Cauchy-Hadamard Theorem definition of Cauchy-Hadamard Theorem in the Free Online Encyclopedia.

Cauchy-Hadamard theorem [kō·shē ′had·ə·mär ‚thir·əm]

(mathematics)

The theorem that the radius of convergence of a Taylor series in the complex variablezis the reciprocal of the limit superior, asnapproaches infinity, of thenth root of the absolute value of the coefficient ofzⁿ.

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) + … a_n(z − z₀)ⁿ + …

where a₀, a₁, …, a_n 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.

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