I am curious if there is a particular name for this, $\int\limits_{-\infty}^\infty e^{i\xi^2}d\xi$. I think it might be related the Fresnel integral but I cannot see it, any suggestions?
2026-02-23 03:47:26.1771818446
Name of a particular improper integral
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If $\omega$ is a square root of $-i$, $$\int_{-R}^R \exp(ix^2)\; dx = \dfrac{\sqrt{\pi}}{\omega} \text{erf}(\omega R)$$ The limit as $R \to \infty$ is then $(1+i) \sqrt{\pi/2}$.