Talk:Cauchy–Hadamard theorem

What the fuck is this: "The proof can be found in the book Introduction to Complex Analysis Part II functions in several Variables by B.V.Shabat"

I don't thinks this is the place for publicity

Multivariable version reference?

Could we get a reference for the multivariable version of the theorem? 218.208.8.112 (talk) 02:50, 28 September 2020 (UTC)

I added a proof 121.121.60.56 (talk) 04:48, 1 October 2020 (UTC)

The proof does not include the cases when R=0 or R=+infinity. 160.216.151.232 (talk) 17:36, 20 April 2023 (UTC)

What is a multi-index raised to a multi-index power?

The section Theorem for several complex variables contains this passage:

"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"

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

to the multidimensional power series

$$\sum _{\alpha \geq 0}c_{\alpha }(z-a)^{\alpha }:=\sum _{\alpha _{1}\geq 0,\ldots ,\alpha _{n}\geq 0}c_{\alpha _{1},\ldots ,\alpha _{n}}(z_{1}-a_{1})^{\alpha _{1}}\cdots (z_{n}-a_{n})^{\alpha _{n}}$$

"

I can try to guess what 𝝆^𝛼 might be.

But it is much, much, much better if the article explained exactly what 𝝆^𝛼 means, where both 𝝆 and 𝛼 are multi-indexes.

I hope someone knowledgeable about this subject can insert a definition.

I have added a definition of both. I removed the reference to "multi-index" and replaced it with the concept of a vector, which I hope is more precise. I changed the notation from $|\alpha |$ to $||\alpha ||$ to avoid the confusion with $|c_{a}|$ which mean different things as I believe $c_{a}$ is not a vector. I found this notation used in Fuks Theory of Analytic Functions of Several Complex Variables which you can find in the openlibrary pp. 48. It is an old text, so I hope the notation is not out of date. I am not 100% sure that $\alpha$ is a vector of natural numbers. Dom walden (talk) 08:05, 7 April 2024 (UTC)