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Showing 22 results of 22

 Re: [Maxima-discuss] question about function identification From: Dimiter Prodanov - 2014-03-12 21:42:27 Attachments: Message as HTML ```Dear Richard, Thank you for the clarification. I define a formal operator for the Fourier transform: FT declare (FT, linear)\$ with the usual linear properties. In the body of the forward transform method I invoke the method to break down the transform into terms and isolate the usual constants. ft(expr,t,f) := .... ret:FT(expr), ... Then I expect to invoke a function identifying 'special' functions that that can't be directly evaluated by the Maxima integral call. Every function such as that I provide in a tabular form, like in your partition.mac program. For example if isOp(expr, _rect) then ( ret:apply1(ret, ft_rect), return (ret) ), in this case the ft_rect is given as a rule of the form defrule(ft_rect, FT(_rect(t, T)), 1/abs(T)*'sinc(f,1/T)), I end up with 2 functions doing the same 'rect' because of the transparency which you clarified. rect(t,T):=funmake (_rect, [t ,T])\$ _rect(t,T):= block( if abs (t)<= T/2 then return(1) else return (0) )\$ isOp(V, oper):= block( if atom(V) then return (false) else return (is(op(V) = oper)) )\$ While this construct works it is somehow cumbersome and I hope to unify it in a 1 functional definition. Indeed I wish to tag objects as you nicely put it . >> If you wish to describe an object by tagging it as rect(...) then that is not a function call. So in this case the transparency of evaluation is not useful. best regards, Dimiter ```
 Re: [Maxima-discuss] Maxima-discuss Digest, Vol 2, Issue 21 From: Leo Butler - 2014-03-12 21:13:23 ```> Hi, > I have recently started to learn Maxima. I have spent quite some time on > Mathematica and I do like Maxima as much as well. Welcome to Maxima, Pankaj. > There is a > communityfor mathematica where > its better reached and presented. Will it not be a > good idea if we had something similar as > Maxima;. > Please join this community, so that we can better manage all the > interactions. This is a good idea. Stackexchange has created a pretty good q&a forum. > I have also posted 3-4 questions, please post a few so that we can attract > more and get going with it. I have taken the liberty of including my answers to these questions. 1. How does Maxima combine features like macros, operators creation, of lisp in Maxima code ? Maxima has buildq and ::= to create and define macros. Lisp code can also be loaded into Maxima, there is an escape to the lisp repl (to_lisp) and the one-line escape :lisp (...) 2. Maxima calls GNUPlot to accomplish graphics, so is it that it allows snippets of GNUPlot as it is or are there some minor changes required ? Maxima has two interfaces to gnuplot (note the spelling: gnuplot is not part of GNU) and both allow the user to pass as much or as little gnuplot code as they want. The draw package, in particular, provides a good abstraction layer that extends gnuplot considerably. If you find you are needing to add gnuplot code, it is likely that the functionality is provided by draw but you are not aware of it. 3. Maxima supports funtional programming but is it possible to create macros that will do it in Haskell style curry ? Or does haskell monad help here ? How about (%i1) curry(f,args,extras):=buildq([f:f,args:args,extras:extras], lambda(args,apply(f,append(args,extras))))\$ (%i2) curry(lambda([a,b],a+b),[a],[2]); (%o2) lambda([a],apply(lambda([a,b],a+b),append([a],[2]))) (%i3) %o2(1); (%o3) 3 Maxima also has parameterized functions. The legacy implementation is tricky, and arguably buggy, but f[b](a) := a+b; creates an array f so that f[2] is equivalent to %o2. (%i4) f[b](a) := a+b \$ (%i5) f[2]; (%o5) lambda([a],a+2) Leo ```
 Re: [Maxima-discuss] Maxima-discuss Digest, Vol 2, Issue 21 From: Pankaj Sejwal - 2014-03-12 19:45:28 Attachments: Message as HTML ```On Wed, Mar 12, 2014 at 11:38 PM, < maxima-discuss-request@...> wrote: > maxima-discuss@... Hi, I have recently started to learn Maxima. I have spent quite some time on Mathematica and I do like Maxima as much as well. There is a communityfor mathematica where its better reached and presented. Will it not be a good idea if we had something similar as Maxima;. Please join this community, so that we can better manage all the interactions. I have also posted 3-4 questions, please post a few so that we can attract more and get going with it. Regards, Pankaj Sejwal ```
 Re: [Maxima-discuss] question about function identification From: Stavros Macrakis - 2014-03-12 18:26:47 Attachments: Message as HTML ```I am not sure how ex1:'rect(...) can break your package but ex2:funmake('_rect, ...) does not. Perhaps the missing piece of information is that expanding out the function definition in ex1 requires the 'nouns' modifier: ev(ex1,nouns)? -s On Wed, Mar 12, 2014 at 10:45 AM, Dimiter Prodanov wrote: > No it does not. This will break the architecture of the package. > I can submit the full code if you want. > > best regards, > > Dimiter > > > On Wed, Mar 12, 2014 at 3:22 PM, Gunter Königsmann wrote: > >> Would >> u:'rect(t,T); >> Solve your problem? >> >> Gunter. >> >> On 12. März 2014 15:13:16 MEZ, Dimiter Prodanov >> wrote: >> >>> Hi Stavros, >>> >>> Thanks for the reply: >>> >>> [image: Inline image 1] >>> I wan to have op= rect >>> >>> best regards, >>> >>> Dimiter >>> >>> On Wed, Mar 12, 2014 at 2:54 PM, Stavros Macrakis >> > wrote: >>> >>>> Not sure exactly what you're trying to do, but if you write 'rect(...), >>>> you will quote the function application. Also, a simpler definition of your >>>> function is >>>> >>>> rect(t,T) := if abs(t) <= T/2 then 1 else 0\$ >>>> >>>> -s >>>> >>>> >>>> >>>> On Wed, Mar 12, 2014 at 8:40 AM, Dimiter Prodanov wrote: >>>> >>>>> Dear list, >>>>> >>>>> I am writing a maxima package for symbolic Fourier transform. >>>>> All well and good but I have some issues >>>>> >>>>> I would like to define some special functions and later identify them >>>>> using an op() call >>>>> So far only this approach works: >>>>> >>>>> rect(t,T):=funmake (_rect, [t ,T])\$ >>>>> _rect(t,T):= block( >>>>> if abs (t)<= T/2 then return(1) >>>>> else return (0) >>>>> )\$ >>>>> >>>>> >>>>> but this is rather awkward. Instead I would like to be able to >>>>> identify a function call like >>>>> >>>>> if (is(op(expr) = rect) then ( >>>>> ret:apply1(ret, ft_rect), " apply Fourier rect rule", >>>>> return (ret) >>>>> ), >>>>> >>>>> If I define rect only in a block statement then op(rect) actually >>>>> returns "if" which is totally useless. >>>>> >>>>> Am I missing something or Maxima script has some weird assumptions? >>>>> >>>>> best regards, >>>>> >>>>> Dimiter >>>>> >>>>> >>>>> >>>>> >>>>> >>>>> >>>>> >>>>> >>>>> >>>>> ------------------------------------------------------------------------------ >>>>> Learn Graph Databases - Download FREE O'Reilly Book >>>>> "Graph Databases" is the definitive new guide to graph databases and >>>>> their >>>>> applications. Written by three acclaimed leaders in the field, >>>>> this first edition is now available. Download your free book today! >>>>> http://p.sf.net/sfu/13534_NeoTech >>>>> _______________________________________________ >>>>> Maxima-discuss mailing list >>>>> Maxima-discuss@... >>>>> https://lists.sourceforge.net/lists/listinfo/maxima-discuss >>>>> >>>>> >>>> >>> ------------------------------ >>> >>> Learn Graph Databases - Download FREE O'Reilly Book >>> "Graph Databases" is the definitive new guide to graph databases and their >>> >>> >>> applications. Written by three acclaimed leaders in the field, >>> this first edition is now available. Download your free book today! >>> http://p.sf.net/sfu/13534_NeoTech >>> >>> ------------------------------ >>> >>> Maxima-discuss mailing list >>> Maxima-discuss@... >>> >>> https://lists.sourceforge.net/lists/listinfo/maxima-discuss >>> >>> >> -- >> Diese Nachricht wurde von meinem Android-Mobiltelefon mit K-9 Mail >> gesendet. >> > > ```
 Re: [Maxima-discuss] Maxima and 1 + 2 + 3 + .... From: Mike Valenzuela - 2014-03-12 18:18:22 Attachments: Message as HTML ```I apologize for some of my bad grammar in my previous email. I used the incorrect term "partial sums" when I meant "numerically calculating the result from finite sums." I am sure I made more slip-ups too. I was trying to write the email quickly. Also, I meant to include the following link as well: https://www.youtube.com/watch?annotation_id=annotation_3085392237&feature=iv&src_vid=w-I6XTVZXww&v=E-d9mgo8FGk On Wed, Mar 12, 2014 at 11:08 AM, Mike Valenzuela wrote: > So the source that I originally heard all this from is: > > > https://www.youtube.com/results?search_query=analytical%20continuation%20zeta%20function&sm=3 > and > > https://www.youtube.com/watch?v=w-I6XTVZXww&index=2&list=PLt5AfwLFPxWK2zCU-4X1iuuu5m8hf6L1B > > I'm also talking about the Riemann zeta function which was extended to the > entire complex domain and asks the question known as the Riemann > hypothesis. The Riemann zeta function is one way to explain the casimir > effect. > > I'm possibly least qualified to make the following explanations (I'm just > an engineer, not a physicist nor mathematician), but it was thought I had > in passing. In my mind, here a few conceivable explanations: > > (1) lim(n--> inf) sum(s,s,1,n) # sum(s,s,1,inf) > Where # is maxima notation for not equals. The LHS assumed a finite value > of *n* at first. Maxima will even simplify this to (*n*^2+*n*)/2, which > only makes sense if *n* is finite. The limit need not equal the true > value! Imagine a function where a single point is displaced (*e.g.*, x/x, > but defined to be 0 at x=0). The RHS never assumes finiteness. You would > never be able to complete the sum of the RHS. Even if given until the > universe grinds to a halt, one would never be able to calculate the RHS > using partial sums. > > (2) Physics might never use limits as mathematicians do. Perhaps physics > only ever uses psuedo-limits. The Cesaro mean can be used to produce a > pseudo-limit of the series (+1 -1 +1 -1 +1 -1...) ( > http://en.wikipedia.org/wiki/Ces%C3%A0ro_mean). A fair number of > techniques produce a value "at infinity" when the limit does not exist. > > (3) A little more research shows that what this entire issue is known as > regularization (http://en.wikipedia.org/wiki/Zeta_regularization). Simply > assigning values to divergent sums (or products). The Riemann zeta function > converges everywhere except s=1, due to analytic continuation. This perhaps > agrees with (2), Might physics somehow avoid infinities by using > regularization? > > > Anyways, those are my thoughts after sleeping on this problem. > > > > On Wed, Mar 12, 2014 at 9:27 AM, Richard Fateman wrote: > >> Based on absolutely no discussion with a human who believes the "-1/12", >> here's my take on it. >> >> There are other situations in which a mathematical result has two parts, >> one of which is >> infinite (or grows without limit as some parameter increases) and one >> is finite >> in the same circumstances. >> >> I encountered this in implementing programs for asymptotic series >> (see "method of multiple scales") >> >> You do some calculation and get another term in the series. You look at >> it >> with great suspicion because it doesn't have the right properties, but >> part of it >> looks possible. What is done is the new term is separated into two >> parts the "secular" part are separated and the other. >> The next step is to throw out the secular part because it grows without >> bound, and ruins the >> whole idea. (example: as t-> oo, t*cos(t) is secular. cos(t) is >> not. ) >> >> So my take on this "-1/12" is that the conventional answer is "infinity >> - 1/12" -- that is, it diverges -- but for >> convenience in matching boundary conditions or some such excuse, the >> infinity is thrown out. I >> think the excuses probably are different from physicists and from >> (non-physicist) mathematicians >> who want a number for zeta(-1), and don't particularly care about string >> theory. >> RJF >> >> >> >> >> >> On 3/12/2014 7:11 AM, Dudley, Jeremy wrote: >> > So, if I pay someone £1 today, £4, tomorrow, £9 the day after, and >> continue until all eternity, they will owe me nothing when the time comes >> to repay me? And by extension, if I stop paying them before eternity is >> reached I will owe them the difference between what I have paid and zero? >> > >> > I saw in the quoted Wordpress article that the zeta(0) value (1 + 1 + 1 >> ...) being -1/12 is used in string theory, to help normalise infinity. But >> the article goes on to point out that the 'analytic continuation' answers >> for z(0), z(-1), and so on become inconsistent as they are manipulated - >> never mind being inconsistent with 'normal' mathematics. >> > >> > -----Original Message----- >> > From: andre maute [mailto:andre.maute@...] >> > Sent: Wednesday, March 12, 2014 11:33 AM >> > To: maxima-discuss@... >> > Subject: Re: [Maxima-discuss] Maxima and 1 + 2 + 3 + .... >> > >> > On 03/12/2014 10:08 AM, Mike Valenzuela wrote: >> >> I encountered some odd results from some numberphil videos and >> >> apparently, even in physics: >> >> sum(s,s,1,inf) = -1/12 >> >> >> >> Maxima claims the sum is divergent. >> >> >> >> Is this incorrect or does the result -1/12 require some unstated >> >> assumptions, such as complex domain? Do not all physicists or >> >> mathematicians agree that the sum is well defined? >> >> >> >> Apparently all the outputs from the riemann zeta function converge >> >> except for the value 1.0. This is surprising as 1+2+3+4+... is just >> >> zeta(-1). The sum of all squares, up "through" infinity is zero. The >> >> sum of all cubes is 1/120. >> >> >> > Perhaps you should read >> > >> > >> http://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/ >> > >> > Andre >> > >> > >> > >> ------------------------------------------------------------------------------ >> > Learn Graph Databases - Download FREE O'Reilly Book "Graph Databases" >> is the definitive new guide to graph databases and their applications. >> Written by three acclaimed leaders in the field, this first edition is now >> available. Download your free book today! >> > http://p.sf.net/sfu/13534_NeoTech >> > _______________________________________________ >> > Maxima-discuss mailing list >> > Maxima-discuss@... >> > https://lists.sourceforge.net/lists/listinfo/maxima-discuss >> > >> > >> ------------------------------------------------------------------------------------------ >> > To read our latest newsletter visit http://www.wrcplc.co.uk/news.aspx- Keeping our clients up-to-date with WRc's business developments. >> > >> ------------------------------------------------------------------------------------------ >> > Visit our website http://www.wrcplc.co.uk as well as http://www.waterportfolio.comfor collaborative research projects. >> > >> ------------------------------------------------------------------------------------------ >> > Follow us on Twitter: https://twitter.com/WRcplc, LinkedIn: >> http://www.linkedin.com/company/wrc-plc and >> > Like Us on Facebook: >> http://www.facebook.com/pages/WRc-plc/288329784573893. >> > >> ------------------------------------------------------------------------------------------ >> > The Information in this e-mail is confidential and is intended solely >> for the addressee. Access to this e-mail by any other party is >> unauthorised. If you are not the intended recipient, any disclosure, >> copying, distribution or any action taken in reliance on the information >> contained in this e-mail is prohibited and maybe unlawful. When addressed >> to WRc Clients, any opinions or advice contained in this e-mail are subject >> to the terms and conditions expressed in the governing WRc Client Agreement. >> > >> ------------------------------------------------------------------------------------------ >> > WRc plc is a company registered in England and Wales. Registered office >> address: Frankland Road, Blagrove, Swindon, Wiltshire SN5 8YF. Company >> registration number 2262098. VAT number 527 1804 53. >> > >> ------------------------------------------------------------------------------------------ >> > >> > >> ------------------------------------------------------------------------------ >> > Learn Graph Databases - Download FREE O'Reilly Book >> > "Graph Databases" is the definitive new guide to graph databases and >> their >> > applications. Written by three acclaimed leaders in the field, >> > this first edition is now available. Download your free book today! >> > http://p.sf.net/sfu/13534_NeoTech >> > _______________________________________________ >> > Maxima-discuss mailing list >> > Maxima-discuss@... >> > https://lists.sourceforge.net/lists/listinfo/maxima-discuss >> >> >> >> ------------------------------------------------------------------------------ >> Learn Graph Databases - Download FREE O'Reilly Book >> "Graph Databases" is the definitive new guide to graph databases and their >> applications. Written by three acclaimed leaders in the field, >> this first edition is now available. Download your free book today! >> http://p.sf.net/sfu/13534_NeoTech >> _______________________________________________ >> Maxima-discuss mailing list >> Maxima-discuss@... >> https://lists.sourceforge.net/lists/listinfo/maxima-discuss >> > > ```
 Re: [Maxima-discuss] Maxima and 1 + 2 + 3 + .... From: Mike Valenzuela - 2014-03-12 18:08:37 Attachments: Message as HTML ```So the source that I originally heard all this from is: https://www.youtube.com/results?search_query=analytical%20continuation%20zeta%20function&sm=3 and https://www.youtube.com/watch?v=w-I6XTVZXww&index=2&list=PLt5AfwLFPxWK2zCU-4X1iuuu5m8hf6L1B I'm also talking about the Riemann zeta function which was extended to the entire complex domain and asks the question known as the Riemann hypothesis. The Riemann zeta function is one way to explain the casimir effect. I'm possibly least qualified to make the following explanations (I'm just an engineer, not a physicist nor mathematician), but it was thought I had in passing. In my mind, here a few conceivable explanations: (1) lim(n--> inf) sum(s,s,1,n) # sum(s,s,1,inf) Where # is maxima notation for not equals. The LHS assumed a finite value of *n* at first. Maxima will even simplify this to (*n*^2+*n*)/2, which only makes sense if *n* is finite. The limit need not equal the true value! Imagine a function where a single point is displaced (*e.g.*, x/x, but defined to be 0 at x=0). The RHS never assumes finiteness. You would never be able to complete the sum of the RHS. Even if given until the universe grinds to a halt, one would never be able to calculate the RHS using partial sums. (2) Physics might never use limits as mathematicians do. Perhaps physics only ever uses psuedo-limits. The Cesaro mean can be used to produce a pseudo-limit of the series (+1 -1 +1 -1 +1 -1...) ( http://en.wikipedia.org/wiki/Ces%C3%A0ro_mean). A fair number of techniques produce a value "at infinity" when the limit does not exist. (3) A little more research shows that what this entire issue is known as regularization (http://en.wikipedia.org/wiki/Zeta_regularization). Simply assigning values to divergent sums (or products). The Riemann zeta function converges everywhere except s=1, due to analytic continuation. This perhaps agrees with (2), Might physics somehow avoid infinities by using regularization? Anyways, those are my thoughts after sleeping on this problem. On Wed, Mar 12, 2014 at 9:27 AM, Richard Fateman wrote: > Based on absolutely no discussion with a human who believes the "-1/12", > here's my take on it. > > There are other situations in which a mathematical result has two parts, > one of which is > infinite (or grows without limit as some parameter increases) and one > is finite > in the same circumstances. > > I encountered this in implementing programs for asymptotic series > (see "method of multiple scales") > > You do some calculation and get another term in the series. You look at it > with great suspicion because it doesn't have the right properties, but > part of it > looks possible. What is done is the new term is separated into two > parts the "secular" part are separated and the other. > The next step is to throw out the secular part because it grows without > bound, and ruins the > whole idea. (example: as t-> oo, t*cos(t) is secular. cos(t) is > not. ) > > So my take on this "-1/12" is that the conventional answer is "infinity > - 1/12" -- that is, it diverges -- but for > convenience in matching boundary conditions or some such excuse, the > infinity is thrown out. I > think the excuses