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 Bivariate quasi-holomorphic function T(x,y) for complex x and y Ciera_ΩMega Junior Fellow Posts: 2 Threads: 1 Joined: Sep 2010 09/04/2010, 05:25 PM (This post was last modified: 09/08/2010, 08:45 AM by Ciera_ΩMega.) Consider a bivariate function T(x,y) for complex x and y that satisfies the following ① T(x,0) = 1 for x≠0 and abs(x)≠1 ② T(x,y+1) = x^T(x,y) ③ For constant x, x≠0 and abs(x)≠1, T(x,y) is bounded on the strip with Re(y)∈[0,1) and Im(y)∈(-∞,∞) ④ For constant real y, y≥0, Re(x)=0 and Im(x)≠0, Re(T(x,y)) and T(x,y)/x are bounded ⑤ T(x,y) is a function of two complex variables holomorphic outside from singularities and branch cuts Conditions ① and ② are basic conditions for tetration; ③ ensures that base-x tetrational cannot diverge for extreme values of Im(y), much as exp(y) is bounded for constant Re(y) and variable Im(y); ④ is derived from the fact that for Re(x)=0 and Im(y)=0, the iterated exponential of x as x→∞ⅈ and y→∞, approaches the 3-cycle {x, 0, 1}, which clearly has Re(T(x,y)) bounded and T(x,y)/x also bounded, and ⑤ ensures smoothness and analyticity of T(x,y) except on cuts and at singularities. The idea for a function described above comes from the thread (tid=380) about base holomorphic tetration for fixed height, and (tid=377) which describes tetration of complex bases to complex heights. How about a bivariate function that is quasi-holomorphic on both base and height, much like how complex bases can be raised to complex powers with quasi-holomorphism on base and exponent? What ideas can be made about uniqueness conditions for such a bivariate function? « Next Oldest | Next Newest »

 Messages In This Thread Bivariate quasi-holomorphic function T(x,y) for complex x and y - by Ciera_ΩMega - 09/04/2010, 05:25 PM RE: Bivariate quasi-holomorphic function T(x,y) for complex x and y - by bo198214 - 09/07/2010, 03:58 PM RE: Bivariate quasi-holomorphic function T(x,y) for complex x and y - by tommy1729 - 09/07/2010, 07:50 PM RE: Bivariate quasi-holomorphic function T(x,y) for complex x and y - by bo198214 - 09/08/2010, 02:23 AM RE: Bivariate quasi-holomorphic function T(x,y) for complex x and y - by mike3 - 09/08/2010, 05:20 AM

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