Elliptic Integrals and arithmetic-geometric mean

In this video, we show that the arithmetic-geometric mean of two positive real numbers can be evaluated using an elliptic integral.

This result was first conjectured by Gaus after intensive calculations and he correctly predicted in his diary that it would open up a whole new field of analysis.

TIMESTAMPS

00:00 Introduction
02:39 Existence of AM-GM mean
06:27 Invariance of integral under AM-GM substitution
14:48 Finishing the proof