probably are different from physicists and from > (non-physicist) mathematicians > who want a number for zeta(-1), and don't particularly care about string > theory. > RJF > > > > > > On 3/12/2014 7:11 AM, Dudley, Jeremy wrote: > > So, if I pay someone £1 today, £4, tomorrow, £9 the day after, and > continue until all eternity, they will owe me nothing when the time comes > to repay me? And by extension, if I stop paying them before eternity is > reached I will owe them the difference between what I have paid and zero? > > > > I saw in the quoted Wordpress article that the zeta(0) value (1 + 1 + 1 > ...) being -1/12 is used in string theory, to help normalise infinity. But > the article goes on to point out that the 'analytic continuation' answers > for z(0), z(-1), and so on become inconsistent as they are manipulated - > never mind being inconsistent with 'normal' mathematics. > > > > -----Original Message----- > > From: andre maute [mailto:andre.maute@...] > > Sent: Wednesday, March 12, 2014 11:33 AM > > To: maxima-discuss@... > > Subject: Re: [Maxima-discuss] Maxima and 1 + 2 + 3 + .... > > > > On 03/12/2014 10:08 AM, Mike Valenzuela wrote: > >> I encountered some odd results from some numberphil videos and > >> apparently, even in physics: > >> sum(s,s,1,inf) = -1/12 > >> > >> Maxima claims the sum is divergent. > >> > >> Is this incorrect or does the result -1/12 require some unstated > >> assumptions, such as complex domain? Do not all physicists or > >> mathematicians agree that the sum is well defined? > >> > >> Apparently all the outputs from the riemann zeta function converge > >> except for the value 1.0. This is surprising as 1+2+3+4+... is just > >> zeta(-1). The sum of all squares, up "through" infinity is zero. The > >> sum of all cubes is 1/120. > >> > > Perhaps you should read > > > > > http://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/ > > > > Andre > > > > > > > ------------------------------------------------------------------------------ > > Learn Graph Databases - Download FREE O'Reilly Book "Graph Databases" is > the definitive new guide to graph databases and their applications. Written > by three acclaimed leaders in the field, this first edition is now > available. Download your free book today! > > http://p.sf.net/sfu/13534_NeoTech > > _______________________________________________ > > Maxima-discuss mailing list > > Maxima-discuss@... > > https://lists.sourceforge.net/lists/listinfo/maxima-discuss > > > > > ------------------------------------------------------------------------------------------ > > To read our latest newsletter visit http://www.wrcplc.co.uk/news.aspx - > Keeping our clients up-to-date with WRc's business developments. > > > ------------------------------------------------------------------------------------------ > > Visit our website http://www.wrcplc.co.uk as well as http://www.waterportfolio.comfor collaborative research projects. > > > ------------------------------------------------------------------------------------------ > > Follow us on Twitter: https://twitter.com/WRcplc, LinkedIn: > http://www.linkedin.com/company/wrc-plc and > > Like Us on Facebook: > http://www.facebook.com/pages/WRc-plc/288329784573893. > > > ------------------------------------------------------------------------------------------ > > The Information in this e-mail is confidential and is intended solely > for the addressee. Access to this e-mail by any other party is > unauthorised. If you are not the intended recipient, any disclosure, > copying, distribution or any action taken in reliance on the information > contained in this e-mail is prohibited and maybe unlawful. When addressed > to WRc Clients, any opinions or advice contained in this e-mail are subject > to the terms and conditions expressed in the governing WRc Client Agreement. > > > ------------------------------------------------------------------------------------------ > > WRc plc is a company registered in England and Wales. Registered office > address: Frankland Road, Blagrove, Swindon, Wiltshire SN5 8YF. Company > registration number 2262098. VAT number 527 1804 53. > > > ------------------------------------------------------------------------------------------ > > > > > ------------------------------------------------------------------------------ > > Learn Graph Databases - Download FREE O'Reilly Book > > "Graph Databases" is the definitive new guide to graph databases and > their > > applications. Written by three acclaimed leaders in the field, > > this first edition is now available. Download your free book today! > > http://p.sf.net/sfu/13534_NeoTech > > _______________________________________________ > > Maxima-discuss mailing list > > Maxima-discuss@... > > https://lists.sourceforge.net/lists/listinfo/maxima-discuss > > > > ------------------------------------------------------------------------------ > Learn Graph Databases - Download FREE O'Reilly Book > "Graph Databases" is the definitive new guide to graph databases and their > applications. Written by three acclaimed leaders in the field, > this first edition is now available. Download your free book today! > http://p.sf.net/sfu/13534_NeoTech > _______________________________________________ > Maxima-discuss mailing list > Maxima-discuss@... > https://lists.sourceforge.net/lists/listinfo/maxima-discuss > ```
 Re: [Maxima-discuss] question about function identification From: Robert Dodier - 2014-03-12 17:50:44 ```On 2014-03-12, Dimiter Prodanov wrote: > No it does not. This will break the architecture of the package. > I can submit the full code if you want. Yes, please do -- I would be interested to see it. best, Robert Dodier ```
 Re: [Maxima-discuss] Maxima and 1 + 2 + 3 + .... From: Richard Fateman - 2014-03-12 16:27:18 ```Based on absolutely no discussion with a human who believes the "-1/12", here's my take on it. There are other situations in which a mathematical result has two parts, one of which is infinite (or grows without limit as some parameter increases) and one is finite in the same circumstances. I encountered this in implementing programs for asymptotic series (see "method of multiple scales") You do some calculation and get another term in the series. You look at it with great suspicion because it doesn't have the right properties, but part of it looks possible. What is done is the new term is separated into two parts the "secular" part are separated and the other. The next step is to throw out the secular part because it grows without bound, and ruins the whole idea. (example: as t-> oo, t*cos(t) is secular. cos(t) is not. ) So my take on this "-1/12" is that the conventional answer is "infinity - 1/12" -- that is, it diverges -- but for convenience in matching boundary conditions or some such excuse, the infinity is thrown out. I think the excuses probably are different from physicists and from (non-physicist) mathematicians who want a number for zeta(-1), and don't particularly care about string theory. RJF On 3/12/2014 7:11 AM, Dudley, Jeremy wrote: > So, if I pay someone £1 today, £4, tomorrow, £9 the day after, and continue until all eternity, they will owe me nothing when the time comes to repay me? And by extension, if I stop paying them before eternity is reached I will owe them the difference between what I have paid and zero? > > I saw in the quoted Wordpress article that the zeta(0) value (1 + 1 + 1 ...) being -1/12 is used in string theory, to help normalise infinity. But the article goes on to point out that the 'analytic continuation' answers for z(0), z(-1), and so on become inconsistent as they are manipulated - never mind being inconsistent with 'normal' mathematics. > > -----Original Message----- > From: andre maute [mailto:andre.maute@...] > Sent: Wednesday, March 12, 2014 11:33 AM > To: maxima-discuss@... > Subject: Re: [Maxima-discuss] Maxima and 1 + 2 + 3 + .... > > On 03/12/2014 10:08 AM, Mike Valenzuela wrote: >> I encountered some odd results from some numberphil videos and >> apparently, even in physics: >> sum(s,s,1,inf) = -1/12 >> >> Maxima claims the sum is divergent. >> >> Is this incorrect or does the result -1/12 require some unstated >> assumptions, such as complex domain? Do not all physicists or >> mathematicians agree that the sum is well defined? >> >> Apparently all the outputs from the riemann zeta function converge >> except for the value 1.0. This is surprising as 1+2+3+4+... is just >> zeta(-1). The sum of all squares, up "through" infinity is zero. The >> sum of all cubes is 1/120. >> > Perhaps you should read > > http://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/ > > Andre > > > ------------------------------------------------------------------------------ > Learn Graph Databases - Download FREE O'Reilly Book "Graph Databases" is the definitive new guide to graph databases and their applications. Written by three acclaimed leaders in the field, this first edition is now available. Download your free book today! > http://p.sf.net/sfu/13534_NeoTech > _______________________________________________ > Maxima-discuss mailing list > Maxima-discuss@... > https://lists.sourceforge.net/lists/listinfo/maxima-discuss > > ------------------------------------------------------------------------------------------ > To read our latest newsletter visit http://www.wrcplc.co.uk/news.aspx - Keeping our clients up-to-date with WRc's business developments. > ------------------------------------------------------------------------------------------ > Visit our website http://www.wrcplc.co.uk as well as http://www.waterportfolio.com for collaborative research projects. > ------------------------------------------------------------------------------------------ > Follow us on Twitter: https://twitter.com/WRcplc, LinkedIn: http://www.linkedin.com/company/wrc-plc and > Like Us on Facebook: http://www.facebook.com/pages/WRc-plc/288329784573893. > ------------------------------------------------------------------------------------------ > The Information in this e-mail is confidential and is intended solely for the addressee. Access to this e-mail by any other party is unauthorised. If you are not the intended recipient, any disclosure, copying, distribution or any action taken in reliance on the information contained in this e-mail is prohibited and maybe unlawful. When addressed to WRc Clients, any opinions or advice contained in this e-mail are subject to the terms and conditions expressed in the governing WRc Client Agreement. > ------------------------------------------------------------------------------------------ > WRc plc is a company registered in England and Wales. Registered office address: Frankland Road, Blagrove, Swindon, Wiltshire SN5 8YF. Company registration number 2262098. VAT number 527 1804 53. > ------------------------------------------------------------------------------------------ > > ------------------------------------------------------------------------------ > Learn Graph Databases - Download FREE O'Reilly Book > "Graph Databases" is the definitive new guide to graph databases and their > applications. Written by three acclaimed leaders in the field, > this first edition is now available. Download your free book today! > http://p.sf.net/sfu/13534_NeoTech > _______________________________________________ > Maxima-discuss mailing list > Maxima-discuss@... > https://lists.sourceforge.net/lists/listinfo/maxima-discuss ```
 Re: [Maxima-discuss] question about function identification From: Richard Fateman - 2014-03-12 15:27:00 ```I think there are two separate concepts that are partially merged in a computer algebra system that are confusing you. "function call" and "object type" or operator. If you wish to describe an object by tagging it as rect(...) then that is not a function call but a compound object of type "rect". If you define a function rect( ...):= def.... then in ordinary circumstances you will no longer see rect (...) in the answer, but the evaluation of its definition. Yes, Maxima uses the same functional notation for both cases. As does mathematics. This is embroidered upon in Maxima with concepts of verb and noun. It is messy, but I think it is inherited from mathematics' ambiguous usage. We don't object to seeing cos(x) --- even though some systems might give an error message "Sorry, x is a non-numeric argument to cos()." In less ordinary circumstances, the definition will punt and might return an answer that includes an "unevaluated" or "noun form" of rect( ). [a good example: an integral only partly evaluated, with a piece left over that is an integral...] Since Maxima and other symbolic systems also tend to have a user accessible representation of programs, it is sometimes possible to manipulate the inner parts of a definition, where program calls to functions exist, in a suspended state waiting to be applied/evaluated. While this kind of manipulation is sometimes appealing, the circumstances in which it is actually the "right" way to compute something are rather rare. Sometimes the best way of making Maxima do what you want, is to re-think the problem a little and change what you want so that it corresponds to common Maxima usage. ```
 Re: [Maxima-discuss] question about function identification From: Robert Dodier - 2014-03-12 15:10:12 ```On 2014-03-12, Dimiter Prodanov wrote: > if (is(op(expr) = rect) then ( > ret:apply1(ret, ft_rect), " apply Fourier rect rule", > return (ret) > ), I may have misunderstood what's the problem here. Bear in mind that apply1 only applies ft_rect to subexpressions of expr which match according to ft_rect. So you do not have to test to see if the operator is the right one (in fact, you should not, because as it stands, you will miss things like 1 + rect(x) in which rect is not the top-level operator). best Robert Dodier ```
