# Tetration Forum

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In my view, "arc" refers to the length of a curve ("curve" as distinctive from a straight line).
But that's already a criticism to the use of "arc" in the context of superlogarithm: the length of the piece of trajectory, (according to the difference of heights at two distinct points (x1,f(x1)) ... (x2,f(x2)) is not meant by the superlogarithm, for which another name was sought. So "arc" seems to be not a special good choice for the inverse of tetration. (in the case of a circle it is the length of the curve, proportional to the angle)

Gottfried
(08/05/2009, 06:15 PM)bo198214 Wrote: [ -> ].. the "arc" prefix for trigonometric functions comes from measuring the length of the arc on the unit circle (which is the angle).
So etymologically this would count against a usage as "inverse"-prefix.
How about arcsinh, arccosh,..? Now the prefix "arc" means "inverse function of".
If I ask for "ArcGamma", everybody understands what do I mean, but if I say "inverse of Gamma", the GNU offers f(z)=1/Gamma(z) instead.

I know that slog already appears in some texts as inverse of tetrational; sorry about that. But I do not extend this confusion to the new texts.

I suggest the rule: if there is some holomorphic function Func, then
the inverse is called ArcFunc,
the superfunction of Func is called SuperFunc,
and its inverse function is called ArcSuperFunc.

For example, if a superfunction of tetrational, we already have names for it:
pentational=pen=SuperTetrational
and its inverse should be called arcpen=ArcSuperTetrational

We need two prefices to indicate the inverse function and to indicate the superfincton.
"arc" and "super" are best.
(08/06/2009, 10:11 AM)Ansus Wrote: [ -> ]Dmitry, there is no 'arcsinh', there is 'arsinh' with 'ar' stands for area (of a sector).
I type
series(arcsinh(x), x)
x-1/6*x^3+3/40*x^5+O(x^6)
This is how I had envisioned the iterational terminology working:
• superfunction of F = iterational of F
• super-exponential = iter-exponential
• super-logarithm = inverse iter-exponential = Abel function of exponential
• super-F = iter-F
By using the "iter-" prefix instead of the "super-" prefix, there is no confusion with previous terminology, and we can continue using "super-logarithm" to mean what it has always meant. And to keep up with the current writings, I suppose these terms could be prefixed with mappings, like (0 -> 1) iter-cosine or something.
Quote:there is no 'arcsinh'
...
(08/06/2009, 02:52 PM)Ansus Wrote: [ -> ]Interesting.
Nothing special. Not only Maple, but also C++ and Mathematica recognize arcsinh as inverse function of sinh. You may try also arccosh, arctanh, arcsech, etc.
Unfortunately, arcGamma is not yet implemented.

(08/10/2009, 05:24 AM)andydude Wrote: [ -> ]This is how I had envisioned the iterational terminology working:
• superfunction of F = iterational of F
• super-exponential = iter-exponential
• super-logarithm = inverse iter-exponential = Abel function of exponential
• super-F = iter-F
By using the "iter-" prefix instead of the "super-" prefix, there is no confusion with previous terminology, and we can continue using "super-logarithm" to mean what it has always meant. And to keep up with the current writings, I suppose these terms could be prefixed with mappings, like (0 -> 1) iter-cosine or something.
We already have the SuperFactorial and ArcSuperFactorial.
Should we rename them to Iter-factorial and Abel-factorial?

I see, Iter and Abel are couple, like integration and differentiation. Soon we may have also pen=pentational=(0 -> 1)Iter-tet, arcpen=ArcPentational=(1 -> 0)Abel-tet;
Iter-Bessel, Abel-Bessel; Abel-Mathieu; Abel-Legendre; Abel-(Hermit-Gaussian)...
and some special prefixes for hyper-operations:

for the two functions x[N]a and a[N]x, add greek prefixes for N.

just say "N-ation" for the hyper-N-operation itself,
"N-power" for x[N]a; "N-root" for the inverse of N-power;
"N-exponential" for a[N]x, "N-logarithm" for the inverse of N-exponential

example:
tetra-power, tetra-exponential
tetra-root, tetra-logarithm

if you don't wanna say "heptaconta(70)-exponential", don't worry, humanity probably won't ever get there or

just make iter[] an operator on functions. then we can square it like any other operators. iter^2 means doing iter twice.
iter^-1[] is the inverse of iter[], it extracts the function you built the iter-function out of. so iter^-1[tetra-exponential]= exponential. this is more generalizable.
also, abel[f] := [iter(f)]^-1, iter[f] considered as a function.
examples:
we may be able to do away with all the hyper-n-operator s*it.

so that for kouznetsov, we can have the mapping information under the iter operator. (it will look a bit like the limit operator)
(08/10/2009, 05:24 AM)andydude Wrote: [ -> ]superfunction of F = iterational of F
Iterational sounds good, however we dont have a problem with superfunction, but with the super- prefix.

Quote:super-exponential = iter-exponential
Imho this doesnt sound very convincing. Additionally if you pronounce iter as i-ter, then this sounds like the German word for pus.

Quote:super-logarithm = inverse iter-exponential = Abel function of exponential
What was needed here was a prefix, both alternatives are not a prefix.
If we abuse Abel as a prefix, like Abel-exponential, then it rather sounds like a particular kind of an exponential, rather than something completely different from an exponential.

btw. in the textbook of Milnor, featured in some other thread, they call the Abel function also Fatou function, and the Schröder function also Kœnigs function.
Quote:mho this doesnt sound very convincing. Additionally if you pronounce iter as i-ter, then this sounds like the German word for pus.

then don't pronounce it that way. say "itter", (rhymes with litter) as in "iteration"
(08/10/2009, 05:51 PM)Tetratophile Wrote: [ -> ]then don't pronounce it that way. say "itter", (rhymes with litter) as in "iteration"

That was not the main reason, as I said "additionally".
The main reason is, that it is just not a proper prefix. Proper prefixes are rather super, hyper, ultra, tetra, etc.
so make iter[] an operator like i said. how are we gonna pronounce it anyway.

Tetratophile Wrote:just make iter[] an operator on functions. then we can square it like any other operators. iter^2 means doing iter twice.
iter^-1[] is the inverse of iter[], it extracts the function you built the iter-function out of. so iter^-1[tetra-exponential]= exponential. this is more generalizable.
also, abel[f] := [iter(f)]^-1, iter[f] considered as a function.
examples: