What do we get if we take the unit circle x² + y² = 1 and complexify it? We get the 3D object (manifold) in 4D space given by the equation
z² + w² = 1
where z, w are complex numbers. Here's one way to see its #geometry. (A slowmo animation to follow.) #calculus
z² + w² = 1
where z, w are complex numbers. Here's one way to see its #geometry. (A slowmo animation to follow.) #calculus
Here is a slow animation of how circles of various radii (centered at the origin) are transformed by the complexified circle function. #math #geometry #calculus
N.B., the reason we see two disconnected loops for the image of circles of small radii is one loop corresponds to the + and other to the - choices in the function u(r,θ). (And, yes, the function w is analytic.)
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