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 TPID 4 tommy1729 Ultimate Fellow Posts: 1,372 Threads: 336 Joined: Feb 2009 08/23/2012, 04:26 PM (This post was last modified: 08/24/2012, 03:10 PM by tommy1729.) here i give a ( nonunique short ) proof of TPID 4. remember that entire taylor series are coo everywhere. ( infinitely differentiable for all finite complex ) let f(z,1) be an entire periodic function with f(0,1)=f(1,1)=1 and period 1. and f(z,1) is not identically 1 for all z. we will prove that for complex b with arg(b) <> 0 , the only solution to the equations is f(z,1) * b^z and hence the proof follows. let k and n be positive integers. f(0) = 1 f(z+k) = b^k f(z) f = entire then take the derivative of the equation f(z+k) = b^k f(z) on both sides f ' (z+k) = b^k f ' (z) again f '' (z+k) = b^k f '' (z) and in general f^(n) (z+k) = b^k f^(n) (z) hence because of taylors theorem we must conclude f(z) = f(0) * f(z,1) * b^z in the neighbourhood of 0. but since f is entire it must be true everywhere and f(0) = 1 hence f(z) = f(z,1) b^z for all z. if arg(b) <> 0 then the period of b^z does not have Re <> 0 and hence b^z is unbounded on the strip. if f(z) needs to be bounded and b^z is not bounded , this implies that f(z,1) needs to be bounded. but this is impossible since f(z,1) has a real period and is entire , it must be unbounded on the strip. ( remember f(z,1) =/= 1 everywhere by definition ) the product of two functions unbounded in the same region must be unbounded in that region. QED regards tommy1729 * post has been edited * « Next Oldest | Next Newest »

 Messages In This Thread TPID 4 - by tommy1729 - 08/23/2012, 04:26 PM RE: TPID 4 - by tommy1729 - 08/24/2012, 03:12 PM RE: TPID 4 - by tommy1729 - 03/28/2014, 12:04 AM RE: TPID 4 - by sheldonison - 06/15/2014, 06:22 PM RE: TPID 4 - by tommy1729 - 04/26/2014, 12:24 PM RE: TPID 4 - by sheldonison - 04/27/2014, 04:37 AM RE: TPID 4 - by tommy1729 - 04/27/2014, 01:40 PM RE: TPID 4 - by tommy1729 - 06/15/2014, 06:35 PM RE: TPID 4 - by sheldonison - 06/15/2014, 06:42 PM RE: TPID 4 - by tommy1729 - 06/15/2014, 07:09 PM RE: TPID 4 - by sheldonison - 06/15/2014, 07:35 PM RE: TPID 4 - by tommy1729 - 06/15/2014, 08:10 PM RE: TPID 4 - by mike3 - 06/17/2014, 09:30 AM RE: TPID 4 - by tommy1729 - 06/17/2014, 12:21 PM RE: TPID 4 - by sheldonison - 06/17/2014, 06:16 PM RE: TPID 4 - by mike3 - 06/17/2014, 09:48 PM RE: TPID 4 - by sheldonison - 06/17/2014, 11:43 PM RE: TPID 4 - by tommy1729 - 06/18/2014, 12:23 PM RE: TPID 4 - by sheldonison - 06/18/2014, 12:59 PM RE: TPID 4 - by tommy1729 - 06/18/2014, 10:21 PM RE: TPID 4 - by sheldonison - 06/18/2014, 10:41 PM RE: TPID 4 - by tommy1729 - 06/18/2014, 11:15 PM RE: TPID 4 - by tommy1729 - 06/17/2014, 10:46 PM RE: TPID 4 - by tommy1729 - 06/16/2014, 09:21 PM RE: TPID 4 - by tommy1729 - 06/16/2014, 10:45 PM RE: TPID 4 - by tommy1729 - 06/16/2014, 10:49 PM RE: TPID 4 - by tommy1729 - 06/16/2014, 10:57 PM RE: TPID 4 - by tommy1729 - 06/17/2014, 10:48 PM RE: TPID 4 - by tommy1729 - 06/18/2014, 10:38 PM RE: TPID 4 - by tommy1729 - 07/07/2014, 11:56 PM

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