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Double exponential function

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A double exponential function is a <a href="Constant_(mathematics)" title="Constant (mathematics)">constant</a> raised to the power of an <a href="Exponentiation" title="Exponentiation">exponential function</a>. The general formula is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=a^{b^{x}}=a^{(b^{x})}}">

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<annotation encoding="application/x-tex">{\displaystyle f(x)=a^{b^{x}}=a^{(b^{x})}}</annotation>

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</math><img src="./6d686647ff4b19b1669aa6bb6d97fa552cb258dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.084ex; height:3.343ex;" alt="{\displaystyle f(x)=a^{b^{x}}=a^{(b^{x})}}" loading="lazy"> (where a>1 and b>1), which grows much more quickly than an exponential function. For example, if a = b = 10:

<li>f(2) = 10100 = <a href="Googol" title="Googol">googol</a></li>

<li>f(100) = 1010100 = <a href="Googolplex" title="Googolplex">googolplex</a>.</li></ul>

<a href="Factorial" title="Factorial">Factorials</a> grow faster than exponential functions, but much more slowly than double exponential functions. However, <a href="Tetration" title="Tetration">tetration</a> and the <a href="Ackermann_function" title="Ackermann function">Ackermann function</a> grow faster. See <a href="Big_O_notation" title="Big O notation">Big O notation</a> for a comparison of the rate of growth of various functions.

The inverse of the double exponential function is the <a href="Logarithm#double_logarithm" title="Logarithm">double logarithm</a> log(log(x)). The complex double exponential function is <a href="Entire_function" title="Entire function">entire</a>, because it is the composition of two entire functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=a^{x}=e^{x\ln a}}">

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<annotation encoding="application/x-tex">{\displaystyle f(x)=a^{x}=e^{x\ln a}}</annotation>

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</math><img src="./6b5b0907c4a3cd0ce4fd344c70d8c7088b7177cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.288ex; height:3.176ex;" alt="{\displaystyle f(x)=a^{x}=e^{x\ln a}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)=b^{x}=e^{x\ln b}}">

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<annotation encoding="application/x-tex">{\displaystyle g(x)=b^{x}=e^{x\ln b}}</annotation>

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</math><img src="./3fa891946e2ec295e861c876d9cf50223c6ff1c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.729ex; height:3.176ex;" alt="{\displaystyle g(x)=b^{x}=e^{x\ln b}}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Double_exponential_sequences">Double exponential sequences</h2></div>

A sequence of positive integers (or real numbers) is said to have double exponential rate of growth if the function giving the nth term of the sequence is bounded above and below by double exponential functions of n.

Examples include

<ul><li>The <a href="Fermat_number" title="Fermat number">Fermat numbers</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(m)=2^{2^{m}}+1}">

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<annotation encoding="application/x-tex">{\displaystyle F(m)=2^{2^{m}}+1}</annotation>

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<li>The harmonic primes: The primes p, in which the sequence 1/2 + 1/3 + 1/5 + 1/7 + ⋯ + 1/p exceeds 0, 1, 2, 3, …<div class="paragraphbreak" style="margin-top:0.5em"></div>The first few numbers, starting with 0, are 2, 5, 277, 5195977, ... (sequence <a href="https://oeis.org/A016088" class="extiw external" title="oeis:A016088">A016088</a> in the <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>)</li>

<li>The <a href="Double_Mersenne_number" title="Double Mersenne number">Double Mersenne numbers</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle MM(p)=2^{2^{p}-1}-1}">

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<annotation encoding="application/x-tex">{\displaystyle MM(p)=2^{2^{p}-1}-1}</annotation>

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<li>The elements of <a href="Sylvester's_sequence" title="Sylvester's sequence">Sylvester's sequence</a> (sequence <a href="https://oeis.org/A000058" class="extiw external" title="oeis:A000058">A000058</a> in the <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>) <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{n}=\left\lfloor E^{2^{n+1}}+{\frac {1}{2}}\right\rfloor }">

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<annotation encoding="application/x-tex">{\displaystyle s_{n}=\left\lfloor E^{2^{n+1}}+{\frac {1}{2}}\right\rfloor }</annotation>

