Riemann surface

The arcsine function can be viewed as a „multivalued“ map $\mathbb C \to \mathbb C$, or better as a single-valued map from the surface $\{(z,w) \in \mathbb C^2\mid \sin z=w\}$ by projection on the first coordinate.

This understanding of complex functions was Riemann's habilitation in 1855 under the supervision of C.F. Gauss.