 Re: [Maxima-discuss] question about function identification From: Robert Dodier - 2014-03-12 15:02:11 ```On 2014-03-12, Dimiter Prodanov wrote: > but this is rather awkward. Instead I would like to be able to identify a > function call like > > if (is(op(expr) = rect) then ( > ret:apply1(ret, ft_rect), " apply Fourier rect rule", > return (ret) > ), Another way to handle this kind of programming-by-cases problem is to use the user-defined rule system. As you add cases, you make up new rules; it isn't necessary to touch any existing code. In the case given above, the rule could be: matchdeclare (uu, all); tellsimp (ft (rect (uu)), ft_rect (uu)); ft (rect (x)); => ft_rect(x) and then you can write ft_rect appropriately for that one case. You can also assign properties to ft to get the built-in simplifier to help. E.g.: declare(ft, linear); ft (2*rect(z) - foo(y)); => 2*ft_rect(z) - ft(foo(y)) Note that any case not yet handled just remains ft(whatever). Some minor quibbles about the code shown above. No need to put "is" into "if", and no need to put parentheses around the condition. No need to make a block if there's just one expression to evaluate, and no need to use 'return'. Comments are like /* ... */ . So the above could be expressed as: if is (op (expr)) = rect then apply1 (expr, ft_rect); /* apply Fourier rect rule */ best, Robert Dodier ```
 Re: [Maxima-discuss] question about function identification From: Dimiter Prodanov - 2014-03-12 14:45:41 Attachments: Message as HTML ```No it does not. This will break the architecture of the package. I can submit the full code if you want. best regards, Dimiter On Wed, Mar 12, 2014 at 3:22 PM, Gunter Königsmann wrote: > Would > u:'rect(t,T); > Solve your problem? > > Gunter. > > On 12. März 2014 15:13:16 MEZ, Dimiter Prodanov > wrote: > >> Hi Stavros, >> >> Thanks for the reply: >> >> [image: Inline image 1] >> I wan to have op= rect >> >> best regards, >> >> Dimiter >> >> On Wed, Mar 12, 2014 at 2:54 PM, Stavros Macrakis wrote: >> >>> Not sure exactly what you're trying to do, but if you write 'rect(...), >>> you will quote the function application. Also, a simpler definition of your >>> function is >>> >>> rect(t,T) := if abs(t) <= T/2 then 1 else 0\$ >>> >>> -s >>> >>> >>> >>> On Wed, Mar 12, 2014 at 8:40 AM, Dimiter Prodanov wrote: >>> >>>> Dear list, >>>> >>>> I am writing a maxima package for symbolic Fourier transform. >>>> All well and good but I have some issues >>>> >>>> I would like to define some special functions and later identify them >>>> using an op() call >>>> So far only this approach works: >>>> >>>> rect(t,T):=funmake (_rect, [t ,T])\$ >>>> _rect(t,T):= block( >>>> if abs (t)<= T/2 then return(1) >>>> else return (0) >>>> )\$ >>>> >>>> >>>> but this is rather awkward. Instead I would like to be able to identify >>>> a function call like >>>> >>>> if (is(op(expr) = rect) then ( >>>> ret:apply1(ret, ft_rect), " apply Fourier rect rule", >>>> return (ret) >>>> ), >>>> >>>> If I define rect only in a block statement then op(rect) actually >>>> returns "if" which is totally useless. >>>> >>>> Am I missing something or Maxima script has some weird assumptions? >>>> >>>> best regards, >>>> >>>> Dimiter >>>> >>>> >>>> >>>> >>>> >>>> >>>> >>>> >>>> >>>> ------------------------------------------------------------------------------ >>>> Learn Graph Databases - Download FREE O'Reilly Book >>>> "Graph Databases" is the definitive new guide to graph databases and >>>> their >>>> applications. Written by three acclaimed leaders in the field, >>>> this first edition is now available. Download your free book today! >>>> http://p.sf.net/sfu/13534_NeoTech >>>> _______________________________________________ >>>> Maxima-discuss mailing list >>>> Maxima-discuss@... >>>> https://lists.sourceforge.net/lists/listinfo/maxima-discuss >>>> >>>> >>> >> ------------------------------ >> >> Learn Graph Databases - Download FREE O'Reilly Book >> "Graph Databases" is the definitive new guide to graph databases and their >> >> applications. Written by three acclaimed leaders in the field, >> this first edition is now available. Download your free book today! >> http://p.sf.net/sfu/13534_NeoTech >> >> ------------------------------ >> >> Maxima-discuss mailing list >> Maxima-discuss@... >> https://lists.sourceforge.net/lists/listinfo/maxima-discuss >> >> > -- > Diese Nachricht wurde von meinem Android-Mobiltelefon mit K-9 Mail > gesendet. > ```
 Re: [Maxima-discuss] Maxima and 1 + 2 + 3 + .... From: Dudley, Jeremy - 2014-03-12 14:33:31 ```So, if I pay someone £1 today, £4, tomorrow, £9 the day after, and continue until all eternity, they will owe me nothing when the time comes to repay me? And by extension, if I stop paying them before eternity is reached I will owe them the difference between what I have paid and zero? I saw in the quoted Wordpress article that the zeta(0) value (1 + 1 + 1 ...) being -1/12 is used in string theory, to help normalise infinity. But the article goes on to point out that the 'analytic continuation' answers for z(0), z(-1), and so on become inconsistent as they are manipulated - never mind being inconsistent with 'normal' mathematics. -----Original Message----- From: andre maute [mailto:andre.maute@...] Sent: Wednesday, March 12, 2014 11:33 AM To: maxima-discuss@... Subject: Re: [Maxima-discuss] Maxima and 1 + 2 + 3 + .... On 03/12/2014 10:08 AM, Mike Valenzuela wrote: > I encountered some odd results from some numberphil videos and > apparently, even in physics: > sum(s,s,1,inf) = -1/12 > > Maxima claims the sum is divergent. > > Is this incorrect or does the result -1/12 require some unstated > assumptions, such as complex domain? Do not all physicists or > mathematicians agree that the sum is well defined? > > Apparently all the outputs from the riemann zeta function converge > except for the value 1.0. This is surprising as 1+2+3+4+... is just > zeta(-1). The sum of all squares, up "through" infinity is zero. The > sum of all cubes is 1/120. > Perhaps you should read http://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/ Andre ------------------------------------------------------------------------------ Learn Graph Databases - Download FREE O'Reilly Book "Graph Databases" is the definitive new guide to graph databases and their applications. Written by three acclaimed leaders in the field, this first edition is now available. Download your free book today! http://p.sf.net/sfu/13534_NeoTech _______________________________________________ Maxima-discuss mailing list Maxima-discuss@... https://lists.sourceforge.net/lists/listinfo/maxima-discuss ------------------------------------------------------------------------------------------ To read our latest newsletter visit http://www.wrcplc.co.uk/news.aspx - Keeping our clients up-to-date with WRc's business