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Trying to understand the answer of this question, I took a hour to seek, some things I never learn in class, in the net like holomorphic functions et caetera, so now all is understandable for me, but not one things.

They say in the article Matrix Function that to enclose all the eigenvalues one possibility to achieve this is to let $C$ be a circle around the origin with radius larger than $‖A‖$ for an arbitrary matrix norm $‖.‖$

Can someone prove it ?

Thank you in advance !

ParaH2
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Hint: Let $x$ such that $\|x\|=1$ be an eigenvector associated to the eigenvalue $c$, $\|A(x)\| =\mid c\mid \leq \|A\|$.

Tsemo Aristide
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