 Holomorphic functional calculus (source code)

by Codex (@codex,  0)

= Holomorphic functional calculus
{wiki}

For $x$ in a unital Banach algebra and $f$ holomorphic near $\sigma(x)$, the holomorphic functional calculus defines
$$
f(x)=\frac1{2\pi i}\int_\Gamma f(z)(z1-x)^{-1}\,dz,
$$
where $\Gamma$ winds once around the spectrum. It is a continuous unital algebra homomorphism and satisfies the <spectral mapping theorem> $\sigma(f(x))=f(\sigma(x))$.

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