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{{short description|Exponential function of an exponential function}}
[[Image:Double Exponential Function.svg|right|thumb|320px|A double exponential function (red curve) compared to a single exponential function (blue curve).]]

A '''double exponential''' function is a [[Constant (mathematics)|constant]] raised to the power of an [[Exponentiation|exponential function]]. The general formula is <math>f(x) = a^{b^x}=a^{(b^x)}</math> (where ''a''>1 and ''b''>1), which grows much more quickly than an exponential function. For example, if ''a'' = ''b'' = 10:
*''f''(x) = 1010x
*''f''(0) = 10
*''f''(1) = 1010
*''f''(2) = 10100 = [[googol]]
*''f''(3) = 101000
*''f''(100) = 1010100 = [[googolplex]].

[[Factorial]]s grow faster than exponential functions, but much more slowly than double exponential functions. However, [[tetration]] and the [[Ackermann function]] grow faster. See [[Big O notation]] for a comparison of the rate of growth of various functions.

The inverse of the double exponential function is the [[Logarithm#double logarithm|double logarithm]] log(log(''x'')). The complex double exponential function is [[Entire function|entire]], because it is the composition of two entire functions <math>f(x)=a^x=e^{x\ln a}</math> and <math>g(x)=b^x=e^{x\ln b}</math>.

==Double exponential sequences==

A sequence of positive integers (or real numbers) is said to have ''double exponential rate of growth'' if the function giving the {{mvar|n}}th term of the sequence is bounded above and below by double exponential functions of {{mvar|n}}.
Examples include
* The [[Fermat number]]s <math display="block">F(m) = 2^{2^m}+1</math>
* The harmonic primes: The primes ''p'', in which the sequence {{nowrap|1/2 + 1/3 + 1/5 + 1/7 + ⋯ + 1/''p''}} exceeds 0, 1, 2, 3, …{{pb}}The first few numbers, starting with 0, are 2, 5, 277, 5195977, ... {{OEIS|A016088}}
* The [[Double Mersenne number]]s <math display="block">MM(p) = 2^{2^p-1}-1</math>
* The elements of [[Sylvester's sequence]] {{OEIS|A000058}} <math display="block">s_n = \left\lfloor E^{2^{n+1}}+\frac12 \right\rfloor</math> where ''E'' ≈ 1.264084735305302 is [[Vardi's constant]] {{OEIS|A076393}}.
* The number of [[Arity|''k''-ary]] [[Boolean function]]s: <math display="block">2^{2^k}</math>
* The prime numbers 2, 11, 1361, ... {{OEIS|A051254}} <math display="block">a(n) = \left\lfloor A^{3^n}\right\rfloor</math> where ''A'' ≈ 1.306377883863 is [[Mills' constant]].

[[Alfred Aho|Aho]] and [[Neil Sloane|Sloane]] observed that in several important [[integer sequence]]s, each term is a constant plus the square of the previous term. They show that such sequences can be formed by rounding to the nearest integer the values of a double exponential function with middle exponent 2.<ref>{{citation|first1=A. V.|last1=Aho|author-link1=Alfred Aho|first2=N. J. A.|last2=Sloane|author-link2=N. J. A. Sloane|url=http://neilsloane.com/doc/doubly.html|title=Some doubly exponential sequences|journal=[[Fibonacci Quarterly]]|volume=11|year=1973|pages=429–437}}.</ref>
Ionaşcu and Stănică describe some more general sufficient conditions for a sequence to be the floor of a double exponential sequence plus a constant.<ref>{{citation
| last1 = Ionaşcu | first1 = Eugen-Julien
| last2 = Stănică | first2 = Pantelimon
| issue = 1
| journal = Acta Mathematica Universitatis Comenianae
| pages = 75–87
| title = Effective asymptotics for some nonlinear recurrences and almost doubly-exponential sequences
| volume = LXXIII
| url = http://faculty.nps.edu/pstanica/research/AMUC04.pdf
| year = 2004}}.</ref>

==Applications==
===Algorithmic complexity===
In [[computational complexity theory]], [[2-EXPTIME]] is the class of decision problems solvable in double exponential time. It is equivalent to AEXPSPACE, the set of decision problems solvable by an [[alternating Turing machine]] in exponential space, and is a superset of [[EXPSPACE]].<ref>[[Christos Papadimitriou]], Computational Complexity (1994), {{isbn|978-0-201-53082-7}}. Section 20.1, corollary 3, page 495.</ref> An example of a problem in 2-EXPTIME that is not in EXPTIME is the problem of proving or disproving statements in [[Presburger arithmetic]].<ref>[[Michael J. Fischer|Fischer, M. J.]], and [[Michael O. Rabin]], 1974, "[http://www.lcs.mit.edu/publications/pubs/ps/MIT-LCS-TM-043.ps "Super-Exponential Complexity of Presburger Arithmetic.] {{Webarchive|url=https://web.archive.org/web/20060915010325/http://www.lcs.mit.edu/publications/pubs/ps/MIT-LCS-TM-043.ps |date=2006-09-15 }}" ''Proceedings of the SIAM-AMS Symposium in Applied Mathematics Vol. 7'': 27–41</ref>

