What is the integral over the reals of eˆ(-xˆ2)?
The integral you're asking about is known as the Gaussian integral, and it has a well-known solution in terms of the error function, which is denoted as erf(x). The Gaussian integral over the real numbers is given by:

∫e^(-x^2) dx = (1/2) √(π) erf(x) + C

where C is an arbitrary constant of integration. The error function is a special function that arises in a variety of problems in mathematics, physics, and engineering. It is defined as:

erf(x) = (2/√(π)) ∫_0^x e^(-t^2) dt

So, the Gaussian integral can also be expressed in terms of the error function.