Cauchy–Hadamard theorem

Statement of the theorem

Consider the formal power series in one complex variable z of the form

where a , c n ∈ C .

Then the radius of convergence of ƒ 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 of the theorem

Without loss of generality assume that a = 0 . We will show first that the power series ∑ 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 ε > 0 , there exists only a finite number of n such that | c n | n ≥ t + ε . Now | c n | ≤ ( t + ε ) n for all but a finite number of n , so the series ∑ c n z n converges if | z | < 1 / ( t + ε ) . This proves the first part.

Conversely, for ε > 0 , | c n | ≥ ( t − ε ) n for infinitely many c n , so if | z | = 1 / ( t − ε ) > R , we see that the series cannot converge because its nth term does not tend to 0.

Statement of the theorem

Let α be a multi-index (a n-tuple of integers) with | α | = α 1 + ⋯ + α n , then f ( x ) converges with radius of convergence ρ (which is also a multi-index) if and only if

to the multidimensional power series

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