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 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
where
Then the radius of convergence
of f at the point a is given by
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
. We will show first that the power series
converges for
, and then that it diverges for
.
First suppose
. Let
not be
or
For any
, there exists only a finite number of
such that
. Now
for all but a finite number of
, so the series
converges if
. This proves the first part.
Conversely, for
,
for infinitely many
, so if
, we see that the series cannot converge because its nth term does not tend to 0.[5]
Theorem for several complex variables
Let
be an n-dimensional vector of natural numbers (
) with
, then
converges with radius of convergence
with
if and only if
to the multidimensional power series
Proof
From [6]
Set
, 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 }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/10f8cad65a553cf2b2baca310dabe775da64ea15)
This is a power series in one variable
which converges for
and diverges for
. Therefore, by the Cauchy-Hadamard theorem for one variable
![{\displaystyle \limsup _{\mu \to \infty }{\sqrt[{\mu }]{\sum _{||\alpha ||=\mu }|c_{\alpha }|\rho ^{\alpha }}}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/902c0cf2b74280d0995d40bc95e57175733883df)
Setting
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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/61c94b8a139504799d2bfbf8c2800c5a08e13d76)
Because
as
![{\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 )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/104bce860c902bf14f734c0369b27254e4cf1d9a)
Therefore
![{\displaystyle \limsup _{||\alpha ||\to \infty }{\sqrt[{||\alpha ||}]{|c_{\alpha }|\rho ^{\alpha }}}=\limsup _{\mu \to \infty }{\sqrt[{\mu }]{|c_{m}|\rho ^{m}}}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff8d149af14cbba57811ba517e50e151be55d483)
Notes
- ^ Cauchy, A. L. (1821), Analyse algébrique.
- ^ Bottazzini, Umberto (1986), The Higher Calculus: A History of Real and Complex Analysis from Euler to Weierstrass, Springer-Verlag, pp. 116–117, ISBN 978-0-387-96302-0. Translated from the Italian by Warren Van Egmond.
- ^ Hadamard, J., "Sur le rayon de convergence des séries ordonnées suivant les puissances d'une variable", C. R. Acad. Sci. Paris, 106: 259–262.
- ^ Hadamard, J. (1892), "Essai sur l'étude des fonctions données par leur développement de Taylor", Journal de Mathématiques Pures et Appliquées, 4e Série, VIII. Also in Thèses présentées à la faculté des sciences de Paris pour obtenir le grade de docteur ès sciences mathématiques, Paris: Gauthier-Villars et fils, 1892.
- ^ Lang, Serge (2002), Complex Analysis: Fourth Edition, Springer, pp. 55–56, ISBN 0-387-98592-1 Graduate Texts in Mathematics
- ^ Shabat, B.V. (1992), Introduction to complex analysis Part II. Functions of several variables, American Mathematical Society, pp. 32–33, ISBN 978-0821819753
External links
- Weisstein, Eric W. "Cauchy-Hadamard theorem". MathWorld.