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.

Contents

Theory for one complex variable

Statement of the theorem

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 of ƒ at the point a is given by

${\frac {1}{R}}=\limsup _{n\to \infty }{\big (}|c_{n}|^{1/n}{\big )}$

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 of the theorem

Without loss of generality assume that $a=0$. We will show first that the power series $\sum c_{n}z^{n}$ converges for $|z|<R$, and then that it diverges for $|z|>R$.

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

Several complex variables

Statement of the theorem

Let $\alpha$ be a multi-index (a n-tuple of integers) with $|\alpha |=\alpha _{1}+\cdots +\alpha _{n}$, then $f(x)$ converges with radius of convergence $\rho$ (which is also a multi-index) if and only if

$\lim _{|\alpha |\to \infty }{\sqrt[{|\alpha |}]{|c_{\alpha }|\rho ^{\alpha }}}=1$

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 of the theorem

The proof can be found in the book Introduction to Complex Analysis Part II functions in several Variables by B. V. Shabat

External links

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