Proposition 1

Let $a$ be an element of A, let U be an open neighbourhood of Sp($a$), and let $f\in \mathscr{O}(U; A)$. Let V be a compact neighbourhood of Sp($a$) contained in U and such that V is a piece of $\mathbf{C}$.

Then $z\mapsto f(z)(z-a)^{-1}$ is continuous on $\partial V$, the differential form $f(z)(z-a)^{-1}dz$ is integrable on $\partial V$, and one has

Let $h$ be a mapping of $\mathbf{C}$ into $\mathbf{C}$, indefinitely differentiable, equal to 1 in a neighbourhood of Sp($a$) and with compact support contained in the interior of V (lemma 1 of I, p. 52). Let $u$ be a mapping of $\mathbf{C}$ into A such that $(h, u)$ is adapted to $a($cf. example 3 of I, p. 53). The associated differential form is $\omega =du\wedge dz$. We get $f \omega =f du\wedge dz=d(f u dz)$ since $f$ is holomorphic. Moreover, $u(z) = (z-a)^{-1}$ on the boundary of V. On the other hand, the differential form $f u dz$ is of class $C^1$ on U. Hence

by formula (5) and Stokes's formula for the piece V (VAR, R2, p. 47, 11.2.3).