Heaviside Coverup Method

This is a fast method to compute the Partial Fraction Decomposition, introduced in MATH213.

How is this derived? Set

F(s) &= F(s) \\ \frac{s^2 + 1}{(s-1)(s+2)(s-5)(s+7)(s-9)} &= \frac{A}{s-1} + \frac{B}{s + 2} + \frac{C}{s-5} + \frac{D}{s+7} + \frac{E}{s-9} \end{align}$$ Then, for $A$, you multiply each side by $s-1$, so you get $$\frac{s^2 + 1}{(s+2)(s-5)(s+7)(s-9)} = A + \frac{B (s-1)}{s + 2} + \frac{C(s-1)}{s-5} + \frac{D(s-1)}{s+7} + \frac{E(s-1)}{s-9}$$ At $s=1$, you just get $A$, so that's how you compute $A = \frac{s^2 + 1}{(s+2)(s-5)(s+7)(s-9)}$ evaluated at $s=1$.