expXk

Expxk is a name used in some mathematical discussions to denote a family of functions obtained by taking the exponential of a real power. In its canonical form, expxk refers to f_k(x) = exp(x^k), where exp denotes the natural exponential e^y and k is a positive parameter. When k is a positive integer, this definition yields a real-valued function for every real x; for other real values of k, x^k may require domain considerations.

Basic properties and behavior. For k > 0 and integer, the function f_k is differentiable on the real line with derivative f'_k(x) = k x^{k-1} exp(x^k). The sign of the derivative depends on x^{k-1}; if k is odd, k−1 is even and f_k is nondecreasing for all x, with a stationary point at x = 0. If k is even, f_k is not monotone on the entire real line, since the derivative changes sign around x = 0. As x grows in magnitude, exp(x^k) exhibits super-exponential growth: log f_k(x) = x^k, so large x lead to extremely rapid increases. Near x = 0, f_k(x) expands as 1 + x^k + O(x^{2k}).

Special cases and variants. The case k = 1 gives the standard exponential function e^x. Other common instances include f_2(x) = e^{x^2}, f_3(x) = e^{x^3}, and so on. Variants sometimes studied in theory replace the exponent with -x^k or use absolute values, for example g_k(x) = exp(-|x|^k), which yield different symmetry and tail behavior.

Applications and context. Expxk serves mainly as a simple example of a non-polynomial, rapidly growing function in analysis and approximation theory. It can be used to illustrate growth comparisons, to construct test cases in numerical methods, or as a hypothetical activation-like function in theoretical discussions. It is not a standard named function in most reference works, but it provides a convenient, well-defined family for exploring exponential growth patterns.

See also: exponential function, real power, activation function (in neural networks), tail behavior in probability.