Failed to parse (Missing texvc executable; please see math/README to configure.): {}^{n}x , for n = 2, 3, 4, 5, 6 and 7.
In mathematics, tetration (also known as hyper-4) is an iterated exponential, the first hyper operator after exponentiation. The portmanteau word tetration was coined by English mathematician Reuben Louis Goodstein from tetra- (four) and iteration. Tetration is used for the notation of very large numbers. Shown here are examples of the first four hyper operators, with tetration as the fourth:
- addition
- Failed to parse (Missing texvc executable; please see math/README to configure.): {{a + b} \atop \,} {= \atop \,} {a \, + \atop \, } {{\underbrace{1 + 1 + \cdots + 1}} \atop b}
- multiplication
- Failed to parse (Missing texvc executable; please see math/README to configure.): {{a \times b = \ } \atop {\ }} {{\underbrace{a + a + \cdots + a}} \atop b}
- exponentiation
- Failed to parse (Missing texvc executable; please see math/README to configure.): {{a^b = \ } \atop {\ }} {{\underbrace{a \times a \times \cdots \times a}} \atop b}
- tetration
- Failed to parse (Missing texvc executable; please see math/README to configure.): {\ ^{b}a = \ \atop {\ }} {{\underbrace{a^{a^{\cdot^{\cdot^{a}}}}}} \atop b}
where each operation is defined by iterating the previous one. The peculiarity of the tetration among these operations is that the first three (addition, multiplication and exponentiation) are generalized for complex values of Failed to parse (Missing texvc executable; please see math/README to configure.): ~b~ , while for tetration, no such regular generalization is yet established; and tetration is not considered an elementary function.
Addition (a+b) can be thought of as being b iterations of the "add one" function applied to a, multiplication (ab) can be thought of as a chained addition involving b numbers a, and exponentiation (Failed to parse (Missing texvc executable; please see math/README to configure.): a^b ) can be thought of as a chained multiplication involving b numbers a. Analogously, tetration (Failed to parse (Missing texvc executable; please see math/README to configure.): ^{b}a ) can be thought of as a chained power involving b numbers a. The parameter a may be called the base-parameter in the following, while the parameter b in the following may be called the height-parameter (which is integral in the first approach but may be generalized to fractional, real and complex heights, see below)
Iterated powers
Note that when evaluating multiple-level exponentiation, the exponentiation is done at the deepest level first (in the notation, at the highest level). In other words:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \,\!\ ^{4}2 = 2^{2^{2^2}} = 2^{\left(2^{\left(2^2\right)}\right)} = 2^{\left(2^4\right)} = 2^{16} = 65,\!536
The convention for iterated exponentiation is to work from the right to the left. Thus,
- Failed to parse (Missing texvc executable; please see math/README to configure.): \,\!2^{2^{2^2}} \not= \,\! \left({\left(2^2\right)}^2\right)^2 = 2^{2 \cdot 2 \cdot2} = 256
.
To generalize the first case (tetration) above, a new notation is needed (see below); however, the second case can be written as
- Failed to parse (Missing texvc executable; please see math/README to configure.): \,\! \left({\left(2^2\right)}^2\right)^2 = 2^{2 \cdot 2 \cdot 2} = 2^{2^3}
Thus, its general form still uses ordinary exponentiation notation.
In general, we can use Knuth's up-arrow notation to write a power as Failed to parse (Missing texvc executable; please see math/README to configure.): (\uparrow b)(a) = a^b
which allows us to write its general form as:
- Failed to parse (Missing texvc executable; please see math/README to configure.): (\uparrow b)^n(a) = a^{b^{n}}
Terminology
There are many terms for tetration, each of which has some logic behind it, but some have not become commonly used for one reason or another. Here is a comparison of each term with its rationale and counter-rationale.
- The term super-exponentiation is the most proper candidate for a name, however, Bromer published his paper Superexponentiation in 1987, and Goodstein published his paper Transfinite Ordinals in Recursive Number Theory (which coined the term tetration) in 1947, which predates Bromer. So although this is not a misnomer, the shorter and older term has gained more use.
