In this form of the Dirac Delta function $$\delta(x) =\mathcal{F}[1] =\frac{1}{2 \pi}\int_{-\infty}^{\infty}e^{i\zeta x} d\zeta.$$ does $\zeta$ always have to be real valued?
If I make the substitution $\zeta(t) = i \omega(t)$ I think I can get this other form of the Dirac Delta function $$\delta(x) = \frac{1}{2 \pi i}\int_{- i \infty}^{i \infty}e^{-\omega x} d\omega.$$ In this form does $\omega(t)$ always have to be strictly imagnary, or could we think of $\omega$ as a more general path in the complex plane (which starts at $-i\infty$ an ends at $\infty$)?