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A double exponential function is a constant raised to the power of an exponential function. The general formula is * f(−1) ≈ 1.26 Factorials grow faster than exponential functions, but much slower than double-exponential functions. The hyper-exponential function and Ackermann function grow even faster. See Big O notation for a comparison of the rate of growth of various functions. The inverse of a double exponential function is a double logarithm.
Aho and Sloane observed that in several important integer sequences, each term is a constant plus the square of the previous term, and show that such sequences can be formed by rounding to the nearest integer the values of a doubly exponential function in which the middle exponent is two.[1] Integer sequences with this squaring behavior include * The Fermat numbers
* 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,.... The first few numbers, starting with 0, are 2,5,277,5195977,... (sequence A016088 in OEIS) * The Double Mersenne numbers
* The elements of Sylvester's sequence (sequence A000058 in OEIS)
where E ≈ 1.264084735305 is Vardi's constant. * The number of k-ary operators:
More generally, if the nth value of an integer sequences is proportional to a double exponential function of n, Ionascu and Stanica call the sequence "almost doubly-exponential" and describe conditions under which it can be defined as the floor of a doubly-exponential sequence plus a constant.[2] Additional sequences of this type include * The prime numbers 2, 11, 1361, ... (sequence A051254 in OEIS)
where A ≈ 1.306377883863 is Mills' constant.
Algorithmic complexity In computational complexity theory, some algorithms take double-exponential time: * Decision procedures for Presburger arithmetic
Some number theoretical bounds are double exponential. Odd perfect numbers with n distinct prime factors are known to be at most
a result of Nielsen (2003).[5] The largest known prime number in the electronic era has grown roughly as a double exponential function of the year since Miller and Wheeler found a 79-digit prime on EDSAC1 in 1951.[6] Theoretical biology In population dynamics the growth of human population is sometimes supposed to be double exponential. Gurevich and Varfolomeyev[7] experimentally fit
where N(y) is the population in year y in millions. Physics In the oscillator Toda 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.[8] References 1. ^ Aho, A. V.; Sloane, N. J. A. (1973), "Some doubly exponential sequences", Fibonacci Quarterly 11: 429–437, http://www.research.att.com/~njas/doc/doubly.html . Retrieved from "http://en.wikipedia.org/"
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, which grows even faster than an exponential function. For example, if a = b = 10:
