Complex logarithm
Construction

The various branches of log z {\displaystyle \log z} cannot be glued to give a single continuous function log: C ∗ → C {\displaystyle \log \colon \mathbb {C} ^{*}\to \mathbb {C} } because two branches may give different values at a point where both are defined. Compare, for example, the principal branch Log z {\displaystyle \operatorname {Log} z} on C − R ≤ 0 {\displaystyle \mathbb {C} -\mathbb {R} _{\leq 0}} with imaginary part θ {\displaystyle \theta } in (− π, π) {\displaystyle (-\pi,\pi)} and the branch L (z) {\displaystyle \operatorname {L} (z)} on C − R ≥ 0 {\displaystyle \mathbb {C} -\mathbb {R} _{\geq 0}} whose imaginary part θ {\displaystyle \theta } lies in (0, 2 π) {\displaystyle (0,2\pi)}. These agree on the upper half plane, but not on the lower half plane. So it makes sense to glue the domains of these branches only along the copies of the upper half plane. The resulting glued domain is connected, but it has two copies of the lower half plane. Those two copies can be visualized as two levels of a parking garage, and one can get from the Log {\displaystyle {\text{Log}}} level of the lower half plane up to the L {\displaystyle {\text{L}}} level of the lower half plane by going 2 π {\displaystyle 2\pi } radians counterclockwise around 0, first crossing the positive real axis (of the Log {\displaystyle {\text{Log}}} level) into the shared copy of the upper half plane and then crossing the negative real axis (of the L {\displaystyle {\text{L}}} level) into the L {\displaystyle {\text{L}}} level of the lower half plane.

One can continue by gluing branches with imaginary part θ {\displaystyle \theta } in (π, 3 π) {\displaystyle (\pi,3\pi)}, in (2 π, 4 π) {\displaystyle (2\pi,4\pi)}, and so on, and in the other direction, branches with imaginary part θ {\displaystyle \theta } in (− 2 π, 0) {\displaystyle (-2\pi,0)}, in (− 3 π, − π) {\displaystyle (-3\pi,-\pi)}, 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 R {\displaystyle R} associated to log z {\displaystyle \log z}.

A point on R {\displaystyle R} can be thought of as a pair (z, θ) {\displaystyle (z,\theta)} where θ {\displaystyle \theta } is a possible value of the argument of z {\displaystyle z}. In this way, R can be embedded in C × R ≈ R 3 {\displaystyle \mathbb {C} \times \mathbb {R} \approx \mathbb {R} ^{3}}.
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