going radians counterclockwise around , first crossing the positive real axis (of the level) into the shared copy of the upper half plane and then crossing the negative real axis (of the level) into the level of the lower half plane. One can continue by gluing branches with imaginary part in , in , and so on, and in the other direction, branches with imaginary part in , in , and so on. The final result is a connected surface that can be viewed as a spiraling parking garage with infinitely many levels extending both upward and downward. This is the Riemann surface associated to . A point on can be thought of as a pair where is a possible value of the argument of . In this way, can be embedded in . The logarithm function on the Riemann surface Because the domains of the branches were glued only along open sets where their values agreed, the branches glue to give a single well-defined function . It maps each point on to . This process of extending the original branch by gluing compatible holomorphic functions is known as analytic continuation. There is a "projection map" from down to that "flattens" the spiral, sending to . For any , if one takes all the points of lying "directly above" and evaluates at all these points, one gets all the logarithms of . Gluing all branches of log z Instead of gluing only the branches chosen above, one can start with all branches of , and simultaneously glue every pair of branches and along the largest open subset of on which and agree. This yields the same Riemann surface and function as before. This approach, although slightly harder to visualize, is more natural in that it does not require selecting any particular branches. If is an open subset of projecting bijectively to its image in , then the restriction of to corresponds to a branch of defined on . Every branch of arises in this way. The Riemann surface as a universal cover The projection map realizes as a covering space of . In fact, it is a Galois covering with deck transformation group isomorphic to , generated by the homeomorphism sending to . As a complex manifold, is biholomorphic with via . (The inverse map sends to .) This shows that is simply connected, so is the universal cover of