From: bobopi@my-deja.com Subject: Re: sum(exp(-k2/4),k=0..infinity)=sqrt(Pi)+1/2 is true? Date: Fri, 04 Feb 2000 12:33:03 GMT Newsgroups: sci.math Summary: [missing] In article <030220000858322117%edgar@math.ohio-state.edu.nospam>, "G. A. Edgar" wrote: > In article <87beql$udo$1@nnrp1.deja.com>, Robin Chapman > wrote: > > > MAPLE sez > > 2 sqrt(pi)exp(-4 pi^2) = > > 0.0000000000000000253714922922897849044028547417... > > > > Are these the same then? > > you mean... > > sum(exp(-k^2/4),k=0..infinity)=sqrt(Pi)+1/2+2*sqrt(Pi)*exp(-4*Pi^2); > > ?? Well, Maple says they agree to 68 places, but the > difference is 0.9301848302752675113476714217953 e-68 > so they are not equal. And what is the NEXT term for the > theta function? > > -- > Gerald A. Edgar edgar@math.ohio-state.edu > 2*sqrt(Pi)*exp(-16*Pi^2) seems to be the next term It might give a sum like sum(exp(-k^2/4),0..infinity)=sqrt(Pi)+1/2+2*sqrt(Pi )*sum(exp(-4*n^2*Pi^2),1..infinity) ?? I remember that Jon. And Peter Borwein have proved in 1992 that 1/10^5*sum(exp(n^2/(10^10)),n=-infinity..infinity) had the same 42 billion first decimals than sqrt(Pi) but was different of sqrt(Pi). B. Gourevitch 'L'univers de Pi' http://www.multimania.com/bgourevitch Sent via Deja.com http://www.deja.com/ Before you buy. ============================================================================== From: robjohn9@idt.net (Rob Johnson) Subject: Re: sum(exp(-k2/4),k=0..infinity)=sqrt(Pi)+1/2 is true? Date: 4 Feb 2000 17:58:54 GMT Newsgroups: sci.math In article <87egtv$24r$1@nnrp1.deja.com>, bobopi@my-deja.com (B. Gourevitch) wrote: >I remember that Jon. And Peter Borwein have proved >in 1992 that >1/10^5*sum(exp(n^2/(10^10)),n=-infinity..infinity) >had the same 42 billion first decimals than >sqrt(Pi) but was different of sqrt(Pi). if you negate the n^2/10^10 above, then I compute that the error would be about -42863147300 8.67030624142366941518273401837467947687703987 x 10 Rob Johnson robjohn9@idt.net ============================================================================== From: robjohn9@idt.net (Rob Johnson) Subject: Re: sum(exp(-kČ/4),k=0..infinity)=sqrt(Pi)+1/2 is true? Date: 3 Feb 2000 17:45:11 GMT Newsgroups: sci.math In article <87beql$udo$1@nnrp1.deja.com>, Robin Chapman wrote: >In article <87a1lk$k5h@nnrp2.farm.idt.net>, > robjohn9@idt.net (Rob Johnson) wrote: >> In article <878ts3$17h$1@front5.grolier.fr>, >> "Panh" wrote: >> >How prove >> >>sum(f^(2k)(0)/(4^k*k!),k=0..infinity)*sqrt(Pi)=int(f(x)*exp(-x^2),x=-infinit >> >y..infinity) >> >sum(exp(-kČ/4),k=0..infinity)=sqrt(Pi)+1/2 is true? >> >> Just another verification that this is not true: >> >> oo >> --- -k^2/4 >> > e = 2.2724538509055160526696597756309300872004041978... >> --- >> k=0 >> >> sqrt(pi) + 1/2 = 2.2724538509055160272981674833411451827975494561... >> >> difference = 0.0000000000000000253714922922897849044028547417... >> >MAPLE sez >2 sqrt(pi)exp(-4 pi^2) = 0.0000000000000000253714922922897849044028547417... > >Are these the same then? Very interesting. I think I have an identity, which I guess you already knew: oo 1 --- -k^2/4 - - + > e 2 --- k=0 oo 1 --- -4 pi^2 k^2 2 sqrt(pi) ( - - + > e ) 2 --- k=0 Both of which, to 1000 places, are equal to 1.77245385090551605266965977563093008720040419783244 73558033479624324133495907352274215634222925978881 01510399743647666261991173186095649370468172292619 70339053264924560439423259599446678539173096737067 59178820876826363001482535704057834036906584119923 61144362000052574444508990640909697166076583088217 52697998800664028408279151228151224931622799881293 05809699560556584429676577467606944855678649129062 21950705276047259574138127891731960728599841742740 44724936262224065637126305456367643306100178108081 16494098545790686772638445065269611777585792822202 35820137457783698617272589430798332122878640105823 26171978656161224111229711771908803854179793374207 84447382325839748285736555203937153108472453843091 96386215512156486788652271370641977513493603408321 20920321898631178916246090160071705999136412640985 69787719947234325541257727464277184432989349347582 94673435035970585715519940929359536781630524474993 97318324399360539843497184997242676442943136757219 90332745075730562009605210429591704892744224472088... I think I need to read more on theta series and elliptic integrals. Rob Johnson robjohn9@idt.net ============================================================================== From: Raymond Manzoni Subject: Re: sum(exp(-k2/4),k=0..infinity)=sqrt(Pi)+1/2 is true? Date: Thu, 03 Feb 2000 22:10:11 +0100 Newsgroups: sci.math Rob Johnson wrote: > Very interesting. I think I have an identity, which I guess you already > knew: > > oo > 1 --- -k^2/4 > - - + > e > 2 --- > k=0 > oo > 1 --- -4 pi^2 k^2 > 2 sqrt(pi) ( - - + > e ) > 2 --- > k=0 Hi, According to the Borweins ("Pi and the AGM" Wiley 1987 page 38) this was known to Poisson in 1823 (the case x=0). He got by 1827 the more general formula : sum_{n=-infinity}^infinity e^(-s*(n+x)^2*Pi) = s^(-1/2) * sum_{k=-infinity}^infinity e^(2*Pi*i*k*x-Pi*k^2/s) The case x=0 may be written : sqrt(s)*Theta3(s) = Theta3(1/s) And setting further s=1/(4*Pi) you'll find your equation. The demonstration (given by the Borweins) involves this equality : int_{x=0}^infinity e^(-x^2)*cos(2xy) dx = sqrt(Pi)/2*e^(-y^2) All these formulas are very nice indeed.., Raymond Manzoni ============================================================================== From: israel@math.ubc.ca (Robert Israel) Subject: Re: sum(exp(-kČ/4),k=0..infinity)=sqrt(Pi)+1/2 is true? Date: 3 Feb 2000 21:45:07 GMT Newsgroups: sci.math In article <87cer7$7tj@nnrp1.farm.idt.net>, robjohn9@idt.net (Rob Johnson) writes: > Very interesting. I think I have an identity, which I guess you already > knew: > oo > 1 --- -k^2/4 > - - + > e > 2 --- > k=0 > oo > 1 --- -4 pi^2 k^2 > 2 sqrt(pi) ( - - + > e ) > 2 --- > k=0 Yes. It's an identity for theta functions. A special case of the Poisson Summation Formula: sum_{k in Z} f(2 pi k) = (2 pi)^(-1) sum_{n in Z} int_{-infinity}^infinity f(y) exp(-i n y) dy So with f(x) = exp(-c (x-a)^2) where c > 0 you get sum_{k in Z} exp(-c (k-a)^2) = sqrt(pi/c) sum_{k in Z} exp(-2 pi i k a) exp(-pi^2 k^2/c) Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2