This term I am teaching “Funktionentheorie”, i.e. functions of a complex variable. It turned out that it would be useful to have the rule of partial integration and a substitution rule for line integrals in this context. Well, these rules hold true and are not difficult to prove, but for some reason they are not treated in the textbooks we used. I state them here for convenience.
Proposition 1 (Partial integration for line integrals of holomorphic functions) Let and be holomorphic functions on a domain and let be continuous and piecewise differentiable. Then it holds that
Proof: It holds that
Integration this along and using that is obviously an antiderivative of we obtain
Proposition 2 (Substitution rule for line integrals of functions of a complex variable) Let be continuous and piecewise differentiable, be continuous on a neighborhood of and let be bi-holomorphic. Then it holds that
Proof: We simply calculate by substituting the parameterization and using the rule for the derivative of the inverse function:
Remark: It is enough that the transformation is biholomorphic from a neighborhood of onto its image .