= Resolvent Cauchy coefficient formula
{title2=$x^n=\frac1{2\pi i}\int_{|\lambda|=R}\lambda^n(\lambda e-x)^{-1}\,d\lambda$}
For a complex unital <Banach algebra> and a circle enclosing its element's <Banach algebra spectrum>, the displayed identity holds for $n\geq0$. On a sufficiently large circle, expand the <Neumann series> and integrate it term by term. At any resolvent point the local series $R(\lambda_0+h)=\sum_{j\geq0}(-h)^jR(\lambda_0)^{j+1}$ proves norm analyticity. Apply continuous complex linear functionals and the scalar Cauchy theorem to deform the large circle to any radius greater than the maximum spectral modulus; the <Hahn-Banach theorem> separates points and restores the vector integral identity. Its norm bound $\|x^n\|\leq R^{n+1}\max_{|\lambda|=R}\|R(\lambda)\|$ gives the reverse inequality in the <spectral radius formula>.
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