A double exponential function is a constant raised to the power of an exponential function. The general formula is , which grows even faster than an exponential function. For example, if a = b = 10:
* 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. 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. Additional sequences of this type include
* The prime numbers 2, 11, 1361, ... (sequence A051254 in OEIS)
where A ≈ 1.306377883863 is Mills' constant.
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).
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.
In population dynamics the growth of human population is sometimes supposed to be double exponential. Gurevich and Varfolomeyev experimentally fit
where N(y) is the population in year y in millions.
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.
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 .