About Linear Independence of Exponential Functions defining an alternating serie
in [Math]
|
Prev: Cosmic Blackbody Microwave Background Radiation proves Atom Totality and dismisses Big Bang Chapt 3 #149; ATOM TOTALITY
Next: Minus operator and no subtraction below zero
From: Gerry Myerson on 9 Jun 2010 19:27 In article <cc2f6f4e-ffd7-48f0-9191-78006d0349b1(a)w31g2000yqb.googlegroups.com>, zeraoulia <zelhadj12(a)gmail.com> wrote: > On Jun 9, 4:20�am, Robert Israel > <isr...(a)math.MyUniversitysInitials.ca> wrote: > > On Wed, 09 Jun 2010 08:52:14 +1000, Gerry Myerson wrote: > > > In article > > > <6011a022-4c5c-4e44-a011-bce855788...(a)b35g2000yqi.googlegroups.com>, > > > �zeraoulia <zelhad...(a)gmail.com> wrote: > > > > >> Hello to all, > > >> My question is: let the alternating serie � (-1) > > >> **n-1*an*exp(i*ɿ*ln(n))=0, where (an) is real and i is the complex > > >> number, ln(n) is the logaritm of n. The sum is taken over all natural > > >> numbers n>=1. �I wonder if the above equation (the serie=0) implies > > >> that all the coefficents an=0 for all n. I appreciate if one can help > > >> me to find an answer with some references. > > > > > 1. Don't use special characters in a text-based newsgroup. > > > What looked like a beta to you looks like gobbledygook once > > > my machine has had its way. > > > > > 2. It's not an alternating series. That term makes sense only > > > for real series. > > > > > 3. The (-1)^(n - 1) seems to be superfluous, as it can be absorbed > > > into the a n, so we have sum a n exp( i b log n) = 0. > > > > > 4. What is beta? Is the equation supposed to hold for all beta? > > > or just for one fixed value of beta? is beta real? > > > > Writing the series as sum a n n^(i beta), what we have is a Dirichlet > > series. �If it ever converges, there is some r such that the series > > converges uniformly on U = {beta in C: Im(beta) > r}, and the sum is > > analytic on that set. If that analytic function is 0 for beta in some � > > set that has a limit point in U, it must be 0 for all beta in U. > > And then by a standard result on uniqueness for Dirichlet series, the > > answer is yes. �See e.g. Theorem 3D in > > <http://rutherglen.ics.mq.edu.au/wchen/lndpnfolder/dpn03-ds.pdf> > > > > -- > > Robert Israel � � � � � � �isr...(a)math.MyUniversitysInitials.ca > > Department of Mathematics � � � �http://www.math.ubc.ca/~israel > > University of British Columbia � � � � � �Vancouver, BC, Canada- Hide > > quoted text - > > > > Thank you very much I think when RI writes, "if that analytic function is 0 for beta in some set that has a limit point in U," he means "if that analytic function is 0 for ALL beta in some set that has a limit point in U." Since you've written elsewhere that you are interested in a single, fixed value of beta, this hypothesis doesn't apply to your case, so Theorem 3D doesn't apply, and your question remains unanswered. And concerning your suggestion elsewhere that (-1)^(n - 1) isn't superfluous, if you let c_n = (-1)^(n - 1) a_n, then c_n is real, and your series is sum c_n exp(i beta log n), and c_n are all zero if and only if a_n are all zero, so what's the use of (-1)^(n - 1)? Is there something about the a_n that you've failed to tell us? -- Gerry Myerson (gerry(a)maths.mq.edi.ai) (i -> u for email)
From: zeraoulia on 10 Jun 2010 01:45 On Jun 10, 1:27 am, Gerry Myerson <ge...(a)maths.mq.edi.ai.i2u4email> wrote: > In article > <cc2f6f4e-ffd7-48f0-9191-78006d034...(a)w31g2000yqb.googlegroups.com>, > > > > > >  zeraoulia <zelhad...(a)gmail.com> wrote: > > On Jun 9, 4:20 am, Robert Israel > > <isr...(a)math.MyUniversitysInitials.ca> wrote: > > > On Wed, 09 Jun 2010 08:52:14 +1000, Gerry Myerson wrote: > > > > In article > > > > <6011a022-4c5c-4e44-a011-bce855788...(a)b35g2000yqi.googlegroups.com>, > > > >  zeraoulia <zelhad...(a)gmail.com> wrote: > > > > >> Hello to all, > > > >> My question is: let the alternating serie Å (-1) > > > >> **n-1*an*exp(i*ÿ*ln(n))=0, where (an) is real and i is the complex > > > >> number, ln(n) is the logaritm of n. The sum is taken over all natural > > > >> numbers n>=1.  