Residue of Quotient

Theorem
Let $f$ and $g$ be functions holomorphic on some region containing $a$.

Let $g$ have a zero of multiplicity $1$ at $a$.

Then:


 * $\displaystyle \operatorname{Res} \left({\frac f g, a}\right) = \frac {f \left({a}\right)} {g' \left({a}\right)}$

Proof
As $g$ has a zero of multiplicity $1$ at $a$, $\dfrac f g$ has a simple pole at $a$ by definition.

So: