03/12/2023, 06:56 AM
I'm going to hold off on talking about this too much at the moment. But I believe I have a way to relate "the most beautiful" extension--as you so called it. And the manner I refer to this as, is we can reconstruct \(f(z)\) for \(|z|>1\) using \(|z| <1\), and vice versa, using nothing but Cauchy's integral theorem...
I don't want to muddy the pond too much by discussing details I haven't worked out yet. I'm about 8 pages in to a well written write up (expecting about 30-35 pages); but for restricted cases: \(L(z) : \mathbb{C}/\mathcal{U} \to \mathbb{C}\) where \(\mathcal{U}\) is the unit disk, I believe I've found your "beautiful" extension. I cannot speak of the more modular functions which have poles within the disk. Though I imagine it won't take much to generalize the work.
Also, super nice pictures
I love graphing complex functions! I wish Pari/gp had better onboard graphing protocols. Though, I've made mike3's graphing program fairly similar in philosophy to the desmos protocol you are using.
Super exciting topic, Caleb. Very happy you've started to discuss this! I'm learning a lot and having the time of my life. Reminding me of my number theory days!
Sincere, Regards
I don't want to muddy the pond too much by discussing details I haven't worked out yet. I'm about 8 pages in to a well written write up (expecting about 30-35 pages); but for restricted cases: \(L(z) : \mathbb{C}/\mathcal{U} \to \mathbb{C}\) where \(\mathcal{U}\) is the unit disk, I believe I've found your "beautiful" extension. I cannot speak of the more modular functions which have poles within the disk. Though I imagine it won't take much to generalize the work.
Also, super nice pictures
I love graphing complex functions! I wish Pari/gp had better onboard graphing protocols. Though, I've made mike3's graphing program fairly similar in philosophy to the desmos protocol you are using. Super exciting topic, Caleb. Very happy you've started to discuss this! I'm learning a lot and having the time of my life. Reminding me of my number theory days!
Sincere, Regards