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 Real and complex behaviour of the base change function (was: The "cheta" function) bo198214 Administrator Posts: 1,386 Threads: 90 Joined: Aug 2007 08/16/2009, 11:15 PM Ok, my explanation for the phenomenon is roughly: The orbit of a point $z_0$ from the upper halfplane under $\log$ is a right winding spiral around the primary fixed point in the upper halfplane. This spiral gets bigger the farther $z_0$ is located from the origin. The spiral getting bigger means the lower part of the spiral gets closer to the real axis. Indeed already the second iteration of $z$ (though it is usually not the minimum of the spiral) converges to the real axis when the imaginary part of $z_0$ goes to infinity. $z_0 = x_0 + iy_0$ $x_1 = \ln(|z|)$ $y_1 = \arctan(y_0/x_0)$ $y_2 = \arctan(\arctan(y_0/x_0)/\ln(\sqrt{x_0^2+y_0^2})$ for big arguments the logarithm grows faster than the arctan, thatswhy if we increase a big $y_0$ the quotient $\arctan(y_0/x_0)/\ln(|z|)$ decreases and hence $y_2$ decreases. As long as the radius of a number $z_n$ with $0<\Im(z_n)<\pi$ is too big, the imaginary part of the logarithm $\Im(z_{n+1})=\Im(\log(z_n))$ (which is the angle of $z_n$) will be smaller than the imaginary part of $z_n$. But doesnt this mean that the singularities that come close to the real axis somehow accumulate around 0? At least the minimum of the spiral will be around 0. I think I could even calculate bounds on the real axis around 0 in which the minimum of every such spiral has to live. « Next Oldest | Next Newest »

 Messages In This Thread Real and complex behaviour of the base change function (was: The "cheta" function) - by bo198214 - 08/12/2009, 08:59 PM RE: Real and complex behaviour of the base change function (was: The "cheta" function) - by jaydfox - 08/15/2009, 12:54 AM RE: Real and complex behaviour of the base change function (was: The "cheta" function - by jaydfox - 08/15/2009, 05:00 PM RE: Real and complex behaviour of the base change function (was: The "cheta" function - by bo198214 - 08/15/2009, 05:36 PM RE: Real and complex behaviour of the base change function (was: The "cheta" function - by jaydfox - 08/15/2009, 06:40 PM RE: Real and complex behaviour of the base change function (was: The "cheta" function - by jaydfox - 08/15/2009, 07:13 PM RE: Real and complex behaviour of the base change function (was: The "cheta" function - by bo198214 - 08/15/2009, 09:44 PM RE: Real and complex behaviour of the base change function (was: The "cheta" function - by jaydfox - 08/15/2009, 10:40 PM RE: Real and complex behaviour of the base change function (was: The "cheta" function - by jaydfox - 08/15/2009, 10:46 PM RE: Real and complex behaviour of the base change function (was: The "cheta" function - by jaydfox - 08/15/2009, 11:02 PM RE: Real and complex behaviour of the base change function (was: The "cheta" function - by jaydfox - 08/15/2009, 11:20 PM RE: Real and complex behaviour of the base change function (was: The "cheta" function - by bo198214 - 08/16/2009, 11:15 PM RE: Real and complex behaviour of the base change function (was: The "cheta" function - by jaydfox - 08/16/2009, 11:38 PM RE: Real and complex behaviour of the base change function (was: The "cheta" function - by bo198214 - 08/17/2009, 08:50 AM RE: Real and complex behaviour of the base change function (was: The "cheta" function - by jaydfox - 08/17/2009, 12:07 PM RE: Real and complex behaviour of the base change function (was: The "cheta" function - by sheldonison - 08/17/2009, 04:01 PM RE: Real and complex behaviour of the base change function (was: The "cheta" function - by bo198214 - 08/17/2009, 04:30 PM RE: Real and complex behaviour of the base change function (was: The "cheta" function - by jaydfox - 08/17/2009, 05:26 PM RE: Real and complex behaviour of the base change function (was: The "cheta" function - by sheldonison - 08/18/2009, 04:37 AM RE: Real and complex behaviour of the base change function (was: The "cheta" function - by jaydfox - 08/17/2009, 05:47 PM RE: Real and complex behaviour of the base change function (was: The "cheta" function) - by bo198214 - 08/17/2009, 02:40 PM base change with decremented exponential - by bo198214 - 08/18/2009, 08:47 AM singularities of base change eta -> e - by bo198214 - 08/18/2009, 06:51 PM RE: singularities of base change eta -> e - by bo198214 - 08/20/2009, 10:28 AM RE: Does the limit converge in the complex plane? - by sheldonison - 08/13/2009, 12:49 AM RE: Does the limit converge in the complex plane? - by bo198214 - 08/13/2009, 07:17 AM RE: Does the limit converge in the complex plane? - by sheldonison - 08/13/2011, 10:32 AM RE: Does the limit converge in the complex plane? - by bo198214 - 08/13/2011, 06:33 PM RE: Does the limit converge in the complex plane? - by sheldonison - 08/13/2009, 06:48 PM

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