Research proof index

Suppose $f$ is holomorphic and nonzero on a punctured neighborhood of $p$, and $f(s)=\frac{A}{s-p}+O(1)$ there with $A\ne0$. Then the logarithmic derivative has principal part $-1/(s-p)$: $\frac{f'(s)}{f(s)}+\frac{1}{s-p}=O(1)$ as $s\to p$. In particular, the residue of $f'/f$ at this simple pole is $-1$.

Let $f$ be holomorphic on a punctured neighborhood of $p$. If $(s-p)f(s)\longrightarrow A\qquad(s\to p),$ then $f$ differs from its principal part $A/(s-p)$ by a bounded function on some punctured neighborhood of $p$. In asymptotic notation, $f(s)=\frac{A}{s-p}+O(1).$