\[\large \displaystyle \int_{-\infty}^{\infty}{\dfrac{e^{-x^{2}}}{\sqrt{\pi}}} \,dx - e^{i \pi} = ~ ?\]

**Notations:**

- \(e \approx 2.71828\) is the Euler's number.
- \(i = \sqrt{-1}\) is the imaginary unit.

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