Expx2

Expx2 is the mathematical shorthand for the exponential of the square of the argument, defined by f(x) = e^{x^2}. It is distinct from the Gaussian e^{-x^2}, though it shares the same quadratic exponent form and appears in related contexts.

Analytical properties and basic facts: Expx2 is an entire function, meaning it is analytic everywhere on the complex plane. It is even, since f(-x) = f(x). The first derivative is f'(x) = 2x e^{x^2}, and higher derivatives follow from the product rule. The function has no zeros on the real line. Its power series expansion around x = 0 is e^{x^2} = ∑_{n=0}^∞ x^{2n} / n!. As x grows, exp(x^2) increases faster than any exponential e^{a x} with fixed a, leading to rapid growth and potential overflow in finite-precision arithmetic. The function can be related to the imaginary error function erfi via ∫ e^{t^2} dt = (√π/2) erfi(t), and erfi'(x) = (2/√π) e^{x^2}.

Context and applications: Expx2 arises in various mathematical and physical settings where expressions involve e^{x^2}. It is often encountered in differential equations, power-series methods, and theoretical analyses alongside the more common Gaussian e^{-x^2}. In integrals and probability theory, erfi and related functions capture the antiderivative behavior of exp(x^2).

Computational notes: In numerical work, exp(x^2) can overflow for sufficiently large |x| in standard floating-point formats (roughly |x| above sqrt(709) in double precision). Practical approaches include performing calculations in logarithmic form, using erfi for integrals, or employing scaled representations to mitigate overflow. Some software libraries provide explicit expx2 or e^(x^2) functionality, while others rely on generic exp and power functions.