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</math> where E ≈ 1.264084735305302 is Vardi's constant (sequence <a href="https://oeis.org/A076393" class="extiw external" title="oeis:A076393">A076393</a> in the <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>).</li>

<li>The number of <a href="Arity" title="Arity">k-ary</a> <a href="Boolean_function" title="Boolean function">Boolean functions</a>: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{2^{k}}}">

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<annotation encoding="application/x-tex">{\displaystyle 2^{2^{k}}}</annotation>

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<li>The prime numbers 2, 11, 1361, ... (sequence <a href="https://oeis.org/A051254" class="extiw external" title="oeis:A051254">A051254</a> in the <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>) <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a(n)=\left\lfloor A^{3^{n}}\right\rfloor }">

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<annotation encoding="application/x-tex">{\displaystyle a(n)=\left\lfloor A^{3^{n}}\right\rfloor }</annotation>

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</math> where A ≈ 1.306377883863 is <a href="Mills'_constant" title="Mills' constant">Mills' constant</a>.</li></ul>

<a href="Alfred_Aho" title="Alfred Aho">Aho</a> and <a href="Neil_Sloane" title="Neil Sloane">Sloane</a> observed that in several important <a href="Integer_sequence" title="Integer sequence">integer sequences</a>, 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.<a href="#cite_note-1">[1]</a>

Ionaşcu and Stănică describe some more general sufficient conditions for a sequence to be the floor of a double exponential sequence plus a constant.<a href="#cite_note-2">[2]</a>

<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Algorithmic_complexity">Algorithmic complexity</h3></div>

In <a href="Computational_complexity_theory" title="Computational complexity theory">computational complexity theory</a>, <a href="2-EXPTIME" title="2-EXPTIME">2-EXPTIME</a> is the class of decision problems solvable in double exponential time. It is equivalent to AEXPSPACE, the set of decision problems solvable by an <a href="Alternating_Turing_machine" title="Alternating Turing machine">alternating Turing machine</a> in exponential space, and is a superset of <a href="EXPSPACE" title="EXPSPACE">EXPSPACE</a>.<a href="#cite_note-3">[3]</a> An example of a problem in 2-EXPTIME that is not in EXPTIME is the problem of proving or disproving statements in <a href="Presburger_arithmetic" title="Presburger arithmetic">Presburger arithmetic</a>.<a href="#cite_note-4">[4]</a>

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 <a href="Chan's_algorithm" title="Chan's algorithm">Chan's algorithm</a> for computing <a href="Convex_hull" title="Convex hull">convex hulls</a>, which performs a sequence of computations using test values hi = 22i (estimates for the eventual output size), taking time O(n log hi) 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.<a href="#cite_note-5">[5]</a>

<div class="mw-heading mw-heading3"><h3 id="Number_theory">Number theory</h3></div>

Some <a href="Number_theory" title="Number theory">number theoretical</a> bounds are double exponential. <a href="Odd_perfect_number" class="mw-redirect" title="Odd perfect number">Odd perfect numbers</a> with n distinct prime factors are known to be at most <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{4^{n}}}">

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<annotation encoding="application/x-tex">{\displaystyle 2^{4^{n}}}</annotation>

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</math><img src="./69f0fa63c02614549b40f2bc329f4880fad12904.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.182ex; height:2.676ex;" alt="{\displaystyle 2^{4^{n}}}" loading="lazy">, a result of Nielsen (2003).<a href="#cite_note-6">[6]</a>

The maximal volume of a <a href="Polytope" title="Polytope">polytope</a> in a d-dimensional <a href="Integer_lattice" title="Integer lattice">integer lattice</a> with k ≥ 1 <a href="Integer_points_in_convex_polyhedra" title="Integer points in convex polyhedra">interior lattice points</a> is at most

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<annotation encoding="application/x-tex">{\displaystyle k\cdot (8d)^{d}\cdot 15^{d\cdot 2^{2d+1}},}</annotation>

