About Linear Independence of Exponential Functions defining an alternating serie
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From: zeraoulia on 8 Jun 2010 14:21 Hello to all, My question is: let the alternating serie â (-1) **n-1*an*exp(i*β*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.
From: Gerry Myerson on 8 Jun 2010 18:52 In article <6011a022-4c5c-4e44-a011-bce855788c73(a)b35g2000yqi.googlegroups.com>, zeraoulia <zelhadj12(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? -- Gerry Myerson (gerry(a)maths.mq.edi.ai) (i -> u for email)
From: Robert Israel on 8 Jun 2010 22:20 On Wed, 09 Jun 2010 08:52:14 +1000, Gerry Myerson wrote: > In article > <6011a022-4c5c-4e44-a011-bce855788c73(a)b35g2000yqi.googlegroups.com>, > zeraoulia <zelhadj12(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 israel(a)math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada
From: zeraoulia on 9 Jun 2010 08:14 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
From: zeraoulia on 9 Jun 2010 08:20
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*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? > > 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 - > > - Show quoted text - Thank you very much |