- The term hyperpower is a natural combination of hyper and power, which aptly describes tetration. The problem lies in the meaning of hyper with respect to the hyper operator hierarchy. When considering hyper operators, the term hyper refers to all ranks, and the term super refers to rank 4, or tetration. So under these considerations hyperpower is a misnomer, since it is only referring to tetration.
- The term power tower has a nice ring to it, but it is a misnomer in two different manners. The first way is that tetration has nothing to do with power functions, as it is a special case of iterated exponentials (see the section iterated powers above). The second way is that the term tower is used to describe nested exponentials which are much more general expressions than tetration. Since power tower is a misnomer in more than one way, it has fallen out of use.
Tetration is often confused with closely related functions and expressions. This is because many of the terminology that is used with them can be used with tetration. Here are a few related terms:
-
| Form |
Terminology |
| Failed to parse (Missing texvc executable; please see math/README to configure.): a^{a^{\cdot^{\cdot^{a^a}}}} |
Tetration |
| Failed to parse (Missing texvc executable; please see math/README to configure.): a^{a^{\cdot^{\cdot^{a^x}}}} |
Iterated exponentials |
| Failed to parse (Missing texvc executable; please see math/README to configure.): a_1^{a_2^{\cdot^{\cdot^{a_n}}}} |
Nested exponentials (also towers) |
| Failed to parse (Missing texvc executable; please see math/README to configure.): a_1^{a_2^{a_3^{\cdot^{\cdot^\cdot}}}} |
Infinite exponentials (also towers) |
In the first two expressions a is the base, and the number of as is the height (add one for x). In the third expression, n is the height, but each of the bases are different.
Care must be taken when referring to iterated exponentials, as it is common to call expressions of this form iterated exponentiation, which is ambiguous, as this can either mean iterated powers or iterated exponentials.
Notation
The notations in which tetration can be written (some of which allow even higher levels of iteration) include:
-
| Name |
Form |
Description |
| Standard notation |
Failed to parse (Missing texvc executable; please see math/README to configure.): \,{}^{b}a |
First used by Hans Maurer; Rudy Rucker's book Infinity and the Mind popularized the notation. |
| Knuth's up-arrow notation |
Failed to parse (Missing texvc executable; please see math/README to configure.): a {\uparrow\uparrow} b |
Allows extension by putting more arrows, or, even more powerfully, an indexed arrow. |
| Conway chained arrow notation |
Failed to parse (Missing texvc executable; please see math/README to configure.): a \rightarrow b \rightarrow 2 |
Allows extension by increasing the number 2 (equivalent with the extensions above), but also, even more powerfully, by extending the chain |
| Ackermann function |
Failed to parse (Missing texvc executable; please see math/README to configure.): {}^{b}2 = \operatorname{A}(4, b - 3) + 3 |
Allows the special case Failed to parse (Missing texvc executable; please see math/README to configure.): a=2
to be written in terms of the Ackermann function.
|
| Iterated exponential notation |
Failed to parse (Missing texvc executable; please see math/README to configure.): {}^{b}a = \exp_a^b(1) |
Allows simple extension to iterated exponentials from initial values other than 1. |
| Hyper operator notation |
Failed to parse (Missing texvc executable; please see math/README to configure.): a^{(4)}b |
Allows extension by increasing the number 4; this gives the family of hyper operators |
| ASCII notation |
a^^b |
Since the up-arrow is used identically to the caret (^), the tetration operator may be written as (^^). |
One notation above shows that tetration can be written as an iterated exponential function where the initial value is one. As a reminder, iterated exponentials have the general form:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \exp_a^n(x) = a^{a^{\cdot^{\cdot^{a^x}}}}
with n a's.
