Analytical Calculation of the Impulse Response of an Arbitrary Filter This is a rather general question.  Suppose I have an arbitrary filter, $F$, defined in the Laplace domain;
$$
F=k\frac{\prod_i (s+z_i)}{\prod_j(s+p_j)}.
$$
How do I calculate the impulse response of this filter analytically?  As a specific simple example, consider the first order low pass filter
$$
F_0=\frac{1}{s+p}.
$$
To be clear, what I want to get from this transformation is a function which describes the delta function response from $t=0$ to $t\rightarrow\infty$ where the delta function is applied at $t=0$.
Your expression is the Laplace transform of the exponential function $e^{-pt}$. The Laplace transform of the dirac unit is $1$. So the inverse Laplace transform is exactly $e^{-pt}$. Maybe the fact that this function is noncausal adds a little complexity here.