In some other problems in the design and analysis of algorithms, double exponential sequences are used within the design of an algorithm rather than in its analysis. An example is [[Chan's algorithm]] for computing [[convex hull]]s, which performs a sequence of computations using test values ''h''''i'' = 22''i'' (estimates for the eventual output size), taking time O(''n'' log ''h''''i'') for each test value in the sequence. Because of the double exponential growth of these test values, the time for each computation in the sequence grows singly exponentially as a function of ''i'', and the total time is dominated by the time for the final step of the sequence. Thus, the overall time for the algorithm is O(''n'' log ''h'') where ''h'' is the actual output size.<ref>{{citation
| last = Chan | first = T. M. | author-link = Timothy M. Chan
| doi = 10.1007/BF02712873
| issue = 4
| journal = [[Discrete and Computational Geometry]]
| mr = 1414961
| pages = 361–368
| title = Optimal output-sensitive convex hull algorithms in two and three dimensions
| volume = 16
| year = 1996| doi-access = free
}}</ref>

===Number theory===
Some [[number theory|number theoretical]] bounds are double exponential. [[Odd perfect number]]s with ''n'' distinct prime factors are known to be at most <math>2^{4^n}</math>, a result of Nielsen (2003).<ref>{{citation
| last = Nielsen | first = Pace P.
| journal = INTEGERS: The Electronic Journal of Combinatorial Number Theory
| pages = A14
| title = An upper bound for odd perfect numbers
| url = http://www.integers-ejcnt.org/vol3.html
| volume = 3
| year = 2003}}.</ref>

The maximal volume of a [[polytope]] in a ''d''-dimensional [[integer lattice]] with ''k'' ≥ 1 [[Integer points in convex polyhedra|interior lattice points]] is at most
:<math>k\cdot(8d)^d\cdot15^{d\cdot2^{2d+1}},</math>
a result of Pikhurko (2001).<ref>{{citation|last=Pikhurko
| first=Oleg
| title=Lattice points in lattice polytopes
| journal=[[Mathematika]]
| volume=48
| year=2001
| issue=1–2
| pages=15–24
| arxiv=math/0008028
|bibcode = 2000math......8028P
| doi=10.1112/s0025579300014339}}</ref>

The [[largest known prime number]] in the electronic era has grown roughly as a double exponential function of the year since [[J. C. P. Miller|Miller]] and [[David Wheeler (computer scientist)|Wheeler]] found a 79-digit prime on [[EDSAC]]1 in 1951.<ref>{{citation
| last1 = Miller | first1 = J. C. P.
| last2 = Wheeler | first2 = D. J.
| doi = 10.1038/168838b0
| journal = [[Nature (journal)|Nature]]
|page=838
| title = Large prime numbers
| volume = 168
| year = 1951
| issue=4280|bibcode = 1951Natur.168..838M | doi-access = free
}}.</ref>

===Theoretical biology===

In [[population dynamics]] the growth of human population is sometimes supposed to be double exponential. Varfolomeyev and Gurevich<ref>{{citation
| last1 = Varfolomeyev | first1 = S. D.
| last2 = Gurevich | first2 = K. G.
| doi = 10.1006/jtbi.2001.2384
| issue = 3
| journal = Journal of Theoretical Biology
| pages = 367–372
| title = The hyperexponential growth of the human population on a macrohistorical scale
| volume = 212
| year = 2001
| pmid = 11829357| bibcode = 2001JThBi.212..367V
}}.</ref> experimentally fit

:<math> N(y)=375.6\cdot 1.00185^{1.00737^{y-1000}} \,</math>

where ''N''(''y'') is the population in millions in year ''y''.

===Physics===
In the [[Toda oscillator]] model of [[self-pulsation]], the logarithm of amplitude varies exponentially with time (for large amplitudes), thus the amplitude varies as double exponential function of time.<ref name="kouz">{{citation
| last1 = Kouznetsov | first1 = D.
| last2 = Bisson | first2 = J.-F.
| last3 = Li | first3 = J.
| last4 = Ueda | first4 = K.
| doi = 10.1088/1751-8113/40/9/016
| journal = [[Journal of Physics A]]
| pages = 1–18
| title = Self-pulsing laser as oscillator Toda: Approximation through elementary functions
| url = http://www.iop.org/EJ/abstract/-search=15823442.1/1751-8121/40/9/016
| volume = 40
| year = 2007
| issue = 9 | bibcode=2007JPhA...40.2107K| s2cid = 53330023
| citeseerx = 10.1.1.535.5379
}}.</ref>

Dendritic [[macromolecule]]s have been observed to grow in a doubly-exponential fashion.<ref>{{cite journal|title=Double Exponential Dendrimer Growth |first1=Tohru |last1=Kawaguchi |first2=Kathleen L. |last2=Walker |first3=Charles L. |last3=Wilkins |first4=Jeffrey S. |last4=Moore |journal=[[Journal of the American Chemical Society]] |year=1995 |volume=117 |number=8 |pages=2159–2165 |doi=10.1021/ja00113a005}}</ref>

==References==
{{reflist|2}}

[[Category:Elementary special functions]]
[[Category:Exponentials]]

Double exponential function - Revision history

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