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:


 * $\Res {\dfrac f g} a = \dfrac {\map f a} {\map {g'} a}$

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

So: