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, 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 $$f(z)=\sum _{n=0}^{\infty }c_{n}(z-a)^{n}$$ where $a,c_{n}\in \mathbb {C} .$

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

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

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

Theorem for several complex variables

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

Proof

From

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

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

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

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

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

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

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

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

Therefore

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

External links

• Weisstein, Eric W. "Cauchy-Hadamard theorem". MathWorld.