developments. ------------------------------------------------------------------------------------------ Visit our website http://www.wrcplc.co.uk as well as http://www.waterportfolio.com for collaborative research projects. ------------------------------------------------------------------------------------------ Follow us on Twitter: https://twitter.com/WRcplc, LinkedIn: http://www.linkedin.com/company/wrc-plc and Like Us on Facebook: http://www.facebook.com/pages/WRc-plc/288329784573893. ------------------------------------------------------------------------------------------ The Information in this e-mail is confidential and is intended solely for the addressee. Access to this e-mail by any other party is unauthorised. If you are not the intended recipient, any disclosure, copying, distribution or any action taken in reliance on the information contained in this e-mail is prohibited and maybe unlawful. When addressed to WRc Clients, any opinions or advice contained in this e-mail are subject to the terms and conditions expressed in the governing WRc Client Agreement. ------------------------------------------------------------------------------------------ WRc plc is a company registered in England and Wales. Registered office address: Frankland Road, Blagrove, Swindon, Wiltshire SN5 8YF. Company registration number 2262098. VAT number 527 1804 53. ------------------------------------------------------------------------------------------ ```
 Re: [Maxima-discuss] question about function identification From: Gunter Königsmann - 2014-03-12 14:23:24 Attachments: Message as HTML ```Would u:'rect(t,T); Solve your problem? Gunter. On 12. März 2014 15:13:16 MEZ, Dimiter Prodanov wrote: >Hi Stavros, > >Thanks for the reply: > >[image: Inline image 1] >I wan to have op= rect > >best regards, > >Dimiter > >On Wed, Mar 12, 2014 at 2:54 PM, Stavros Macrakis >wrote: > >> Not sure exactly what you're trying to do, but if you write >'rect(...), >> you will quote the function application. Also, a simpler definition >of your >> function is >> >> rect(t,T) := if abs(t) <= T/2 then 1 else 0\$ >> >> -s >> >> >> >> On Wed, Mar 12, 2014 at 8:40 AM, Dimiter Prodanov >wrote: >> >>> Dear list, >>> >>> I am writing a maxima package for symbolic Fourier transform. >>> All well and good but I have some issues >>> >>> I would like to define some special functions and later identify >them >>> using an op() call >>> So far only this approach works: >>> >>> rect(t,T):=funmake (_rect, [t ,T])\$ >>> _rect(t,T):= block( >>> if abs (t)<= T/2 then return(1) >>> else return (0) >>> )\$ >>> >>> >>> but this is rather awkward. Instead I would like to be able to >identify a >>> function call like >>> >>> if (is(op(expr) = rect) then ( >>> ret:apply1(ret, ft_rect), " apply Fourier rect rule", >>> return (ret) >>> ), >>> >>> If I define rect only in a block statement then op(rect) actually >returns >>> "if" which is totally useless. >>> >>> Am I missing something or Maxima script has some weird assumptions? >>> >>> best regards, >>> >>> Dimiter >>> >>> >>> >>> >>> >>> >>> >>> >>> >>> >------------------------------------------------------------------------------ >>> Learn Graph Databases - Download FREE O'Reilly Book >>> "Graph Databases" is the definitive new guide to graph databases and >their >>> applications. Written by three acclaimed leaders in the field, >>> this first edition is now available. Download your free book today! >>> http://p.sf.net/sfu/13534_NeoTech >>> _______________________________________________ >>> Maxima-discuss mailing list >>> Maxima-discuss@... >>> https://lists.sourceforge.net/lists/listinfo/maxima-discuss >>> >>> >> > > >------------------------------------------------------------------------ > >------------------------------------------------------------------------------ >Learn Graph Databases - Download FREE O'Reilly Book >"Graph Databases" is the definitive new guide to graph databases and >their >applications. Written by three acclaimed leaders in the field, >this first edition is now available. Download your free book today! >http://p.sf.net/sfu/13534_NeoTech > >------------------------------------------------------------------------ > >_______________________________________________ >Maxima-discuss mailing list >Maxima-discuss@... >https://lists.sourceforge.net/lists/listinfo/maxima-discuss -- Diese Nachricht wurde von meinem Android-Mobiltelefon mit K-9 Mail gesendet.```
 Re: [Maxima-discuss] question about function identification From: Dimiter Prodanov - 2014-03-12 14:13:45 Attachments: Message as HTML     image.png
 Re: [Maxima-discuss] question about function identification From: Stavros Macrakis - 2014-03-12 13:55:30 Attachments: Message as HTML ```One thing to be careful about: the noun/verb system. You might need if op(x)=nounify('rect) or =verbify('rect) depending on exactly what you're doing. On Wed, Mar 12, 2014 at 9:54 AM, Stavros Macrakis wrote: > Not sure exactly what you're trying to do, but if you write 'rect(...), > you will quote the function application. Also, a simpler definition of your > function is > > rect(t,T) := if abs(t) <= T/2 then 1 else 0\$ > > -s > > > > On Wed, Mar 12, 2014 at 8:40 AM, Dimiter Prodanov wrote: > >> Dear list, >> >> I am writing a maxima package for symbolic Fourier transform. >> All well and good but I have some issues >> >> I would like to define some special functions and later identify them >> using an op() call >> So far only this approach works: >> >> rect(t,T):=funmake (_rect, [t ,T])\$ >> _rect(t,T):= block( >> if abs (t)<= T/2 then return(1) >> else return (0) >> )\$ >> >> >> but this is rather awkward. Instead I would like to be able to identify a >> function call like >> >> if (is(op(expr) = rect) then ( >> ret:apply1(ret, ft_rect), " apply Fourier rect rule", >> return (ret) >> ), >> >> If I define rect only in a block statement then op(rect) actually returns >> "if" which is totally useless. >> >> Am I missing something or Maxima script has some weird assumptions? >> >> best regards, >> >> Dimiter >> >> >> >> >> >> >> >> >> >> ------------------------------------------------------------------------------ >> Learn Graph Databases - Download FREE O'Reilly Book >> "Graph Databases" is the definitive new guide to graph databases and their >> applications. Written by three acclaimed leaders in the field, >> this first edition is now available. Download your free book today! >> http://p.sf.net/sfu/13534_NeoTech >> _______________________________________________ >> Maxima-discuss mailing list >> Maxima-discuss@... >> https://lists.sourceforge.net/lists/listinfo/maxima-discuss >> >> > ```
 Re: [Maxima-discuss] question about function identification From: Stavros Macrakis - 2014-03-12 13:54:11 Attachments: Message as HTML ```Not sure exactly what you're trying to do, but if you write 'rect(...), you will quote the function application. Also, a simpler definition of your function is rect(t,T) := if abs(t) <= T/2 then 1 else 0\$ -s On Wed, Mar 12, 2014 at 8:40 AM, Dimiter Prodanov wrote: > Dear list, > > I am writing a maxima package for symbolic Fourier transform. > All well and good but I have some issues > > I would like to define some special functions and later identify them > using an op() call > So far only this approach works: > > rect(t,T):=funmake (_rect, [t ,T])\$ > _rect(t,T):= block( > if abs (t)<= T/2 then return(1) > else return (0) > )\$ > > > but this is rather awkward. Instead I would like to be able to identify a > function call like > > if (is(op(expr) = rect) then ( > ret:apply1(ret, ft_rect), " apply Fourier rect rule", > return (ret) > ), > > If I define rect only in a block statement then op(rect) actually returns > "if" which is totally useless. > > Am I missing something or Maxima script has some weird assumptions? > > best regards, > > Dimiter > > > > > > > > > > ------------------------------------------------------------------------------ > Learn Graph Databases - Download FREE O'Reilly Book > "Graph Databases" is the definitive new guide to graph databases and their > applications. Written by three acclaimed leaders in the field, > this first edition is now available. Download your free book today! > http://p.sf.net/sfu/13534_NeoTech > _______________________________________________ > Maxima-discuss mailing list > Maxima-discuss@... > https://lists.sourceforge.net/lists/listinfo/maxima-discuss > > ```
 [Maxima-discuss] question about function identification From: Dimiter Prodanov - 2014-03-12 12:41:03 Attachments: Message as HTML ```Dear list, I am writing a maxima package for symbolic Fourier transform. All well and good but I have some issues I would like to define some special functions and later identify them using an op() call So far only this approach works: rect(t,T):=funmake (_rect, [t ,T])\$ _rect(t,T):= block( if abs (t)<= T/2 then return(1) else return (0) )\$ but this is rather awkward. Instead I would like to be able to identify a function call like if (is(op(expr) = rect) then ( ret:apply1(ret, ft_rect), " apply Fourier rect rule", return (ret) ), If I define rect only in a block statement then op(rect) actually returns "if" which is totally useless. Am I missing something or Maxima script has some weird assumptions? best regards, Dimiter ```
 Re: [Maxima-discuss] Maxima and 1 + 2 + 3 + .... From: Gunter Königsmann - 2014-03-12 11:37:04 Attachments: Message as HTML ```There were some german newspapers that showed that if you add up the numbers from one to ten, then rearrange everything in a way you end up in a equation equivalent to that and then extend this new scheme to infinity you can get all odd kind of results. Since these results always included - 1/12 I assume the source might be youtube.com/watch?v=w-I6XTVZXww On 12. März 2014 11:41:11 MEZ, "José Carlos Santos" wrote: >On 12-03-2014 9:08, Mike Valenzuela wrote: > >> I encountered some odd results from some numberphil videos and >> apparently, even in physics: >> sum(s,s,1,inf) = -1/12 >> >> Maxima claims the sum is divergent. > >Because it is. > >> Is this incorrect or does the result -1/12 require some unstated >> assumptions, such as complex domain? Do not all physicists or >> mathematicians agree that the sum is well defined? > >Only in sense that they all agree that the sum is +oo. > >> Apparently all the outputs from the riemann zeta function converge >> except for the value 1.0. This is surprising as 1+2+3+4+... is just >> zeta(-1). > >No, it is not. The equality zeta(s) = 1/1^s + 1/2^s + 1/3^s + ... is >only valid when Re(s) > 0. > >> The sum of all squares, up "through" infinity is zero. The sum >> of all cubes is 1/120. > >Where did you get this? > >Best regards, > >Jose Carlos Santos > > >------------------------------------------------------------------------------ >Learn Graph Databases - Download FREE O'Reilly Book >"Graph Databases" is the definitive new guide to graph databases and >their >applications. Written by three acclaimed leaders in the field, >this first edition is now available. Download your free book today! >http://p.sf.net/sfu/13534_NeoTech >_______________________________________________ >Maxima-discuss mailing list >Maxima-discuss@... >https://lists.sourceforge.net/lists/listinfo/maxima-discuss -- Diese Nachricht wurde von meinem Android-Mobiltelefon mit K-9 Mail gesendet.```
 Re: [Maxima-discuss] Maxima and 1 + 2 + 3 + .... From: andre maute - 2014-03-12 11:33:01 ```On 03/12/2014 10:08 AM, Mike Valenzuela wrote: > I encountered some odd results from some numberphil videos and apparently, > even in physics: > sum(s,s,1,inf) = -1/12 > > Maxima claims the sum is divergent. > > Is this incorrect or does the result -1/12 require some unstated > assumptions, such as complex domain? Do not all physicists or > mathematicians agree that the sum is well defined? > > Apparently all the outputs from the riemann zeta function converge except > for the value 1.0. This is surprising as 1+2+3+4+... is just zeta(-1). The > sum of all squares, up "through" infinity is zero. The sum of all cubes is > 1/120. > Perhaps you should read http://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/ Andre ```
 Re: [Maxima-discuss] Maxima and 1 + 2 + 3 + .... From: José Carlos Santos - 2014-03-12 11:07:05 ```On 12-03-2014 9:08, Mike Valenzuela wrote: > I encountered some odd results from some numberphil videos and > apparently, even in physics: > sum(s,s,1,inf) = -1/12 > > Maxima claims the sum is divergent. Because it is. > Is this incorrect or does the result -1/12 require some unstated > assumptions, such as complex domain? Do not all physicists or > mathematicians agree that the sum is well defined? Only in sense that they all agree that the sum is +oo. > Apparently all the outputs from the riemann zeta function converge > except for the value 1.0. This is surprising as 1+2+3+4+... is just > zeta(-1). No, it is not. The equality zeta(s) = 1/1^s + 1/2^s + 1/3^s + ... is only valid when Re(s) > 0. > The sum of all squares, up "through" infinity is zero. The sum > of all cubes is 1/120. Where did you get this? Best regards, Jose Carlos Santos ```
 [Maxima-discuss] Maxima and 1 + 2 + 3 + .... From: Mike Valenzuela - 2014-03-12 09:08:59 Attachments: Message as HTML ```I encountered some odd results from some numberphil videos and apparently, even in physics: sum(s,s,1,inf) = -1/12 Maxima claims the sum is divergent. Is this incorrect or does the result -1/12 require some unstated assumptions, such as complex domain? Do not all physicists or mathematicians agree that the sum is well defined? Apparently all the outputs from the riemann zeta function converge except for the value 1.0. This is surprising as 1+2+3+4+... is just zeta(-1). The sum of all squares, up "through" infinity is zero. The sum of all cubes is 1/120. ```

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