DOUBLE EXPONENTIAL FUNCTION

:''This article is about double exponential functions. For the double exponential distribution, see Laplace distribution.''
A 'double exponential' function is a constant raised to the power of an exponential function. The general formula is $f(x)\; =\; a^\{b^x\}$, which grows even faster than an exponential function. For example, if ''a'' = ''b'' = 10:

★ ''f''(−1) ≈ 1.26

★ ''f''(0) = 10

★ ''f''(1) = 10¹⁰

★ ''f''(2) = 10¹⁰⁰ = googol

★ ''f''(3) = 10¹⁰⁰⁰

★ ''f''(100) = 10^{10^100} = googolplex
Factorials grow faster than exponential functions, but much slower than double-exponential functions. Compare the hyper-exponential function, which grows even faster.

Contents

Doubly exponential sequences

Applications

See also

References

Doubly exponential sequences

The ''n''th value of each of the following integer sequences is proportional to a double exponential function of ''n''. The sequences themselves may not be double exponential functions, but they can be defined using formulas in which such functions appear.

★ Fermat numbers
:$F(m)\; =\; 2^\{2^m\}+1$

★ Double Mersenne numbers
:$MM(p)\; =\; 2^\{2^p-1\}-1$

★ Elements of Sylvester's sequence
::
ight
floor
:where ''E'' ≈ 1.26408 is Vardi's constant .
Aho and Sloane (1973) investigate several other sequences that, like Sylvester's sequence, can be formed by rounding to the nearest integer the values of a doubly exponential function, and provide general conditions on the values of a sequence that cause it to be of this type.

Applications

In computational complexity theory, some algorithms take double-exponential time:

★ Decision procedures for Presburger arithmetic (in the worst case)

★ Finding a complete set of associative-commutative unifiers [1]

★ Satisfying CTL⁺ (which is, in fact, 2-EXPTIME-complete) [2]
Likewise, some number theoretical upper bounds are double exponential. Odd perfect numbers with ''n'' distinct prime factors are known to be at most
:$2^\{4^n\}$
a result of Nielsen (2003).[3]

See also

★ Ackermann function

★ Big O notation

References

★ .