When I was a student, the descriptions of multi-valued complex graphs obtained by suitably-joined slit planes always rankled, at least a little.

[diffgeom@mathstodon.xyz]

When I was a student, the descriptions of multi-valued complex graphs obtained by suitably-joined slit planes always rankled, at least a little.

This unit circle \(z^{2} + w^{2} = 1\) with a portion of the \(z\)-plane at bottom, animated to show three loops lifted continuously, depicts all the Standard Business about holomorphic branches of \(\sqrt{1 - z^{2}}\) in singly- or doubly-slit planes with what accuracy 3-space can accommodate. (The imaginary part of \(w\) is projected away.)

Particularly, the lift of the large loop (encircling both points \(z = \pm 1\)) lies on a holomorphic branch of \(\sqrt{1 - z^{2}}\) defined in the slit plane \(\mathbf{C} \setminus [-1, 1]\).

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