There are not as many notations for iterated exponentials, but here are a few:
-
| Name |
Form |
Description |
| Standard notation |
Failed to parse (Missing texvc executable; please see math/README to configure.): \exp_a^n(x) |
Euler coined the notation Failed to parse (Missing texvc executable; please see math/README to configure.): \exp_a(x) = a^x
, and iteration notation Failed to parse (Missing texvc executable; please see math/README to configure.): f^n(x)
has been around about as long.
|
| Knuth's up-arrow notation |
Failed to parse (Missing texvc executable; please see math/README to configure.): (a{\uparrow})^n(x) |
Allows for super-powers and super-exponential function by increasing the number of arrows; used in the article on large numbers. |
| Galidakis' notation |
Failed to parse (Missing texvc executable; please see math/README to configure.): \,{}^{n}(a, x) |
Allows for large expressions in the base; used by Ioannis Galidakis in On Extending hyper4 ... to the Reals. |
| ASCII (auxiliary) |
a^^n@x |
Based on the view that an iterated exponential is auxiliary tetration. |
| ASCII (standard) |
exp_a^n(x) |
Based on standard notation. |
Examples
In the following table, most values are too large to write in scientific notation, so iterated exponential notation is employed to express them in base 10. The values containing a decimal point are approximate.
-
| Failed to parse (Missing texvc executable; please see math/README to configure.): n |
Failed to parse (Missing texvc executable; please see math/README to configure.): {}^{2}n |
Failed to parse (Missing texvc executable; please see math/README to configure.): {}^{3}n |
Failed to parse (Missing texvc executable; please see math/README to configure.): {}^{4}n |
| 0 |
1 |
0 |
1 |
| 1 |
1 |
1 |
1 |
| 2 |
4 |
16 |
65,536 |
| 3 |
27 |
7,625,597,484,987 |
Failed to parse (Missing texvc executable; please see math/README to configure.): \exp_{10}^3(1.09902) |
| 4 |
256 |
Failed to parse (Missing texvc executable; please see math/README to configure.): \exp_{10}^2(2.18788) |
Failed to parse (Missing texvc executable; please see math/README to configure.): \exp_{10}^3(2.18726) |
| 5 |
3,125 |
Failed to parse (Missing texvc executable; please see math/README to configure.): \exp_{10}^2(3.33931) |
Failed to parse (Missing texvc executable; please see math/README to configure.): \exp_{10}^3(3.33928) |
| 6 |
46,656 |
Failed to parse (Missing texvc executable; please see math/README to configure.): \exp_{10}^2(4.55997) |
Failed to parse (Missing texvc executable; please see math/README to configure.): \exp_{10}^3(4.55997) |
| 7 |
823,543 |
Failed to parse (Missing texvc executable; please see math/README to configure.): \exp_{10}^2(5.84259) |
Failed to parse (Missing texvc executable; please see math/README to configure.): \exp_{10}^3(5.84259) |
| 8 |
16,777,216 |
Failed to parse (Missing texvc executable; please see math/README to configure.): \exp_{10}^2(7.18045) |
Failed to parse (Missing texvc executable; please see math/README to configure.): \exp_{10}^3(7.18045) |
| 9 |
387,420,489 |
Failed to parse (Missing texvc executable; please see math/README to configure.): \exp_{10}^2(8.56784) |
Failed to parse (Missing texvc executable; please see math/README to configure.): \exp_{10}^3(8.56784) |
| 10 |
10,000,000,000 |
Failed to parse (Missing texvc executable; please see math/README to configure.): \exp_{10}^3(1) |
Failed to parse (Missing texvc executable; please see math/README to configure.): \exp_{10}^4(1) |
Extensions
Extending Failed to parse (Missing texvc executable; please see math/README to configure.): \,{}^{b}x
to real numbers Failed to parse (Missing texvc executable; please see math/README to configure.): x > 0
is straightforward and gives, for each natural number Failed to parse (Missing texvc executable; please see math/README to configure.): b
, a super-power function Failed to parse (Missing texvc executable; please see math/README to configure.): \,f(x) = {}^{b}x . The term super is sometimes replaced by hyper, but this only applies to tetration with integer height, and is falling out of usage. All other uses of the two prefixes use the convention: hyper for all ranks of hyper operators, and super for the rank 4 hyper operator, known as tetration.
