Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
Bivariate quasi-holomorphic function T(x,y) for complex x and y
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?

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

Possibly Related Threads...
Thread Author Replies Views Last Post
  Ueda - Extension of tetration to real and complex heights MphLee 2 72 12/03/2021, 01:23 AM
Last Post: JmsNxn
  A Holomorphic Function Asymptotic to Tetration JmsNxn 2 1,131 03/24/2021, 09:58 PM
Last Post: JmsNxn
  New mathematical object - hyperanalytic function arybnikov 4 6,154 01/02/2020, 01:38 AM
Last Post: arybnikov
  Complex Tetration, to base exp(1/e) Ember Edison 7 9,876 08/14/2019, 09:15 AM
Last Post: sheldonison
  Is there a function space for tetration? Chenjesu 0 2,189 06/23/2019, 08:24 PM
Last Post: Chenjesu
  Can we get the holomorphic super-root and super-logarithm function? Ember Edison 10 15,032 06/10/2019, 04:29 AM
Last Post: Ember Edison
  Degamma function Xorter 0 2,606 10/22/2018, 11:29 AM
Last Post: Xorter
  An explicit series for the tetration of a complex height Vladimir Reshetnikov 13 24,195 01/14/2017, 09:09 PM
Last Post: Vladimir Reshetnikov
  holomorphic binary operators over naturals; generalized hyper operators JmsNxn 15 28,243 08/22/2016, 12:19 AM
Last Post: JmsNxn
  Should tetration be a multivalued function? marraco 17 31,598 01/14/2016, 04:24 AM
Last Post: marraco

Users browsing this thread: 1 Guest(s)