Cauchy–Hadamard theorem

Short description: A theorem that determines the radius of convergence of a power series.

In mathematics, the Cauchy–Hadamard theorem is a result in complex analysis named after the France 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 [math]\displaystyle{ f(z) = \sum_{n = 0}^{\infty} c_{n} (z-a)^{n} }[/math] where [math]\displaystyle{ a, c_n \in \Complex. }[/math]

Then the radius of convergence [math]\displaystyle{ R }[/math] of f at the point a is given by [math]\displaystyle{ \frac{1}{R} = \limsup_{n \to \infty} \left( | c_{n} |^{1/n} \right) }[/math] 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 are 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 [math]\displaystyle{ a=0 }[/math]. We will show first that the power series [math]\displaystyle{ \sum_n c_n z^n }[/math] converges for [math]\displaystyle{ |z|\lt R }[/math], and then that it diverges for [math]\displaystyle{ |z|\gt R }[/math].

First suppose [math]\displaystyle{ |z|\lt R }[/math]. Let [math]\displaystyle{ t=1/R }[/math] not be [math]\displaystyle{ 0 }[/math] or [math]\displaystyle{ \pm\infty. }[/math] For any [math]\displaystyle{ \varepsilon \gt 0 }[/math], there exists only a finite number of [math]\displaystyle{ n }[/math] such that [math]\displaystyle{ \sqrt[n]{|c_n|} \geq t+\varepsilon }[/math]. Now [math]\displaystyle{ |c_n| \leq (t+\varepsilon)^n }[/math] for all but a finite number of [math]\displaystyle{ c_n }[/math], so the series [math]\displaystyle{ \sum_n c_n z^n }[/math] converges if [math]\displaystyle{ |z| \lt 1/(t+\varepsilon) }[/math]. This proves the first part.

Conversely, for [math]\displaystyle{ \varepsilon \gt 0 }[/math], [math]\displaystyle{ |c_n|\geq (t-\varepsilon)^n }[/math] for infinitely many [math]\displaystyle{ c_n }[/math], so if [math]\displaystyle{ |z|=1/(t-\varepsilon) \gt R }[/math], we see that the series cannot converge because its nth term does not tend to 0.

Theorem for several complex variables

Let [math]\displaystyle{ \alpha }[/math] be a multi-index (a n-tuple of integers) with [math]\displaystyle{ |\alpha|=\alpha_1+\cdots+\alpha_n }[/math], then [math]\displaystyle{ f(x) }[/math] converges with radius of convergence [math]\displaystyle{ \rho }[/math] (which is also a multi-index) if and only if [math]\displaystyle{ \limsup_{|\alpha|\to\infty} \sqrt[|\alpha|]{|c_\alpha|\rho^\alpha}=1 }[/math] to the multidimensional power series [math]\displaystyle{ \sum_{\alpha\geq0}c_\alpha(z-a)^\alpha := \sum_{\alpha_1\geq0,\ldots,\alpha_n\geq0}c_{\alpha_1,\ldots,\alpha_n}(z_1-a_1)^{\alpha_1}\cdots(z_n-a_n)^{\alpha_n} }[/math]

Proof

From

Set [math]\displaystyle{ z = a + t\rho }[/math] [math]\displaystyle{ (z_i = a_i + t\rho_i) }[/math], then

[math]\displaystyle{ \sum_{\alpha \geq 0} c_\alpha (z - a)^\alpha = \sum_{\alpha \geq 0} c_\alpha \rho^\alpha t^{|\alpha|} = \sum_{\mu \geq 0} \left( \sum_{|\alpha| = \mu} |c_\alpha| \rho^\alpha \right) t^\mu }[/math]

This is a power series in one variable [math]\displaystyle{ t }[/math] which converges for [math]\displaystyle{ |t| \lt 1 }[/math] and diverges for [math]\displaystyle{ |t| \gt 1 }[/math]. Therefore, by the Cauchy-Hadamard theorem for one variable

[math]\displaystyle{ \limsup_{\mu \to \infty} \sqrt[\mu]{\sum_{|\alpha| = \mu} |c_\alpha| \rho^\alpha} = 1 }[/math]

Setting [math]\displaystyle{ |c_m| \rho^m = \max_{|\alpha| = \mu} |c_\alpha| \rho^\alpha }[/math] gives us an estimate

[math]\displaystyle{ |c_m| \rho^m \leq \sum_{|\alpha| = \mu} |c_\alpha| \rho^\alpha \leq (\mu + 1)^n |c_m| \rho^m }[/math]

Because [math]\displaystyle{ \sqrt[\mu]{(\mu + 1)^n} \to 1 }[/math] as [math]\displaystyle{ \mu \to \infty }[/math]

[math]\displaystyle{ \sqrt[\mu]{|c_m| \rho^m} \leq \sqrt[\mu]{\sum_{|\alpha| = \mu} |c_\alpha| \rho^\alpha} \leq \sqrt[\mu]{|c_m| \rho^m} \implies \sqrt[\mu]{\sum_{|\alpha| = \mu} |c_\alpha| \rho^\alpha} = \sqrt[\mu]{|c_m| \rho^m} \qquad (\mu \to \infty) }[/math]

Therefore

[math]\displaystyle{ \limsup_{|\alpha|\to\infty} \sqrt[|\alpha|]{|c_\alpha|\rho^\alpha} = \limsup_{\mu \to \infty} \sqrt[\mu]{|c_m| \rho^m} = 1 }[/math]

External links

• Weisstein, Eric W.. "Cauchy-Hadamard theorem". http://mathworld.wolfram.com/Cauchy-HadamardTheorem.html.