Res_{z=z_0}f(z)= \frac{1}{(k-1)!} \lim_{z\to z_0} \frac{d^{k-1}}{dz^{k-1}} \left[(z-z_0)^k f(z)\right]

Res_{z=z_0}f(z)= \lim_{z\to z_0} \frac{d^{k-1}}{dz^{k-1}} \left[(z-z_0)^k f(z)\right]

Res_{z=z_0}f(z)= \frac{1}{(k+1)!} \lim_{z\to z_0} \frac{d^k}{dz^k} \left[(z-z_0)^{k-1} f(z)\right]

Res_{z=z_0}f(z)= \frac{1}{k!} \lim_{z\to z_0} \frac{d^k}{dz^k} \left[(z-z_0)^k f(z)\right]