Cauchy–Hadamard theorem

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,^[1] but remained relatively unknown until Hadamard rediscovered it.^[2] Hadamard's first publication of this result was in 1888;^[3] he also included it as part of his 1892 Ph.D. thesis.^[4]

Theorem for one complex variable

Consider the formal power series in one complex variable z of the form $${\displaystyle f(z)=\sum _{n=0}^{\infty }c_{n}(z-a)^{n}}$$ where ${\displaystyle a,c_{n}\in \mathbb {C} .}$

Then the radius of convergence ${\displaystyle R}$ of f at the point a is given by $${\displaystyle {\frac {1}{R}}=\limsup _{n\to \infty }\left(|c_{n}|^{1/n}\right)}$$ 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 ${\displaystyle a=0}$. We will show first that the power series ${\textstyle \sum _{n}c_{n}z^{n}}$ converges for ${\displaystyle |z|<R}$, and then that it diverges for ${\displaystyle |z|>R}$.

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

Conversely, for ${\displaystyle \varepsilon >0}$, ${\displaystyle |c_{n}|\geq (t-\varepsilon )^{n}}$ for infinitely many ${\displaystyle c_{n}}$, so if ${\displaystyle |z|=1/(t-\varepsilon )>R}$, we see that the series cannot converge because its nth term does not tend to 0.^[5]

Theorem for several complex variables

Let ${\displaystyle \alpha }$ be an n-dimensional vector of natural numbers (${\displaystyle \alpha =(\alpha _{1},\cdots ,\alpha _{n})\in \mathbb {N} ^{n}}$) with ${\displaystyle \|\alpha \|:=\alpha _{1}+\cdots +\alpha _{n}}$, then ${\displaystyle f(z)}$ converges with radius of convergence ${\displaystyle \rho =(\rho _{1},\cdots ,\rho _{n})\in \mathbb {R} ^{n}}$, ${\displaystyle \rho ^{\alpha }=\rho _{1}^{\alpha _{1}}\cdots \rho _{n}^{\alpha _{n}}}$ if and only if $${\displaystyle \limsup _{\|\alpha \|\to \infty }{\sqrt[{\|\alpha \|}]{|c_{\alpha }|\rho ^{\alpha }}}=1}$$ of the multidimensional power series $${\displaystyle f(z)=\sum _{\alpha \geq 0}c_{\alpha }(z-a)^{\alpha }:=\sum _{\alpha _{1}\geq 0,\ldots ,\alpha _{n}\geq 0}c_{\alpha _{1},\ldots ,\alpha _{n}}(z_{1}-a_{1})^{\alpha _{1}}\cdots (z_{n}-a_{n})^{\alpha _{n}}.}$$

Proof

From ^[6]

Set ${\displaystyle z=a+t\rho }$ ${\displaystyle (z_{i}=a_{i}+t\rho _{i}).}$ Then

${\displaystyle \sum _{\alpha \geq 0}c_{\alpha }(z-a)^{\alpha }=\sum _{\alpha \geq 0}c_{\alpha }\rho ^{\alpha }t^{\|\alpha \|}=\sum _{\mu \geq 0}\left(\sum _{\|\alpha \|=\mu }|c_{\alpha }|\rho ^{\alpha }\right)t^{\mu }.}$

This is a power series in one variable ${\displaystyle t}$ which converges for ${\displaystyle |t|<1}$ and diverges for ${\displaystyle |t|>1}$. Therefore, by the Cauchy–Hadamard theorem for one variable

${\displaystyle \limsup _{\mu \to \infty }{\sqrt[{\mu }]{\sum _{\|\alpha \|=\mu }|c_{\alpha }|\rho ^{\alpha }}}=1.}$

Setting ${\displaystyle |c_{m}|\rho ^{m}=\max _{\|\alpha \|=\mu }|c_{\alpha }|\rho ^{\alpha }}$ gives us an estimate

${\displaystyle |c_{m}|\rho ^{m}\leq \sum _{\|\alpha \|=\mu }|c_{\alpha }|\rho ^{\alpha }\leq (\mu +1)^{n}|c_{m}|\rho ^{m}.}$

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

${\displaystyle {\sqrt[{\mu }]{|c_{m}|\rho ^{m}}}\leq {\sqrt[{\mu }]{\sum _{\|\alpha \|=\mu }|c_{\alpha }|\rho ^{\alpha }}}\leq {\sqrt[{\mu }]{|c_{m}|\rho ^{m}}}\implies {\sqrt[{\mu }]{\sum _{\|\alpha \|=\mu }|c_{\alpha }|\rho ^{\alpha }}}={\sqrt[{\mu }]{|c_{m}|\rho ^{m}}}\qquad (\mu \to \infty ).}$

Therefore

${\displaystyle \limsup _{\|\alpha \|\to \infty }{\sqrt[{\|\alpha \|}]{|c_{\alpha }|\rho ^{\alpha }}}=\limsup _{\mu \to \infty }{\sqrt[{\mu }]{|c_{m}|\rho ^{m}}}=1.}$