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.



Dirac Delta function with a complex argument
I am afraid this is due to a misunderstanding of what Dirac meant. He does not write "for any $y\in \mathbb{C}$" but he refers to a "c-number". The c stands for "classical" as opposed to quantum, and what Dirac means is that $y$ is a real number and not a Hermitian operator. Dirac never considered the delta function of a complex argument, only of real numbers.


When working with a complex number $z$, you can introduce the product of the delta function of the real and imaginary parts of $z$, and if you wish you can call that $\delta(z)\equiv \delta(\Re z)\delta(\Im z)$. So ultimately the fundamental object remains the delta function of a real number.


Concerning `Mathematica`: I am not able to reproduce your finding that Mathematica would return the delta function of a complex number. if I input



```
Integrate[Exp[I*lambda*x]*Exp[c*x]/Sqrt[2*Pi],{x,-Infinity,Infinity}]

```

into the online `Mathematica` interface at [Wolfram Alpha](http://www.wolframalpha.com) it returns "integral does not converge", which seems to me to be the only sensible answer (without further information on $\lambda$ and $c$).


