I have noticed that there are two main relationships between the mathematical constants $\pi$ and $e$:
- The Gaussian integral, $\int_{-\infty}^{+\infty} e^{-x^2} dx = \sqrt{\pi}$
- Euler's identity, $e^{i\pi}+1 = 0$
I know how to prove each of these formulas, but I am curious if there is a conditional proof that can directly prove one statement from the other.
In other words, can the Gaussian integral imply Euler's identity, or vice versa?
Thanks.