Extension to infinitesimal bases
Sometimes, Failed to parse (Missing texvc executable; please see math/README to configure.): 0^0
is taken to be an undefined quantity. In this case, values for Failed to parse (Missing texvc executable; please see math/README to configure.): \,{}^{k}0
cannot be defined directly. However, Failed to parse (Missing texvc executable; please see math/README to configure.): \lim_{n\rightarrow0} {}^{k}n
is well defined, and exists:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \lim_{n\rightarrow0} {}^{k}n = \begin{cases} 1, & k \mbox{ even} \\ 0, & k \mbox{ odd} \end{cases}
This limit holds for negative Failed to parse (Missing texvc executable; please see math/README to configure.): n , as well. Failed to parse (Missing texvc executable; please see math/README to configure.): \,{}^{k}0
could be defined in terms of this limit and this would agree with a definition of Failed to parse (Missing texvc executable; please see math/README to configure.): 0^0 = 1
. This limit definition holds for Failed to parse (Missing texvc executable; please see math/README to configure.): {}^{2}0 = 1
because 2 is even, and holds for Failed to parse (Missing texvc executable; please see math/README to configure.): {}^{0}0 = 1
because 0 is even.
Extension to complex bases
Since complex numbers can be raised to powers, tetration can be applied to bases of the form Failed to parse (Missing texvc executable; please see math/README to configure.): z = a + bi , where Failed to parse (Missing texvc executable; please see math/README to configure.): i
is the square root of −1. For example, Failed to parse (Missing texvc executable; please see math/README to configure.): {}^{k}z
where Failed to parse (Missing texvc executable; please see math/README to configure.): z=i
, tetration is achieved by using the principal branch of the natural logarithm, and using Euler's formula we get the relation:
- Failed to parse (Missing texvc executable; please see math/README to configure.): i^{a+bi} = e^{{i\pi \over 2} (a+bi)} = e^{-{b\pi \over 2}} \left(\cos{a\pi \over 2} + i \sin{a\pi \over 2}\right)
This suggests a recursive definition for Failed to parse (Missing texvc executable; please see math/README to configure.): {}^{(k+1)}i = a'+b'i
given any Failed to parse (Missing texvc executable; please see math/README to configure.): {}^{k}i = a+bi
- Failed to parse (Missing texvc executable; please see math/README to configure.): a' = e^{-{b\pi \over 2}} \cos{a\pi \over 2}
- Failed to parse (Missing texvc executable; please see math/README to configure.): b' = e^{-{b\pi \over 2}} \sin{a\pi \over 2}
The following approximate values can be derived:
-
| Failed to parse (Missing texvc executable; please see math/README to configure.): {}^{k}i |
Approximate Value |
| Failed to parse (Missing texvc executable; please see math/README to configure.): {}^{1}i = i |
i |
| Failed to parse (Missing texvc executable; please see math/README to configure.): {}^{2}i = i^{\left({}^{1}i\right)} |
Failed to parse (Missing texvc executable; please see math/README to configure.): 0.2079 |
| Failed to parse (Missing texvc executable; please see math/README to configure.): {}^{3}i = i^{\left({}^{2}i\right)} |
Failed to parse (Missing texvc executable; please see math/README to configure.): 0.9472 + 0.3208i |
| Failed to parse (Missing texvc executable; please see math/README to configure.): {}^{4}i = i^{\left({}^{3}i\right)} |
Failed to parse (Missing texvc executable; please see math/README to configure.): 0.0501 + 0.6021i |
| Failed to parse (Missing texvc executable; please see math/README to configure.): {}^{5}i = i^{\left({}^{4}i\right)} |
Failed to parse (Missing texvc executable; please see math/README to configure.): 0.3872 + 0.0305i |
| Failed to parse (Missing texvc executable; please see math/README to configure.): {}^{6}i = i^{\left({}^{5}i\right)} |
