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.
[4] Consider the formal power series in one complex variable z of the form
Then the radius of convergence
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.
Without loss of generality assume that
We will show first that the power series
, there exists only a finite number of
for all but a finite number of
This proves the first part.
, we see that the series cannot converge because its nth term does not tend to 0.
α
be an n-dimensional vector of natural numbers (
α = (
α
α
‖ α ‖ :=
α
α
converges with radius of convergence
α
α
α
‖ α ‖ → ∞
of the multidimensional power series
Then This is a power series in one variable
Therefore, by the Cauchy–Hadamard theorem for one variable Setting
‖ α ‖ = μ