We call each interval θ ∈ [ 2πn, 2πn+2π ) over which ln(z) is well defined a "branch"
Different branches correspond to different 𝑅𝑖𝑒𝑚𝑎𝑛𝑛 𝑆𝑢𝑟𝑓𝑎𝑐𝑒𝑠pic.twitter.com/1aoNsZOvaz
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The Natural Logarithm is a strange object in the complex plane
For a given value 𝑧, ln(𝑧) takes on an infinite number of values
If we restrict θ ∈ [0, 2π) then ln(𝑧) is well defined and is said to live in the "principle branch"pic.twitter.com/aXmViDTtC2
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Riemann surfaces can be thought of as deformations of the complex plane
locally, they look like patches of the complex plane, but their global topology can be very different.
Below is the Riemann surface for f(z) = √z.
(images not mine)pic.twitter.com/CQc1RrWqq7
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