Let $g:\mathbb{C} \mapsto \mathbb{C}^{n \times n}$. How does one define the integral $$\int_\Gamma g(z) dz,$$ for some closed curve $\Gamma \subset \mathbb{C}$?
Is it understood entry-wise or as an special case of the Bochner Integral? I'm trying to understand the Matrix Cauchy integral formula for the definition of matrix functions.
$$f(A) = \frac{1}{2\pi i}\int_\Gamma f(z)(z I - A)^{-1}dz$$ where $f$ is analytic on and inside a closed contour $\Gamma$ that encloses $\Lambda(A)$.