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=== Cauchy integral ===
[[Cauchy's integral formula]] from [[complex analysis]] can also be used to generalize scalar functions to matrix functions. Cauchy's integral formula states that for any [[analytic function]] {{mvar|f}} defined on a set {{math|''D'' ⊂ ℂ}}, one has
:<math>f(x) = \frac{1}{2\pi i} \
where {{mvar|C}} is a closed simple curve inside the ___domain {{mvar|D}} enclosing {{mvar|x}}.
Now, replace {{mvar|x}} by a matrix {{mvar|A}} and consider a path {{mvar|C}} inside {{mvar|D}} that encloses all [[eigenvalue]]s of {{mvar|A}}. One possibility to achieve this is to let {{mvar|C}} be a circle around the [[origin (mathematics)|origin]] with [[radius]] larger than ‖{{mvar|A}}‖ for an arbitrary [[matrix norm]] ‖•‖. Then, {{math|''f''(''A'')}} is definable by
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