichlet Theorem to f, then you'll getS= pi*cos(pi/8)/8= pi*sqrt(2-sqrt(2))/16 instead of pi*sqrt(4)/16. A funny fact is that even #wolframalpha can find this closed form for this series .... #mathematics
(DIR) Post #AVacqiWYQL22WBf1fc by uxor@mastodon.xyz
2023-05-12T20:19:28Z
1 likes, 0 repeats
@ai @Hyolobrika @jeffcliff waouh beautiful, looks harder than using the Fourier series expansion but it works ! Do not apologies, I'm happy that this post finally interested someone ;-) the bug report on sourceforge is still open but no one seems to work on this subject https://sourceforge.net/p/maxima/bugs/3630/I recently found another bug of the same type, again with a Fourier series expansion. I will post it soon ...
(DIR) Post #AVeQ8cnbvT2RCfgfOC by uxor@mastodon.xyz
2023-05-14T12:01:37Z
2 likes, 1 repeats
@ai @Hyolobrika @jeffcliff here is another example where #maxima and #sagemath fail to simplify a summation . Wolframalpha.com also fails to find the pi/16 value but gives different closed forms and a numerical value compatible with pi/16sum(1.0*sin(n*pi/3)^3*cos(n*pi/3)/n,n,1,Infinity)### sage false result S= 2.4150948914066888 ## exact value S_exact= 1/16*pi = 0.19634954084936207## partial sum agree with exact valueS_num= 0.11348739774386768*sqrt(3) = 0.1965659389111564#mathematics
(DIR) Post #AVeQ8kV7I6zx5d60fo by uxor@mastodon.xyz
2023-05-14T12:02:00Z
2 likes, 0 repeats
@ai @Hyolobrika @jeffcliff you can check it with with sage and the following source code :var("n")S=sum(sin(n*pi/3)^3*cos(n*pi/3)/n,n,1,Infinity);print("sum(1.0*sin(n*pi/3)^3*cos(n*pi/3)/n,n,1,Infinity)")print("S=",float(S))S_exact=pi/16print("S_exact=",S_exact,"=",float(S_exact))S_num=sum(1.0*sin(n*pi/3)^3*cos(n*pi/3)/n,n,1,1000);print("S_num=",S_num,"=",float(S_num))
(DIR) Post #AVeQelE48BoMbZNvBQ by jeffcliff@shitposter.club
2023-05-14T16:25:02.809738Z
0 likes, 0 repeats
did you report this bug?
(DIR) Post #AVeTd5Kai1LgFAKUxU by uxor@mastodon.xyz
2023-05-14T16:58:11Z
1 likes, 0 repeats
@jeffcliff @Hyolobrika @ai I've added the informations on the original report (bug 3630) and reported on maxima mailing list.
(DIR) Post #AVeZEqYEbmQnBBXL8q by ai@cawfee.club
2023-05-14T18:01:10.858483Z
1 likes, 0 repeats
@uxor @Hyolobrika @jeffcliff It's strange that these programs fail because I feel like there is almost an algorithm for evaluating sums like this. Here's my work, which is similar to my previous post:
(DIR) Post #AVeg6xLc9f2lcXonqa by uxor@mastodon.xyz
2023-05-14T18:25:59Z
2 likes, 0 repeats
@ai @Hyolobrika @jeffcliff such computation relies on the Gosper's algorithm wihich can solve any summations of hypergeometric terms. Gosper worked himself on Macsyma during the seventees !https://en.wikipedia.org/wiki/Gosper%27s_algorithm?wprov=sfla1 https://en.wikipedia.org/wiki/Gosper%27s_algorithm?wprov=sfla1
(DIR) Post #AVegDQotUTjQlTbLeq by chillanarchist01@liberdon.com
2023-05-14T19:19:21Z
0 likes, 0 repeats
@uxor @ai @Hyolobrika @jeffcliff seventies*