09/04/2010, 05:25 PM

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?

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?