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  "body": "\n## Summary\n\nLet $a$ be an element of a unital [[Banach algebra|9KYWN94QDN]], and let $f$ be a [[holomorphic function|0NAHXR1FRH]] on a neighbourhood of the [[spectrum|EH0CZ19RKP]] of $a$.\nThe holomorphic functional calculus gives a meaning to $f(a)$.\nIt is defined by a Cauchy integral of $f$ against the [[resolvent|J5YNNKSB9P]] of $a$.\nThe map $f \\mapsto f(a)$ is an algebra homomorphism that extends the evaluation of polynomials, and it satisfies the spectral mapping theorem $\\sigma(f(a)) = f(\\sigma(a))$.\nIt works for all bounded operators on a Banach space, not only normal ones, and it gives spectral projections, logarithms and square roots of operators.\n\n## Explanations\n\n### L1 human\n\nFor a polynomial $p$ and a square matrix $A$, the matrix $p(A)$ is clear: replace $z$ by $A$.\nThe holomorphic functional calculus does the same for any function given locally by power series, using Cauchy's integral formula instead of a formula for $f$.\nThe value $f(\\lambda)$ is replaced by the integral of $f(\\lambda)$ times $(\\lambda - a)^{-1}$ around the spectrum.\nFor example, if $A = \\begin{pmatrix} 2 \u0026 0 \\\\ 0 \u0026 3 \\end{pmatrix}$ and $f = \\exp$, the calculus gives the diagonal matrix with entries $e^2$ and $e^3$.\nIf the spectrum splits into two parts, the function equal to $1$ near one part and $0$ near the other gives a projection onto the matching invariant subspace.\n",
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