fake id(x) for better 2sinh method. - Printable Version +- Tetration Forum (https://math.eretrandre.org/tetrationforum) +-- Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1) +--- Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3) +--- Thread: fake id(x) for better 2sinh method. (/showthread.php?tid=871) fake id(x) for better 2sinh method. - tommy1729 - 05/20/2014 For improving the 2sinh method so that it works for all bases > eta and other improvements I consider : 2sinh(fake id(x)) fake id(x) behaves like e/2 x for small x, like x for large real and like 0 for large imaginary part. SO what is the best fake id(x) ? Remember we only want a single fixpoint (2sinh(fake id(x))=x) for any base. maybe id(x) = exp(fakelog(x)) ... Thinking... regards tommy1729 RE: fake id(x) for better 2sinh method. - tommy1729 - 06/04/2014 There is a problem with this particular type of fake function. Although it may be fixed as stated in post 1 it seems a big problem. Perhaps the biggest problem of " fake function theory ". id(x) as desired in post nr 1 : " fake id(x) behaves like e/2 x for small x, like x for large real and like 0 for large imaginary part " is in contradiction with the well known LindelĂ¶f theorem. It seems the natural fix is : " fake id(z) behaves like e/2 z for small z and like z for large z " regards tommy1729 RE: fake id(x) for better 2sinh method. - tommy1729 - 06/04/2014 Hmm this new id(z) also seems in trouble because id(z) -z goes to 0 for large z ... but stays bounded for small z : so id(z) - z is not entire. So maybe use a function that grows like exp(x) near the real line and grows like z otherwise ? Notice if id(z) is not entire , then neither is id(exp(z)) or even id(fakeexp(z)). Hmm. Seems really an issue. regards tommy1729 RE: fake id(x) for better 2sinh method. - tommy1729 - 06/04/2014 Wait a sec , it seems carlson's theorem applies here ! id(z) must grow fast enough otherwise id(z) - z is "close" to 0 for integer z , hence to avoid id(z) being flat it seems id(z) must grow fast enough in the imaginary direction. The problem is " close " is not the exact condition of the theorem (exactly 0 is). To make ln(id(z)/z) an entire function this also suggests id(z) is of exp type. Hence I write id(z) = z exp(t(z)) where t(z) is a taylor series. Now id(z)/z IS NEVER ALLOWED TO BE EQUAL TO 1. and exp(t(x)) for real x is a fake id(1) function. Thus t(x) is a fake id(0) function. However saying t(z) = id(z) - z brings us back to the Original problem ... Hmm. This is getting stranger by the sec. Maybe t(z) should be a fake id(0) for reals and a fake exp^[1/2](z) in the upper half plane ( in terms of absolute value ). But that brings us back again to carlson so I guess its better to have f(z) = fake id(0) around the real axis. f(z) = fake exp(z) near the imag axis. I finally see less objections. But what is f(z) ? regards tommy1729 RE: fake id(x) for better 2sinh method. - tommy1729 - 06/04/2014 (06/04/2014, 11:10 PM)tommy1729 Wrote: Now id(z)/z IS NEVER ALLOWED TO BE EQUAL TO 1. and exp(t(x)) for real x is a fake id(1) function. This is still a big objection. so exp(t(z)) is not allowed to be 0 nor 1. This violates the little picard theorem. Hmm Seems all theorems of complex analysis are against me ! regards tommy1729 RE: fake id(x) for better 2sinh method. - tommy1729 - 06/05/2014 A good id(z) has been found ! id(z)/z = 1 sometimes but that is no big problem. id(z) = z * fake(1) on the real line id(z) = z * fake(exp(exp(i z))^O(z)) on the imag line. id(z) is of the type z * f(z i) where f(z i) is similar too $f(z i)=\int_0^\infty \frac{e^{zt i}}{t^t}\mathrm{d}t$ Yes the same function that occured in the fake exp^[1/2] thread. It seems " fake function theory " is developping fast. regards tommy1729