Unlocking Complex Functions: The Power of Analytic Continuation 🔍

Where do complex functions come from? In this video we explore the idea of analytic continuation, a powerful technique which allows us to extend functions such as sin(x) from the real numbers into the complex plane. Using analytic continuation we can finally define the zeta function for complex inputs and make sense of what it is the Riemann Hypothesis is claiming.

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Links:
Blog post by Terry Tao: https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/

Mathologer: Ramanujan: Making sense of 1+2+3...=-1/12 and Co: https://www.youtube.com/watch?v=jcKRGpMiVTw

Chapters:
00:00 zetamath does puzzles
00:23 Recap
02:40 Bombelli and the cubic formula
08:45 Evaluating real functions at complex numbers
12:33 Maclaurin series
21:22 Taylor series
27:19 Analytic continuation
35:57 What goes wrong
48:19 Next time

Animations in this video were created using Manim Community. For more information, visit https://manim.community

Thanks to Keith Welker for our theme music.