Complex Analysis: Theory and Applications · Problem 7.2.11

Identity (7.6) typically involves a residue calculation or Laurent series expansion around a singularity. The conditions $z \neq z_0$ and $z \neq \alpha$ ensure that Lemma 7.1's assumptions hold: you're working in a punctured neighborhood where the function remains analytic and the lemma's machinery (whether it's a contour integral, series convergence, or algebraic manipulation) doesn't break down. By carefully examining the domain where Identity (7.6) is stated or applied—usually excluding the poles or special points—you can verify these exclusions are built into the problem's setup, making the lemma's hypotheses valid throughout.

Identity (7.6) typically involves a residue calculation or Laurent series expansion around a singularity. The conditions $z \neq z_0$ and $z \neq \alpha$ ensure that Lemma 7.1's assumptions hold: you're working in a punctured neighborhood where the function remains analytic and the lemma's machinery (whether it's a contour integral, series convergence, or algebraic manipulation) doesn't break down. By carefully examining the domain where Identity (7.6) is stated or applied—usually excluding the poles or special points—you can verify these exclusions are built into the problem's setup, making the lemma's hypotheses valid throughout.