Tetration

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About the Definitions

The following definitions are based on widely available definitions, including Munafo's definition of the hyper-operators. The only new definition here is the definition of the super-logarithm, which is discussed in more detail in my paper. You can access my paper from the menu above.

Definitions

Definition HY. Hyper-n Operators

x

⁽ⁿ⁾

y

x

n y

`x`⁽ⁿ⁾1 = `x`	if `n` ∈ Z⁺, `n` > 1
`x`⁽ⁿ⁾`y` = `x`^{(n − 1)}`x`⁽ⁿ⁾(`y` − 1)	if `n` ∈ Z⁺, `n` > 0
`x`⁽⁰⁾`y` = `x` ° `y`	"zeration" (the successor function, in most cases)
`x`⁽¹⁾`y` = `x` + `y`	"addition"
`x`⁽²⁾`y` = `x` `y`	"multiplication"
`x`⁽³⁾`y` = `x`^y = ¹`x`^y = e^{y log(x)}	"exponentiation"
`x`⁽⁴⁾`y` = ^y`x` = ^y`x`¹	"tetration"

Definition SL. Super-logarithm

_x

z

x

z

slog_x(`z`) = `y`	if `z` = ^y`x`
slog_x(`z`) = slog_x(`x`^z) − 1	if `x` > 1, `z` ≤ 0
slog_x(`z`) = slog_x(log_x(`z`)) + 1	if `x` > 1, `z` > 1
slog_x(`z`) = −1 + ∑_k=1..∞ `v`_k(`x`) `z`^k / `k`!	if `x` > 1, 0 < `z` ≤ 1 NEW where `v`_k(`x`) is the `k`-th element of the first column of the inverse matrix of: {{ `k` ^j / `k`! − δ_jk log(`x`)^−j }_k=1..∞ }_j=0..∞

Definition T. Tetration

^y

x

y

⁰`x` = 1
^y`x` = ^y`x`¹ = `x`⁽⁴⁾`y`
^y`x` = slog_x⁻¹(`y`)	if `x` > 1, `y` > −2
^y`x` = log_x(^y+1`x`)	if `x` > 1, `y` < −2

Definition IE. Iterated Exponentiation

^y

x

^a

x

y

a

⁰`x`^a = `a`
^y`x`^a = exp_x^y(`a`) = ^{y + slog_x(a)}`x`	if `x` > 1
^1/2e^x = hexp(`x`)	satisfies: hexp(hexp(`x`)) = exp(`x`)
^−1/2e^x = hlog(`x`)	satisfies: hlog(hlog(`x`)) = log(`x`)