Failed to parse (Missing texvc executable; please see math/README to configure.): 0.7823 + 0.5446i |
| Failed to parse (Missing texvc executable; please see math/README to configure.): {}^{7}i = i^{\left({}^{6}i\right)} |
Failed to parse (Missing texvc executable; please see math/README to configure.): 0.1426 + 0.4005i |
| Failed to parse (Missing texvc executable; please see math/README to configure.): {}^{8}i = i^{\left({}^{7}i\right)} |
Failed to parse (Missing texvc executable; please see math/README to configure.): 0.5198 + 0.1184i |
| Failed to parse (Missing texvc executable; please see math/README to configure.): {}^{9}i = i^{\left({}^{8}i\right)} |
Failed to parse (Missing texvc executable; please see math/README to configure.): 0.5686 + 0.6051i |
Solving the inverse relation as in the previous section, yields the expected Failed to parse (Missing texvc executable; please see math/README to configure.): \,{}^{0}i = 1
and Failed to parse (Missing texvc executable; please see math/README to configure.): \,{}^{(-1)}i = 0
, with negative values of Failed to parse (Missing texvc executable; please see math/README to configure.): k
giving infinite results on the imaginary axis. Plotted in the complex plane, the entire sequence spirals to the limit Failed to parse (Missing texvc executable; please see math/README to configure.): 0.4383 + 0.3606i
, which could be interpreted as the value where Failed to parse (Missing texvc executable; please see math/README to configure.): k
is infinite.
Such tetration sequences have been studied since the time of Euler but are poorly understood due to their chaotic behavior. Most published research historically has focused on the convergence of the power tower function. Current research has greatly benefited by the advent of powerful computers with fractal and symbolic mathematics software. Much of what is known about tetration comes from general knowledge of complex dynamics and specific research of the exponential map.
Extension to infinite heights
Tetration can be extended to heights (b in Failed to parse (Missing texvc executable; please see math/README to configure.): {}^{b}a ) that are not finite, but infinite. This is because for bases within a certain interval, tetration converges to a finite value as the height tends to infinity. For example, Failed to parse (Missing texvc executable; please see math/README to configure.): \sqrt{2}^{\sqrt{2}^{\cdot^{\cdot^{\cdot}}}}
converges to 2, and can therefore be said to be equal to 2. The trend towards 2 can be seen by evaluating a small finite tower:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{1.41}}}}} = \sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{1.63}}}} = \sqrt{2}^{\sqrt{2}^{\sqrt{2}^{1.76}}} = \sqrt{2}^{\sqrt{2}^{1.84}} = \sqrt{2}^{1.89} = 1.93
In general, the infinite power tower Failed to parse (Missing texvc executable; please see math/README to configure.): x^{x^{\cdot^{\cdot}}} , defined as the limit of Failed to parse (Missing texvc executable; please see math/README to configure.): {}^{n}x
as n goes to infinity, converges for Failed to parse (Missing texvc executable; please see math/README to configure.): e^{-e} < x < e^{1/e}
, roughly the interval from 0.066 to 1.44. For arbitrary real Failed to parse (Missing texvc executable; please see math/README to configure.): r
with Failed to parse (Missing texvc executable; please see math/README to configure.): 0 < r < e
, let Failed to parse (Missing texvc executable; please see math/README to configure.): x = r^{1/r} , then the limit is Failed to parse (Missing texvc executable; please see math/README to configure.): r
- there is no convergence for Failed to parse (Missing texvc executable; please see math/README to configure.): x > e^{1/e}
since max of Failed to parse (Missing texvc executable; please see math/README to configure.): r^{1/r}
is Failed to parse (Missing texvc executable; please see math/README to configure.): e^{1/e}
.
