Corollary

Let $a\in A$. One has $\Theta_a(1) = 1$.

Let $R> \varrho (a)$ be a real number. Let V be the closed disk with center 0 and radius R, so that Sp($a$)$\subset \mathring{V}$. It is a piece of $\mathbf{C}$ whose boundary $\partial V$ is the circle with center 0 and radius R. For $z\in \mathbf{C}-\mathring{V}$, one has the formula $(z-a)^{-1}=z^{-1}(1-z^{-1}a)^{-1}=\sum^{+\infty}_{j=1}z^{-j}a^{j-1}$. The series converges uniformly for $z\in \partial V$. Hence

since

for every integer $j\not=-1$ and

(VAR, R2, p. 44, 10.4.5, and p. 47, 11.2.1, example).