expexpx

Expexpx is a term used in some mathematical discussions to denote the double exponential function, defined as expexpx(x) = exp(exp(x)) = e^{e^x}. It is not a standard mathematical function name, but the notation emphasizes the composition of the exponential function with itself. In many texts the operation is simply described as the exponential of the exponential.

Basic properties: expexpx is defined for all real x, is strictly increasing, and is smooth. The derivative is expexpx'(x) = exp(x) · exp(exp(x)) = e^x · e^{e^x}. The second derivative is more complex but positive for all x, indicating convexity. For example, expexpx(0) = e ≈ 2.718, and expexpx(1) = e^e ≈ 15.154.

Behavior and range: as x → -∞, expexpx(x) → 1; as x → ∞, expexpx grows faster than any fixed exponential or polynomial. The function's range is (1, ∞). It serves as a canonical example of double exponential growth in theoretical discussions.

Context and usage: expexpx is used mainly in theoretical contexts to illustrate extreme growth and in discussions of algorithms or models that exhibit double exponential behavior. It is also related to tetration as a simple instance of iterated exponentials, though it is not a standard alternative to that broader concept.

See also: exponential function, double exponential, tetration.