This may be extended to complex numbers Failed to parse (Missing texvc executable; please see math/README to configure.): z
with the definition:
- Failed to parse (Missing texvc executable; please see math/README to configure.): {}^{\infty}z = z^{z^{\cdot^{\cdot}}} = -\frac{\mathrm{W}(-\ln{z})}{\ln{z}}
where Failed to parse (Missing texvc executable; please see math/README to configure.): \mathrm{W}(z)
represents Lambert's W function. As the limit Failed to parse (Missing texvc executable; please see math/README to configure.): y={}^{\infty}x
(if existent, i.e. for Failed to parse (Missing texvc executable; please see math/README to configure.): e^{-e} < x < e^{1/e}
) must satisfy Failed to parse (Missing texvc executable; please see math/README to configure.): x^y=y
we see that Failed to parse (Missing texvc executable; please see math/README to configure.): h(x):={}^{\infty}x
is (the lower branch of) the inverse function of Failed to parse (Missing texvc executable; please see math/README to configure.): y\mapsto y^{1/y}
.
Extension to negative heights
Tetration can be extended to heights that are negative. Using the relation:
- Failed to parse (Missing texvc executable; please see math/README to configure.): {}^{k}n = \log_n \left({}^{(k+1)}n\right)
(which follows from the definition of tetration), one can derive (or define) values for Failed to parse (Missing texvc executable; please see math/README to configure.): {}^{k}n
where Failed to parse (Missing texvc executable; please see math/README to configure.): k \in \{-1, 0, 1\}
.
- Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{array}{rclclcccc} {}^{1}n & = & \log_n \left({}^{2}n\right) & = & \log_{n} \left(n^n\right) & = & n \log_{n} n & = & n \\ {}^{0}n & = & \log_{n} \left({}^{1}n\right) & = & \log_{n} n & & & = & 1 \\ {}^{(-1)}n & = & \log_{n} \left({}^{0}n\right) & = & \log_{n} 1 & & & = & 0 \end{array}
This confirms the intuitive definition of Failed to parse (Missing texvc executable; please see math/README to configure.): {}^{1}n
as simply being Failed to parse (Missing texvc executable; please see math/README to configure.): n
. However, no further values can be derived by further iteration in this fashion, as Failed to parse (Missing texvc executable; please see math/README to configure.): \log_n 0
is undefined.
Similarly, since Failed to parse (Missing texvc executable; please see math/README to configure.): \log_{1} 1
is also undefined:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \log_{1} 1 = \frac{\log_n 1}{\log_n 1} = \frac{0}{0}
the derivation above does not hold when Failed to parse (Missing texvc executable; please see math/README to configure.): n
= 1. Therefore, Failed to parse (Missing texvc executable; please see math/README to configure.): {}^{(-1)}1
must remain an undefined quantity as well. (The figure Failed to parse (Missing texvc executable; please see math/README to configure.): {}^{0}1
can safely be defined as 1, however.)
Extension to real heights
Image:Real-tetration.png
Failed to parse (Missing texvc executable; please see math/README to configure.): \,{}^{x}e using linear approximation.
At this time there is no commonly accepted solution to the general problem of extending tetration to the real or complex values of Failed to parse (Missing texvc executable; please see math/README to configure.): b , although it is an active area of research. Various approaches are mentioned below. For an approach that is still disputed until it has been reviewed further, see ultra exponential function.
In general the problem is finding a super-exponential function Failed to parse (Missing texvc executable; please see math/README to configure.): \,f(x) = {}^{x}a
over real Failed to parse (Missing texvc executable; please see math/README to configure.): x > -2
that satisfies:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \,{}^{(-1)}a = 0
- Failed to parse (Missing texvc executable; please see math/README to configure.): \,{}^{0}a = 1
- Failed to parse (Missing texvc executable; please see math/README to configure.): \,{}^{b}a = a^{\left({}^{(b-1)}a\right)}
- A fourth requirement that is usually one of:
-
- A continuity requirement (usually just that Failed to parse (Missing texvc executable; please see math/README to configure.): {}^{x}a
is continuous in both variables for Failed to parse (Missing texvc executable; please see math/README to configure.): x > 0
).
-
- A differentiability requirement (can be once, twice, n times, or infinitely differentiable in x).
