An Infinite Product
in [Math]
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From: Han de Bruijn on 13 Apr 2010 06:57 On 13 apr, 12:21, Tonico <Tonic...(a)yahoo.com> wrote: > On Apr 13, 10:35 am, Han de Bruijn <umum...(a)gmail.com> wrote: > > > > > > > On 12 apr, 20:18, A N Niel <ann...(a)nym.alias.net.invalid> wrote: > > > > In article <20100412134614.421...(a)newsreader.com>, David W. Cantrell > > > > <DWCantr...(a)sigmaxi.net> wrote: > > > > Maury Barbato <mauriziobarb...(a)aruba.it> wrote: > > > > > Hello, > > > > > consider the infinite product > > > > > > product_{n=1}^Infty [1 + (n^(-2))], > > > > > > which is easily seen to converge taking the logarithm. > > > > > Someone knows th exact value of the product? > > > > > sinh(pi)/pi > > > > > David > > > > and, more generally, for any complex z, we have: > > > sin(pi z) = pi z product(n=1 to infinity)(1-z^2/n^2) > > > Wonder if there are OTHER infinite products NOT resemblant to this one > > because I've seen this one so many times .. > > > Han de Bruijn- > > Wonder no more, my dear HdB: > > http://pi.physik.uni-bonn.de/~dieckman/InfProd/InfProd.html#Infinitex... > > You're welcome ( this googling thingy is marvelous...;) ) > > Tonio Hey, that's nice! Thanks, Tonio: Han de Bruijn
From: Maury Barbato on 13 Apr 2010 03:05
A N Niel wrote: > In article <20100412134614.421$wX(a)newsreader.com>, > David W. Cantrell > <DWCantrell(a)sigmaxi.net> wrote: > > > Maury Barbato <mauriziobarbato(a)aruba.it> wrote: > > > Hello, > > > consider the infinite product > > > > > > product_{n=1}^Infty [1 + (n^(-2))], > > > > > > which is easily seen to converge taking the > logarithm. > > > Someone knows th exact value of the product? > > > > sinh(pi)/pi > > > > David > > and, more generally, for any complex z, we have: > sin(pi z) = pi z product(n=1 to infinity)(1-z^2/n^2) Thank you very much for your help, David and Niel. Friendly Regards, Maury Barbato |