I wonder if the above equation (the serie=0) implies > > > >> that all the coefficents an=0 for all n. I appreciate if one can help > > > >> me to find an answer with some references. > > > > > 1. Don't use special characters in a text-based newsgroup. > > > > What looked like a beta to you looks like gobbledygook once > > > > my machine has had its way. > > > > > 2. It's not an alternating series. That term makes sense only > > > > for real series. > > > > > 3. The (-1)^(n - 1) seems to be superfluous, as it can be absorbed > > > > into the a n, so we have sum a n exp( i b log n) = 0. > > > > > 4. What is beta? Is the equation supposed to hold for all beta? > > > > or just for one fixed value of beta? is beta real? > > > > Writing the series as sum a n n^(i beta), what we have is a Dirichlet > > > series.  If it ever converges, there is some r such that the series > > > converges uniformly on U = {beta in C: Im(beta) > r}, and the sum is > > > analytic on that set. If that analytic function is 0 for beta in some  > > > set that has a limit point in U, it must be 0 for all beta in U. > > > And then by a standard result on uniqueness for Dirichlet series, the > > > answer is yes.  See e.g. Theorem 3D in > > > <http://rutherglen.ics.mq.edu.au/wchen/lndpnfolder/dpn03-ds.pdf> > > > > -- > > > Robert Israel        isr...(a)math.MyUniversitysInitials.ca > > > Department of Mathematics     http://www.math.ubc.ca/~israel > > > University of British Columbia       Vancouver, BC, Canada- Hide > > > quoted text - > > > Thank you very much > > I think when RI writes, "if that analytic function is 0 for beta in > some set that has a limit point in U," he means "if that analytic > function is 0 for ALL beta in some set that has a limit point in U." > Since you've written elsewhere that you are interested in a single, > fixed value of beta, this hypothesis doesn't apply to your case, so > Theorem 3D doesn't apply, and your question remains unanswered. > > And concerning your suggestion elsewhere that (-1)^(n - 1) isn't > superfluous, if you let c_n = (-1)^(n - 1) a_n, then c_n is real, > and your series is sum c_n exp(i beta log n), and c_n are all zero > if and only if a_n are all zero, so what's the use of (-1)^(n - 1)? > Is there something about the a_n that you've failed to tell us? > > -- > Gerry Myerson (ge...(a)maths.mq.edi.ai) (i -> u for email)- Hide quoted text - > > - Show quoted text - Yes some numerical evidences show to me that the answer is no. Thank you very much for your kind help.
From: zeraoulia on 10 Jun 2010 01:46
On Jun 10, 3:37 am, Robert Israel <isr...(a)math.MyUniversitysInitials.ca> wrote: > On Wed, 9 Jun 2010 05:14:28 -0700 (PDT), zeraoulia wrote: > > On Jun 9, 12:52 am, Gerry Myerson <ge...(a)maths.mq.edi.ai.i2u4email> > > wrote: > >> In article > >> <6011a022-4c5c-4e44-a011-bce855788...(a)b35g2000yqi.googlegroups.com>, > > >> zeraoulia <zelhad...(a)gmail.com> wrote: > >>> Hello to all, > >>> My question is: let the alternating serie (-1) > >>> **n-1*an*exp(i*É¿*ln(n))=0, where (an) is real and i is the complex > >>> number, ln(n) is the logaritm of n. The sum is taken over all natural > >>> numbers n>=1. I wonder if the above equation (the serie=0) implies > >>> that all the coefficents an=0 for all n. I appreciate if one can help > >>> me to find an answer with some references. > > >> 1. Don't use special characters in a text-based newsgroup. > >> What looked like a beta to you looks like gobbledygook once > >> my machine has had its way. > > >> 2. It's not an alternating series. That term makes sense only > >> for real series. > > >> 3. The (-1)^(n - 1) seems to be superfluous, as it can be absorbed > >> into the a_n, so we have sum a_n exp( i b log n) = 0. > > >> 4. What is beta? Is the equation supposed to hold for all beta? > >> or just for one fixed value of beta? is beta real? > > >> -- > >> Gerry Myerson (ge...(a)maths.mq.edi.ai) (i -> u for email) > > > The coefficient (-1)^(n - 1) is not superfluous in the equation. The > > number beta is real and the equation supposed to hold for one fixed > > value of beta. > > Thank you > > Then the answer is no. Try a_1 = a_2 = 1, all other a_n = 0, > beta = 2 pi/ln(2). > -- > Robert Israel isr...(a)math.MyUniversitysInitials.ca > Department of Mathematics http://www.math.ubc.ca/~israel > University of British Columbia Vancouver, BC, Canada- Hide quoted text - > > - Show quoted text - Yes some numerical evidences show to me that the answer is no. Thank you very much for your kind help. |