For all $a>0$ we have $${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-a{x}^2}=\sqrt{\frac{\pi }{a}}$$. Let's ignore the condition on $a$ and say $a$ is $-1$, then we have $${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{x^2}=\sqrt{\pi }i$$. The left side obviously diverges and is never even slightly imaginary, but the right side is finite and purely imaginary. What is the connection between this integral and it's "value" (I'm pretty sure there is one, like how $1+2+4+\cdots =-1$ makes sense over the 2-adic numbers)? Under what weird interpretation of integration is this correct? How can we find more of these weird integrals and their value?

#math #maths #mathematics #extendingMath #integration #analyticContinuation?