This lecture presents the technical asymptotic results underlying Siefring’s intersection theory for punctured holomorphic curves, including the necessary prerequisites on asymptotic operators and the relations proved by Hofer, Wysocki and Zehnder between winding numbers of asymptotic eigenfunctions and the Conley–Zehnder index. Siefring’s relative asymptotic formulas are stated largely without proof but are motivated in terms of an asymptotic analogue of the similarity principle. The last section then discusses the punctured analogue of the question about holomorphic foliations considered in Lecture 2, which motivates the definition of the normal Chern number for punctured holomorphic curves.