Cauchy-Hadamard Theorem

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Cauchy-Hadamard theorem

[kō·shē ′had·ə·mär ‚thir·əm]
The theorem that the radius of convergence of a Taylor series in the complex variable z is the reciprocal of the limit superior, as n approaches infinity, of the n th root of the absolute value of the coefficient of zn.

Cauchy-Hadamard Theorem


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.