I'm interested in the (obviously divergent) integral
$$ \int_{-\infty}^\infty dx e^{-x f}\ ,$$
where $f$ is real. Is there any way to meaningfully assign a value to this integral? I was thinking of the following procedure:
$$\lim_{\epsilon\rightarrow 0}\int_{-\infty}^\infty dx e^{-x f- \epsilon x^2} = \lim_{\epsilon\rightarrow 0} \sqrt{\frac{\pi}{\epsilon}} e^{\frac{f^2}{4 \epsilon}} = \lim_{\epsilon\rightarrow 0} \sqrt{\frac{\pi}{\epsilon}} e^{-\frac{(if)^2}{4 \epsilon}}\ .$$
The last formula can be recognized as a limit representation of Dirac delta function. Thus the result is $\delta(if)$.
Does this make any sense? If it does, how do I interpret delta function over imaginary argument?
EDIT: Perhaps I should have explained why I'm thinking about this. The reason comes from physics and calculation of partition functions. Suppose we have a partition function involving some field $\psi$, i.e.:
$Z = \int \mathcal{D} \psi e^{-S[\psi]}$
Now suppose we have some restriction for the values of $\psi$, for example we require that $F[\Psi] = c$, where $c$ is a real number. Now we can introduce this restriction in two ways. First is to add a delta function inside the integral, i.e.:
$\delta (F[\psi] -c) = \int dx e^{- i x (F[\psi]-c)}$,
and the partition function becomes
$Z = \int \mathcal{D} \int dx \psi e^{-S[\psi] - ix(F[\psi]-c)}$
Now suppose we want to evaluate this partition function within the saddle point approximation. This amounts to just minimizing the action plus the term that comes from the restriction, i.e:
$\frac{\delta }{\delta \psi} [S[\psi] - ix(F[\psi]-c)] = 0$ .
On the other hand, since this is a minimization problem, we could introduce the restriction as Lagrange multiplier. In this case, we would be minimizing a slightly different expression:
$\frac{\delta }{\delta \psi} [S[\psi] - x(F[\psi]-c)] = 0$ ,
where $x$ is, in this case, the lagrange multiplier. The difference between the two is in the imaginary unit in front of the restriction condition. Now there seems to be some deep connection between the two methods (delta function inside the integral and Lagrange multipliers). In fact, the delta function method would amount exactly to the Lagrange multiplier method if we could interpret the divergent integral I wrote above as the Dirac delta function.