When CAN'T Math Be Generalized? | The Limits of Analytic Continuation

There's often a lot of emphasis in math on generalizing concepts beyond the domains where they were originally defined, but what are the limits of this process? Let's take a look at a small example from complex analysis where we actually have the tools to predict when this is impossible.

This video is a participant in the third Summer of Math Exposition (#SoME3) hosted by 3Blue1Brown to encourage more math content online. To learn more, see this:
3blue1brown.substack.com/p/some3-begins


=Chapters=
0:00 - Intro
1:15 - Extending a Geometric Series
3:39 - Complex Power Series
6:23 - Analytic Continuation
8:30 - Analyzing the Gap Series
11:51 - Visualizing the Gap Series
19:21 - Gap Theorems


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This video was generously supported in part by these patrons on Patreon:
Marshall Harrison, Michael OConnor, Mfriend.

To support future videos, become a patron at www.patreon.com/morphocular
Thank you for your support!


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CREDITS

The music tracks used in this video are (in order of first appearance): Icelandic Arpeggios, Checkmate, Ascending, Orient, Faultlines


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The animations in this video were mostly made with a homemade Python library called "Morpho". It's mostly a personal project, but if you want to play with it, you can find it here:
github.com/morpho-matters/morpholib