 Jump of Laurent coefficients across poles

Suppose a meromorphic function is holomorphic on two concentric annuli about $a$, with only finitely many poles between chosen inner and outer coefficient circles. Let $a_n,b_n$ denote the corresponding Laurent coefficients. Subtract their contour integral formulas and apply the residue theorem to the intervening annular region. It givesThe sum is over the poles crossed, including their multiplicities through the residue. This identity applies to every integer index. Thus changing the annulus can change even the coefficients of nonnegative powers; it is not merely a change in the principal part.

For example, $f(z)=1/\sin z$ has simple poles at $\pm \pi$, each with residue $-1$. Moving a coefficient circle centered at zero from radius below $\pi$ to radius between $\pi$ and $2\pi$ changes its $n$th coefficient by $-{\pi }^{-n-1}-(-\pi {)}^{-n-1}$. Even indices remain unchanged, while odd indices acquire a nonzero correction.