According to:

Dirac, P. A. M. (1927). "The physical interpretation of the quantum dynamics." Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character 113(765), pp.621–641.

For any $y \in \mathbb{C}$, if $f$ is analytic, then $\int_{-\infty}^\infty f(x) \delta(y-x)dx = f(y)$.

And, according to Wolfram Mathematica, the Fourier transform: $$ \int_{-\infty}^\infty \frac{e^{i \lambda x}}{\sqrt{2\pi}} e^{cx} dx = \sqrt{2\pi} \delta(\lambda - i c). $$ which essentially means that for any $\lambda \in \mathbb{C}$ we have $$ \int_{-\infty}^\infty \frac{e^{i \lambda x}}{2\pi}dx = \delta(\lambda). $$ This all seems consistent since $$ e^{cx} = \int_{-\infty}^\infty \frac{e^{-i \lambda x}}{\sqrt{2\pi}} \sqrt{2\pi} \delta(\lambda - i c) d\lambda = e^{-i (i c)x} = e^{cx} . $$ But, I am wondering if anybody knows where I can find some sort of justification for these formal manipulations. I'd like to have a better understanding of what is going on here. And neither Mathematica nor Dirac provide any sort of justification for the above results.