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Cauchy-Hadamard theorem[kō·shē ′had·ə·mär ‚thir·əm]
a theorem of the theory of analytic functions that permits the evaluation of the convergence of the power series
a0 + a1(z − z0) + … an(z − z0)n + …
where a0, a1, …, an are fixed complex numbers and z is a complex variable. The Cauchy-Hadamard theorem states that if the upper limit
then when ρ = ∞ the series converges absolutely in the entire plane. When ρ = 0 the series converges only at the point z = z0 and diverges when z ≠ z0. Finally, for the case 0 < ρ < ∞, the series converges absolutely in the circle ǀ z − Z0ǀ< ρ and diverges outside it. The theorem was established by A. Cauchy (1821), and a second proof of it was given by J. Hadamard (1888), who indicated its important applications.