Let's say I have the following expression:
$$f(n) = \sum_{m=1}^{n-1} \delta(\tau -m )$$
Now, for integer $n$ this is simple enough, but I'm interested in analytically continuing this function to non-integer $n$. Is there some easy way to do this, maybe with the Fourier transformed version of the $\delta$ function? I'm particularly interested in the case where $n \rightarrow 1$.