Cauchy-Hadamard Theorem

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 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) + … 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.

单词	cauchy-hadamard theorem
释义	Cauchy-Hadamard Theorem 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 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) + … 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.
随便看	multipoint command conference multipoint command negating mcs multipoint command symmetrical data-transmission multipoint command visualization multi point communication file system multipoint communication framework multi point communications multipoint conference unit multi point conferencing multipoint conferencing multi-point conferencing unit multipoint conferencing unit multi point constraints multi point controller multi-point control protocol multi-point control protocol data unit multipoint control services multipoint control unit multi-point courtesy inspection multi-point critical flicker fusion frequency multi-point diamond search multipoint disconnect multipoint discrete input output multipoint distribution service multipoint distribution system