bo198214 Wrote:Quote:[*] remark: this argumentation seems to include, that f and g have the same base-parameter
There is no base. \( f \) and \( g \) are arbitrary analytic functions with development and fixed point at 0.
Hmm - first: I didn't expect that someone would reply so fast; so I added my corrections in the statement-list just by updating. Sorry.
second: yes, true; may be the terms "base" and "height" focus the reader to some speciality. What I mean is the following.
I express the operator iteration as
addition: x {+,b} h which means (x +b)+b)+b)...+b) h-times
multiplication: x {*,b} h which means (x *b)*b)*b)...*b) h-times
tetration: x {^,b} h which means b^(...^(b^(b^x)))) h-times
and generally for a function F, expressed as operator, this may perhaps be extended to
function F: x {F,b} h which means then F(F(...F((x)))
Then the formal base-parameter b may be included into the definition of F itself and may be omitted/defaulted. For instance, we are not used to consider the sin()-function in explicite terms of a "base" - but in fact, the powerseries may be parametrized with log(b) getting a sin_b(x), and the conventional sin()-function is then sin_b(x) with b=exp(1)
So, for the iterated sin-function sin_b we may write
function sin_b: x {SIN,b} h which means then sin_b(sin_b(...sin_b(x))) h-times
and then
function sin : x {SIN,exp(1)} h
with the default
function sin : x {sin} h = sin°h(x)
Regarding the "Height" - well, in fact it is better to say "iterate", but the letter "i" for its abbreviation would be *much* confusing. Our usual "t" is somehow anonym, and I used "t" from the beginning of my tetration-discussion as a very basic notation (also in my program-code), so what to do? "Height" for "iteration-height" is at least not completely misleading...
Gottfried Helms, Kassel