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</math><img src="./51fcadf3346461223fec0229c790f763c89d433b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.427ex; height:3.509ex;" alt="{\displaystyle k\cdot (8d)^{d}\cdot 15^{d\cdot 2^{2d+1}},}" loading="lazy"></dd></dl>

a result of Pikhurko (2001).<a href="#cite_note-7">[7]</a>

The <a href="Largest_known_prime_number" title="Largest known prime number">largest known prime number</a> in the electronic era has grown roughly as a double exponential function of the year since <a href="J._C._P._Miller" title="J. C. P. Miller">Miller</a> and <a href="David_Wheeler_(computer_scientist)" title="David Wheeler (computer scientist)">Wheeler</a> found a 79-digit prime on <a href="EDSAC" title="EDSAC">EDSAC</a>1 in 1951.<a href="#cite_note-8">[8]</a>

<div class="mw-heading mw-heading3"><h3 id="Theoretical_biology">Theoretical biology</h3></div>

In <a href="Population_dynamics" title="Population dynamics">population dynamics</a> the growth of human population is sometimes supposed to be double exponential. Varfolomeyev and Gurevich<a href="#cite_note-9">[9]</a> experimentally fit

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<annotation encoding="application/x-tex">{\displaystyle N(y)=375.6\cdot 1.00185^{1.00737^{y-1000}}\,}</annotation>

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</math><img src="./eafde06aaa161da082dbb58935ff71543ec4b369.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.268ex; height:3.509ex;" alt="{\displaystyle N(y)=375.6\cdot 1.00185^{1.00737^{y-1000}}\,}" loading="lazy"></dd></dl>

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

<div class="mw-heading mw-heading3"><h3 id="Physics">Physics</h3></div>

In the <a href="Toda_oscillator" title="Toda oscillator">Toda oscillator</a> model of <a href="Self-pulsation" title="Self-pulsation">self-pulsation</a>, the logarithm of amplitude varies exponentially with time (for large amplitudes), thus the amplitude varies as double exponential function of time.<a href="#cite_note-kouz-10">[10]</a>

Dendritic <a href="Macromolecule" title="Macromolecule">macromolecules</a> have been observed to grow in a doubly-exponential fashion.<a href="#cite_note-11">[11]</a>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFAhoSloane1973" class="citation cs2"><a href="Alfred_Aho" title="Alfred Aho">Aho, A. V.</a>; <a href="N._J._A._Sloane" class="mw-redirect" title="N. J. A. Sloane">Sloane, N. J. A.</a> (1973), <a rel="nofollow" class="external text" href="http://neilsloane.com/doc/doubly.html">"Some doubly exponential sequences"</a>, <a href="Fibonacci_Quarterly" title="Fibonacci Quarterly">Fibonacci Quarterly</a>, 11: 429–437</cite>.

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<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFIonaşcuStănică2004" class="citation cs2">Ionaşcu, Eugen-Julien; Stănică, Pantelimon (2004), <a rel="nofollow" class="external text" href="http://faculty.nps.edu/pstanica/research/AMUC04.pdf">"Effective asymptotics for some nonlinear recurrences and almost doubly-exponential sequences"</a> (PDF), Acta Mathematica Universitatis Comenianae, LXXIII (1): 75–87</cite>.

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<li id="cite_note-3"><a href="#cite_ref-3">^</a> <a href="Christos_Papadimitriou" title="Christos Papadimitriou">Christos Papadimitriou</a>, Computational Complexity (1994), <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-201-53082-7</bdi>. Section 20.1, corollary 3, page 495.

