柯西－阿达马公式

公式陈述

对于单一复数变量“z”的形式幂级数 f(z)=∑ ₙ₌₀^(∞)cₙ(z-a)ⁿ.

• 上式中 a,cₙ∈ C,
• 则该级数收敛半径 R 由下式给出： （公式略）
• 其中 limsup 定义为 （公式略）

其中 sup 为集合的最小上界。

In mathematics, the Cauchy–Hadamard theorem is a result in complex analysis named after the French mathematicians Augustin Louis Cauchy and Jacques Hadamard, describing the radius of convergence of a power series. It was published in 1821 by Cauchy, but remained relatively unknown until Hadamard rediscovered it. Hadamard's first publication of this result was in 1888; he also included it as part of his 1892 Ph.D. thesis.

Theorem for one complex variable

Consider the formal power series in one complex variable z of the form f(z)=∑ ₙ₌₀^(∞)cₙ(z-a)ⁿ

where a,cₙ∈ C.

Then the radius of convergence R of f at the point a is given by （公式略）

where lim sup denotes the limit superior, the limit as n approaches infinity of the supremum of the sequence values after the nth position. If the sequence values is unbounded so that the lim sup is ∞, then the power series does not converge near a, while if the lim sup is 0 then the radius of convergence is ∞, meaning that the series converges on the entire plane.

Proof

Without loss of generality assume that a=0. We will show first that the power series ∑ ₙcₙzⁿ converges for |z|<R, and then that it diverges for |z|>R.

First suppose |z|<R. Let t=1/R not be 0 ± ∞.

For any ε >0, there exists only a finite number of n such that （公式略）.

Now |cₙ|<(t+ε)ⁿ for all but a finite number of cₙ, so the series ∑ ₙcₙzⁿ converges if |z|<1/(t+ε). This proves the first part.

Conversely, for ε >0, |cₙ|≥ (t-ε)ⁿ for infinitely many cₙ, so if |z|=1/(t-ε)>R, we see that the series cannot converge because its nth term does not tend to 0.

Theorem for several complex variables

Let α be an n-dimensional vector of natural numbers ( α =(α ₁,⋯,α ₙ)∈ N ⁿ) with ‖α ‖:=α ₁+⋯ +α ₙ, then f(z) converges with radius of convergence ρ =(ρ ₁,⋯,ρ ₙ)∈ R ⁿ, ρ ^(α)=ρ ₁^α ₁⋯ ρ ₙ^α ₙ if and only if （公式略）

of the multidimensional power series （公式略）

Proof

• From
• Set z=a+tρ
• ( (zᵢ=aᵢ+tρ ᵢ). Then （公式略）

This is a power series in one variable t which converges for |t|<1 and diverges for |t|>1. Therefore, by the Cauchy–Hadamard theorem for one variable （公式略）

Setting |cₘ|ρ ^(m)=max _(‖α ‖=μ)|c_(α)|ρ ^(α) gives us an estimate |cₘ|ρ ^(m)≤ ∑ _(‖α ‖=μ)|c_(α)|ρ ^(α)≤ (μ +1)ⁿ|cₘ|ρ ^(m).

• Because （公式略） μ → ∞
• | （公式略）
• Therefore （公式略）