When we have a tetration with his base "a" between , it tends to an asymptote value such that .

But that depends on the definition . If it were defined , we would get an horizontal line (drawn with red dashes in the graphic down).

The interesting part is when we define a bit larger than , the tetration also converges to the same asymptote to the right, and another asymptote to the left, at the limit , defined by .

We get a Z-shaped tetration function (drawn in green), contained between and , which I call the Z curve.

The problem is that the Z curve is not uniquely defined. It depends on the value choosen for . Any value between is valid, and the only difference this choice make, is the horizontal displacement on the curve. I drew the Z curve matching the origin with his inflection point (roughly).

The upper asymptote is also a non trivial function, and there is also another non trivial possible upper branch, obtained by choosing °a>-oo (drawn in blue line).

The question is, what would be convenient values for the definition of for the green and blue branches?

If we take , may be multivalued at x=0: .

The inflection point in the Z curve is near to 3. Maybe that base has all integer values for °a? (°1,41421356 = (1,2,3,4,5) )

Note how the blue branch resembles the function .

But that depends on the definition . If it were defined , we would get an horizontal line (drawn with red dashes in the graphic down).

The interesting part is when we define a bit larger than , the tetration also converges to the same asymptote to the right, and another asymptote to the left, at the limit , defined by .

We get a Z-shaped tetration function (drawn in green), contained between and , which I call the Z curve.

The problem is that the Z curve is not uniquely defined. It depends on the value choosen for . Any value between is valid, and the only difference this choice make, is the horizontal displacement on the curve. I drew the Z curve matching the origin with his inflection point (roughly).

The upper asymptote is also a non trivial function, and there is also another non trivial possible upper branch, obtained by choosing °a>-oo (drawn in blue line).

The question is, what would be convenient values for the definition of for the green and blue branches?

If we take , may be multivalued at x=0: .

The inflection point in the Z curve is near to 3. Maybe that base has all integer values for °a? (°1,41421356 = (1,2,3,4,5) )

Note how the blue branch resembles the function .

I have the result, but I do not yet know how to get it.