Function whose derivative is itself, but

First we take that the solution to such functional differential equation is of the form

$f(x) = c_1 e^{Ax}$

for some unknown constant $A$ and some arbitrary constant $c_1$ .

Substituting that to the said differential equation,

$c_1 Ae^{Ax} = c_1 e^{A(x-1)}$

$c_1 Ae^{Ax} = c_1 e^{-A} e^{Ax}$

$A = e^{-A}$

$Ae^A =1$

$A= W(1) = \Omega \approx 0.567143$

Where $\Omega$ is the Omega Constant . This means our function is

$f(x) = c_1 e^{W(1) x}$

Now, the Lambert's W function has the property such that

$e^{W(x)} = \frac{1}{W(x)}$

So we can rephrase the function as

$f(x) = c_1 [W(1)]^{-x}$

From which we can see that since $f(\pi) = [W(1)]^{-\pi}$ , so that makes $c_1 =1$ .

From here we can now determine $f(\frac{1}{\Omega})$ .

$f(\frac{1}{\Omega}) = e^{\Omega \times \frac{1}{\Omega}} = e$

And thus, $\lfloor 1000 \times e \rfloor = 2718$ .