ExpEx

ExpEx is a mathematical construct used to describe a hierarchical extension of the exponential function. In this framework, expEx is defined as a family of operators Exp_d for integers d ≥ 1, where d denotes nesting depth. For a real input x, Exp_1(x) equals e^x, and Exp_{d+1}(x) equals e^{ Exp_d(x) }. For a function f, Exp_d(f)(x) is obtained by applying Exp_d pointwise: Exp_d(f)(x) = Exp_d( f(x) ). This formalism treats expEx as a notational device to study nested exponentials with controlled height.

When the input is a real constant a, Exp_d(a) yields a finite-height exponential tower of depth d with base e, namely e^{e^{...^{a}}}, which is often described in relation to tetration. The construct is useful for discussing qualitative properties of ultra-rapid growth, as well as for defining approximations of infinite-height exponentials by truncation.

Key properties include monotonicity under reasonable regularity assumptions, and the fact that height-raising operations rapidly amplify small changes in the inner argument due to the chain rule. The derivative of Exp_d(f) with respect to x involves repeated application of the chain rule, leading to a product of the inner derivatives and the corresponding exponentials.

ExpEx sits primarily in theoretical discussions and is seldom encountered as a standard function in basic analysis. It provides a compact language for describing nested exponentials and their qualitative behavior, and it connects to related notions such as tetration and hyper-operations.