While I was studying about the spin-glass model based on replica theory (Spin-glass theory for pedestrians, page 16, equation 49), I have encountered something like an exponential form of Dirac-delta function: \begin{equation} \delta(q - \hat{q}) = \int_{\mathbb{R}}\exp((q-\hat{q})\lambda)d\lambda \end{equation} where both $q$ and $\hat{q}$ are real numbers and $\lambda$ as well. According to this post, the quantity $\lambda$ is also called as a Lagrange multiplier. I guess the above form of the equation is somewhat connected to an idea of analytic continuation of complex representation of Dirac-delta. The usual complex form Dirac-delta function looks like following. \begin{equation} \delta(q - \hat{q}) = \frac{1}{2\pi}\int_{\mathbb{R}}\exp(i\lambda(q-\hat{q}))d\lambda \end{equation} Anyways, I have tried to obtain the real exponential form of Dirac-delta function using the Laplace transform and the inverse Laplace transform. \begin{align} F(\hat{s}) &= \int_{\mathbb{R}}f(t)\exp(-\hat{s}t)dt \\ &= \int_{\mathbb{R}}\left[\frac{1}{2\pi i}\int_{i\infty}^{i\infty}F(s)\exp(st)ds\right]\exp(-\hat{s}t)dt \\ &= \int_{i\infty}^{i\infty}F(s)\left[\frac{1}{2\pi i}\int_{\mathbb{R}}\exp((s-\hat{s})t)dt\right] ds \end{align} I just found the general form of the inverse Laplace transform from google. Then by definition of Dirac-delta function, I obtain the following. \begin{equation} \delta(s-\hat{s}) = \frac{1}{2\pi i}\int_{\mathbb{R}}\exp((s-\hat{s})t)dt \end{equation}
It looks similar to the first equation, but there exists a complex constant term and both $s$ and $\hat{s}$ are still complex numbers. What do you think? Can you help me to understand the real exponential form of Dirac-delta function?