Analytic function of a matrix: Difference between revisions

Revision as of 02:37, 24 November 2020

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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} \oint_{\!\!\!\!\!C}\! {\frac{f(z)}{z-x}}\, \mathrm{d}z ~,</math>

where {{mvar|C}} is a closed simple curve inside the ___domain {{mvar|D}} enclosing {{mvar|x}}.