- A regularity requirement (implying twice differentiable in x) that:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \left( \frac{d^2}{dx^2}f(x) > 0\right)
for all Failed to parse (Missing texvc executable; please see math/README to configure.): x > 0
The fourth requirement differs from author to author, and between approaches. There are two main approaches to extending tetration to real heights, one is based on the regularity requirement, and one is based on the differentiability requirement. These two approaches seem to be so different that they may not be reconciled, as they produce results inconsistent with each other.
Fortunately, any solution that satisfies one of these in an interval of length one can be extended to a solution for all positive real numbers. When Failed to parse (Missing texvc executable; please see math/README to configure.): \,{}^{x}a
is defined for an interval of length one, the whole function easily follows for all Failed to parse (Missing texvc executable; please see math/README to configure.): x > -2
.
A linear approximation (solution to the continuity requirement, approximation to the differentiability requirement) is given by:
- Failed to parse (Missing texvc executable; please see math/README to configure.): {}^{x}a \approx \begin{cases} \log_a({}^{(x+1)}a) & x \le -1 \\ 1 + x & -1 < x \le 0 \\ a^{\left({}^{(x-1)}a\right)} & x > 0 \end{cases}
hence:
-
| Approximation |
Domain |
| Failed to parse (Missing texvc executable; please see math/README to configure.): \,{}^{x}a \approx x+1 |
for Failed to parse (Missing texvc executable; please see math/README to configure.): -1<x<0 |
| Failed to parse (Missing texvc executable; please see math/README to configure.): \,{}^{x}a \approx a^x |
for Failed to parse (Missing texvc executable; please see math/README to configure.): 0<x<1 |
| Failed to parse (Missing texvc executable; please see math/README to configure.): \,{}^{x}a \approx a^{a^{(x-1)}} |
for Failed to parse (Missing texvc executable; please see math/README to configure.): 1<x<2 |
and so on. However, it is only piecewise differentiable; at integer values of x the derivative is multiplied by Failed to parse (Missing texvc executable; please see math/README to configure.): \ln{a} .
A quadratic approximation (to the differentiability requirement) is given by:
- Failed to parse (Missing texvc executable; please see math/README to configure.): {}^{x}a \approx \begin{cases} \log_a({}^{(x+1)}a) & x \le -1 \\ 1 + \frac{2\log(a)}{1+\log(a)}x - \frac{1-\log(a)}{1+\log(a)}x^2 & -1 < x \le 0 \\ a^{\left({}^{(x-1)}a\right)} & x > 0 \end{cases}
which is differentiable for all Failed to parse (Missing texvc executable; please see math/README to configure.): x > 0 , but not twice differentiable.
Other, more complicated solutions may be smoother and/or satisfy additional properties. When defining Failed to parse (Missing texvc executable; please see math/README to configure.): {}^{x}a
for every a, another possible requirement could be that Failed to parse (Missing texvc executable; please see math/README to configure.): {}^{x}a
is monotonically increasing with a. Other solutions require not just continuity, but differentiability, or even infinite differentiability. Another approach is to define tetration over real heights as the inverse of the super-logarithm, which is its inverse function with respect to the height.
Super-exponential growth
A super-exponential function grows even faster than a double exponential function; for example, if Failed to parse (Missing texvc executable; please see math/README to configure.): a
= 10:
- Failed to parse (Missing texvc executable; please see math/README to configure.): f(-1)=0
- Failed to parse (Missing texvc executable; please see math/README to configure.): f(0)=1
- Failed to parse (Missing texvc executable; please see math/README to configure.): f(1)=10
- Failed to parse (Missing texvc executable; please see math/README to configure.): f(2)=10^{10}
- Failed to parse (Missing texvc executable; please see math/README to configure.): f(2.3)=10^{100}
(googol)
- Failed to parse (Missing texvc executable; please see math/README to configure.): \,\!f(3)=10^{10^{10}}
- Failed to parse (Missing texvc executable; please see math/README to configure.): \,\!f(3.3)=10^{10^{100}}
(googolplex)
- It passes Failed to parse (Missing texvc executable; please see math/README to configure.): \,\!10^{10^x}
at Failed to parse (Missing texvc executable; please see math/README to configure.): x = 2.376
- Failed to parse (Missing texvc executable; please see math/README to configure.): f(x) \approx 4.83 \times 10^{237}
Approaches to inverse functions
The inverse functions of tetration are called the super-root (or hyper-4-root), and the super-logarithm (or hyper-4-logarithm). The square super root Failed to parse (Missing texvc executable; please see math/README to configure.): \mathrm{ssrt}(x)
which is the inverse function of Failed to parse (Missing texvc executable; please see math/README to configure.): x^x
can be represented with the Lambert W function:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \mathrm{ssrt}(x)=e^{W(\mathrm{ln}(x))}=\frac{\mathrm{ln}(x)}{W(\mathrm{ln}(x))}
The super-logarithm Failed to parse (Missing texvc executable; please see math/README to configure.): \mathrm{slog}_a b
is defined for all positive and negative real numbers.
The function Failed to parse (Missing texvc executable; please see math/README to configure.): \mathrm{slog}_a
satisfies:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \mathrm{slog}_a a^b = 1 + \mathrm{slog}_a b
- Failed to parse (Missing texvc executable; please see math/README to configure.): \mathrm{slog}_a b = 1 + \mathrm{slog}_a \log_a b
- Failed to parse (Missing texvc executable; please see math/README to configure.): \mathrm{slog}_a b > -2
Examples:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \,\mathrm{slog}_{10} -3 = -1 + \mathrm{slog}_{10} 0.001 = -1 + -0.999 = -1.999
- Failed to parse (Missing texvc executable; please see math/README to configure.): \,\mathrm{slog}_{10} 3 = \log_{10} 3 = .477
- Failed to parse (Missing texvc executable; please see math/README to configure.): \,\mathrm{slog}_{10} 10^{6\times 10^{23}} = 1 + \mathrm{slog}_{10} 6\times 10^{23} = 2 + \mathrm{slog}_{10} 23.778 = 3 + \mathrm{slog}_{10} 1.376 = 3 + \log_{10} 1.376 = 3.139
See also
References
- Daniel Geisler, tetration.org
- I.N. Galidakis, On extending hyper4 to nonintegers (undated, 2006 or earlier) (A simpler, easier to read review of the next reference)
- I.N. Galidakis, On Extending hyper4 and Knuth's Up-arrow Notation to the Reals (undated, 2006 or earlier).
- Robert Munafo, Extension of the hyper4 function to reals (An informal discussion about extending tetration to the real numbers.)
- Lode Vandevenne, Tetration of the Square Root of Two, (2004). (Attempt to extend tetration to real numbers.)
- I.N. Galidakis, Mathematics, (Definitive list of references to tetration research. Lots of information on the Lambert W function, Riemann surfaces, and analytic continuation.)
- Galidakis, Ioannis and Weisstein, Eric W. Power Tower
- Joseph MacDonell, Some Critical Points of the Hyperpower Function.
- Dave L. Renfro, Web pages for infinitely iterated exponentials (Compilation of entries from questions about tetration on sci.math.)
- Andrew Robbins, Home of Tetration (An infinitely differentiable extension of tetration to real numbers.)
- R. Knobel. "Exponentials Reiterated." American Mathematical Monthly 88, (1981), p. 235-252.
- Hans Maurer. "Über die Funktion Failed to parse (Missing texvc executable; please see math/README to configure.): y=x^{[x^{[x(\cdots)]}]}
für ganzzahliges Argument (Abundanzen)." Mittheilungen der Mathematische Gesellschaft in Hamburg 4, (1901), p. 33-50. (Reference to usage of Failed to parse (Missing texvc executable; please see math/README to configure.): \ ^ba
from Knobel's paper.)
External links
fr:tetration hu:Tetráció
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