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<li id="cite_note-4"><a href="#cite_ref-4">^</a> <a href="Michael_J._Fischer" title="Michael J. Fischer">Fischer, M. J.</a>, and <a href="Michael_O._Rabin" title="Michael O. Rabin">Michael O. Rabin</a>, 1974, "<a rel="nofollow" class="external text" href="http://www.lcs.mit.edu/publications/pubs/ps/MIT-LCS-TM-043.ps">"Super-Exponential Complexity of Presburger Arithmetic.</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20060915010325/http://www.lcs.mit.edu/publications/pubs/ps/MIT-LCS-TM-043.ps">Archived</a> 2006-09-15 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a>" Proceedings of the SIAM-AMS Symposium in Applied Mathematics Vol. 7: 27–41

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<li id="cite_note-5"><a href="#cite_ref-5">^</a> <cite id="CITEREFChan1996" class="citation cs2"><a href="Timothy_M._Chan" title="Timothy M. Chan">Chan, T. M.</a> (1996), "Optimal output-sensitive convex hull algorithms in two and three dimensions", <a href="Discrete_and_Computational_Geometry" class="mw-redirect" title="Discrete and Computational Geometry">Discrete and Computational Geometry</a>, 16 (4): 361–368, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02712873">10.1007/BF02712873</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1414961">1414961</a></cite>

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> <cite id="CITEREFNielsen2003" class="citation cs2">Nielsen, Pace P. (2003), <a rel="nofollow" class="external text" href="http://www.integers-ejcnt.org/vol3.html">"An upper bound for odd perfect numbers"</a>, INTEGERS: The Electronic Journal of Combinatorial Number Theory, 3: A14</cite>.

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> <cite id="CITEREFPikhurko2001" class="citation cs2">Pikhurko, Oleg (2001), "Lattice points in lattice polytopes", <a href="Mathematika" title="Mathematika">Mathematika</a>, 48 (1–2): 15–24, <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0008028">math/0008028</a>, <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2000math......8028P">2000math......8028P</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1112%2Fs0025579300014339">10.1112/s0025579300014339</a></cite>

</li>

<li id="cite_note-8"><a href="#cite_ref-8">^</a> <cite id="CITEREFMillerWheeler1951" class="citation cs2">Miller, J. C. P.; Wheeler, D. J. (1951), "Large prime numbers", <a href="Nature_(journal)" title="Nature (journal)">Nature</a>, 168 (4280): 838, <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1951Natur.168..838M">1951Natur.168..838M</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2F168838b0">10.1038/168838b0</a></cite>.

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<li id="cite_note-9"><a href="#cite_ref-9">^</a> <cite id="CITEREFVarfolomeyevGurevich2001" class="citation cs2">Varfolomeyev, S. D.; Gurevich, K. G. (2001), "The hyperexponential growth of the human population on a macrohistorical scale", Journal of Theoretical Biology, 212 (3): 367–372, <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2001JThBi.212..367V">2001JThBi.212..367V</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1006%2Fjtbi.2001.2384">10.1006/jtbi.2001.2384</a>, <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/11829357">11829357</a></cite>.

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<li id="cite_note-kouz-10"><a href="#cite_ref-kouz_10-0">^</a> <cite id="CITEREFKouznetsovBissonLiUeda2007" class="citation cs2">Kouznetsov, D.; Bisson, J.-F.; Li, J.; Ueda, K. (2007), <a rel="nofollow" class="external text" href="http://www.iop.org/EJ/abstract/-search=15823442.1/1751-8121/40/9/016">"Self-pulsing laser as oscillator Toda: Approximation through elementary functions"</a>, <a href="Journal_of_Physics_A" title="Journal of Physics A">Journal of Physics A</a>, 40 (9): 1–18, <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2007JPhA...40.2107K">2007JPhA...40.2107K</a>, <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.535.5379">10.1.1.535.5379</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F1751-8113%2F40%2F9%2F016">10.1088/1751-8113/40/9/016</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:53330023">53330023</a></cite>.

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<li id="cite_note-11"><a href="#cite_ref-11">^</a> <cite id="CITEREFKawaguchiWalkerWilkinsMoore1995" class="citation journal cs1">Kawaguchi, Tohru; Walker, Kathleen L.; Wilkins, Charles L.; Moore, Jeffrey S. (1995). "Double Exponential Dendrimer Growth". <a href="Journal_of_the_American_Chemical_Society" title="Journal of the American Chemical Society">Journal of the American Chemical Society</a>. 117 (8): 2159–2165. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1021%2Fja00113a005">10.1021/ja00113a005</a>.</cite>

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