Identities
Common Definitions
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• nxa =
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for iterated exponentiation
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• nx =
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for integer tetration
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Simple Definitions
- nxa = expnx(a)
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for iterated exponentiation
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- nx = nx1
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for integer tetration
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Inverse Function Formulas
- z = yx
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tetration
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- y = slogx(z)
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super-logarithm
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- x = srty(z)
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super-root
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- z = yxa
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iterated exponentiation
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- y = slogx(z) − slogx(a)
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inverse iterated exponentiation with respect to y
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- x = srt_(y + slogx(a))(z)
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inverse iterated exponentiation with respect to x
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- a = −yxz
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inverse iterated exponentiation with respect to a
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Integer Values of Tetration
- −nx
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undefined for integer n
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- −2x = ±∞
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- −1x = 0
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- 0x = 1
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- 1x = x
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- 2x = xx
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- ∞x = −W(−log(z)) / log(z)
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in terms of the product-logarithm, the Lambert-W function.
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- −∞x = ∞x
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is a different branch of the infinite iterated exponential.
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Integer Values of Iterated Exponentiation
- −nxa =
lognx(a)
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- −2xa = logx(logx(a))
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- −1xa = logx(a)
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- 0xa = a
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- 1xa = xa
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- 2xa = x^x^a
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the graph of which made this website's logo
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- nxa = expnx(a)
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- ∞xa = ∞x
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in terms of the infinite iterated exponential.
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- −∞xa = ∞x
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is a different branch of the infinite iterated exponential.
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The Infinite Iterated Exponential and Company
This table shows an interesting relationship:
that these functions are so similar, they can all be expressed in terms of each other.
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• ∞x
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= −W(−log(z)) / log(z)
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= 1 / srt2(1 / x)
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the infinite iterated exponential
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• ∞(e−x)x
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= W(x)
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= x / srt2(ex)
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the product-logarithm
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• 1 / ∞(1 / x)
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= log(z) / W(log(z))
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= srt2(x)
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the square super-root
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General Piecewise-Definitions
For tetration:
- yx = logx(y+1x)
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if y < -1, note: this piece iterates!
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- yx = t(x, y)
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if -1 < y ≤ 0
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- yx = x^(y−1x)
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if y > 0, note: this piece iterates!
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where
t(
x,
y) is the
critical function for tetration.
For the super-logarithm:
- slogx(z) =
slogx(xz) − 1
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if z ≤ 0
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- slogx(z) = s(x, z)
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if 0 < z ≤ 1
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- slogx(z) =
slogx(logx(z)) + 1
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if z > 1, note: this piece iterates!
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where
s(
x,
z) is the
critical function for the super-logarithm.
A Simple Extension to Real Numbers
The simplest extension to real numbers is the bare minimum required
to obey the known integer values for tetration and the super-logarithm:
- t(x, y) = y + 1
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if -1 < y ≤ 0, for tetration
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- s(x, z) = z − 1
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if 0 < z ≤ 1, for superlog
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which satisfy the constraints:
- t(x, -1) = 0
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like -1x = 0
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- t(x, 0) = 1
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like 0x = 1
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and
- s(x, 0) = -1
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like slogx(0) = -1
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- s(x, 1) = 0
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like slogx(1) = 0
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Complex Conjugate Properties
Using my extension, these have been verified numerically
for inputs with small imaginary parts between −1/2 and 1/2.
- conj(yx) = conj(y)x
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for tetration
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- conj(slogx(z)) = slogx(conj(z))
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for superlog
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Integration Formulas
- ∫ x=0..1 xx dx =
∑ n=1..∞ (−1)n+1 / nn
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Power Tower
Formula #27 -- From MathWorld
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- ∫ x=0..1 x−x dx =
∑ n=1..∞ 1 / nn
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Power Tower
Formula #29 -- From MathWorld
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Hyper-operator Notation
- x(n)y
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Munafo's hyper-operator notation (de facto standard)
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- x(n)y = x↑(n−2) y
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Knuth's extended up-arrow notation (acedemic standard)
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- x(n)y = x → y → (n − 2)
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Conway's chained-arrow notation
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- x(n)y = { x, y, n }
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Bower's array notation
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- x(n)y = hyper(x, n, y)
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The hyper() function notation
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- x(n)y = hyper-n(x, y)
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The hyper-n() function notation
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Tetration Notation
- yx = "
x^^y"
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ASCII notation for tetration
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- yx = x↑↑y
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Knuth's up-arrow notation
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- yx = x↑2 y
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Knuth's extended up-arrow notation
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- yx = x → y → 2
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Conway's chained-arrow notation
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- yx = x(4)y
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Munafo's hyper-operator notation (de facto standard)
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- yx = hyper(x, 4, y)
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The hyper() function notation
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- yx = hyper4(x, y)
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The hyper-n() function notation
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- yx = twry(x)
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A notation for power tower functions
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- yx = tetx(y)
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A notation for tetrational functions
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Iterated Exponentiation Notation
Galidakis uses the notation
y(
x,
a) for iterated exponentiation,
which is beneficial when
a is complicated.
In ASCII, the same can be expressed using the notation "
x^^y@a",
which can be found in several places around the internet. Interestingly,
some have noticed that the notation "
x^^y.a" is equivalent
to the first-degree approximation of tetration, where
y is the integer part and
a is the fractional part of the value being tetrated.
- yxa = "
x^^y@a"
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ASCII notation for iterated exponentiation
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- yxa = "
x^^y.a"
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ASCII notation for approximate iterated exponentiation
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- yxa = y(x, a)
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Galidakis' notation for iterated exponentiation
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- yxa = expyx(a)
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A functional notation for iterated exponentiation
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Miscellaneous
- xy = ey log(x)
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exponentiation in terms of exp and log
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- xy = 1 + slog_x(y)x
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exponentiation in terms of tet and slog
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- xy = 1xy
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exponentiation in terms of continuously iterated exponentiation
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- yx = yx1
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tetration in terms of continuously iterated exponentiation
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- yxa = y + slog_x(a)x
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continuously iterated exponentiation in terms of tetration
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