Analytic primary matrix functions by Jordan blocks

Let $A\in {\mathbb{C}}^{n\times n}$ with $n\ge 1$, let $\mathrm{\Omega }\subseteq \mathbb{C}$ be open with $\sigma (A)\subseteq \mathrm{\Omega }$, where $\sigma (A)$ is the eigenvalue set from Eigenvalues, eigenvectors, eigenspaces $E_{\lambda }(T)=\ker (T-\lambda I)$, and the spectrum ${\sigma }_F(T)$ of an endomorphism, and let $f:\mathrm{\Omega }\to \mathbb{C}$ be holomorphic in the sense of Complex differentiability at a point, the complex derivative, holomorphic functions, and entire functions. Choose a Jordan representation in the convention of Jordan bases and Jordan canonical forms over the base field, which exists by Jordan form over the base field exists exactly when the characteristic polynomial splits and Fundamental theorem of algebra: every nonconstant complex polynomial has a complex root: $A=Z\mathrm{diag}(J_{d_1}({\lambda }_1),\dots ,J_{d_r}({\lambda }_r))Z^{-1},$ where $J_d(\lambda )=\lambda I_d+N_d$ and $N_d$ has ones on its first superdiagonal and is nilpotent of order at most $d$. Define the Jordan expression $f_J(A):=Z\mathrm{diag}({\sum }_{k=0}^{d_1-1}\frac{f^{(k)}({\lambda }_1)}{k!}{N}_{d_1}^k,\dots ,{\sum }_{k=0}^{d_r-1}\frac{f^{(k)}({\lambda }_r)}{k!}{N}_{d_r}^k)Z^{-1}\mathrm{.}$ The analytic primary matrix function $f(A)$ is this expression, with the same single scalar branch $f$ and its derivatives used in every block at a repeated eigenvalue. Its independence of the chosen Jordan representation is discharged by Jordan, contour, similarity, and normal-functional-calculus formulas agree ↗.