Talk:Radius of Convergence of Power Series over Factorial

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Versions for Complex Numbers

Since I intend to upload some results about the complex exponential functions, I need some theorems that ProofWiki only has proved for real numbers. I intend to generalize them to $\C$. These theorems are:

My question is: Do I just alter the theorems and proofs so they hold for $\C$ as well as $\R$, or should I give each theorem a new page for complex numbers? Of course, you can give a different answer for each theorem. --Anghel (talk) 12:32, 17 January 2013 (UTC)

I suggest an approach like on Definition:Power Series and Definition:Radius of Convergence. Incidentally I like the complex approach on the second page better than the real approach (which seems flawed because it is a theorem that the disc of convergence is actually a disk (whether in one or two dimensions) about $\xi$). It is quite customary to let the radius of convergence attain $+\infty$, and $\overline \R$ gives a rigid foundation for this use of the lemniscate. I suggest implementing it for reals as well, and adjusting/formulating theorems to be compliant with this use.
In conclusion, I think my answer is about mid-way between adapting the proof and making a new page. --Lord_Farin (talk) 12:56, 17 January 2013 (UTC)
I've never come up with a solution which is 100% satisfactory to me, let alone anyone else so I'll defer to L_F here. As long as we still have a proof which is accessible to anyone who wouldn't know what a complex number was if it bit them. --prime mover (talk) 12:58, 17 January 2013 (UTC)
Then, my approach will be to write new pages if the theorem or its proof involve anything more complicated than replacing absolute value with complex modulus. --Anghel (talk) 21:03, 17 January 2013 (UTC)
Fine with me; the Nth Root Test is a good example of your approach working out well. --Lord_Farin (talk) 21:51, 17 January 2013 (UTC)