quality:corruption, structural:expansion
Branches of the complex logarithm

Is there a different way to choose a logarithm of each nonzero complex number so as to make a function L (z) {\displaystyle \operatorname {L} (z)} that is continuous on all of C ∗ {\displaystyle \mathbb {C} ^{*}} ? The answer is no. To see why, imagine tracking such a logarithm function along the unit circle, by evaluating L (e i θ) {\displaystyle \operatorname {L} \left(e^{i\theta }\right)} as θ {\displaystyle \theta } increases from 0 {\displaystyle 0} to 2 π {\displaystyle 2\pi }. If L (z) {\displaystyle \operatorname {L} (z)} is continuous, then so is L (e i θ) − i θ {\displaystyle \operatorname {L} \left(e^{i\theta }\right)-i\theta }, but the latter is a difference of two logarithms of e i θ, {\displaystyle e^{i\theta },} so it takes values in the discrete set 2 π i Z, {\displaystyle 2\pi i\mathbb {Z},} so it is constant. In particular, L (e 2 π i) − 2 π i = L (e 0) − 0 {\displaystyle \operatorname {L} \left(e^{2\pi i}\right)-2\pi i=\operatorname {L} \left(e^{0}\right)-0}, which contradicts L (e 2 π i) − 2 π = L (e 0) − 0 {\displaystyle \operatorname {L} \left(e^{2\pi i}\right)-2\pi =\operatorname {L} \left(e^{0}\right)-0}.

To obtain a continuous logarithm defined on complex numbers, it is hence necessary to restrict the domain to a smaller subset U {\displaystyle U} of the complex plane. Because one of the goals is to be able to differentiate the function, it is reasonable to assume that the function is defined on a neighborhood of each point of its domain; in other words, U {\displaystyle U} should be an open set. Also, it is reasonable to assume that U {\displaystyle U} is connected, since otherwise the function values on different components of U {\displaystyle U} could be unrelated to each other. All this motivates the following definition:

For example, the principal value defines a branch on the open set where it is continuous, which is the set C − R ≤ 0 {\displaystyle \mathbb {C} -\mathbb {R} _{\leq 0}} obtained by removing 0 and all negative real numbers from the complex plane.

Another example: The Mercator series

converges locally uniformly for | u | < 1 {\displaystyle |u|<1}, so setting z = 1 + u {\displaystyle z=1+u} defines a branch of log z {\displaystyle \log z} on the open disk of radius 1 centered at 1. (Actually, this is just a restriction of Log z {\displaystyle \operatorname {Log} z}, as can be shown by differentiating the difference and comparing values at 1.)

Once a branch is fixed, it may be denoted log z {\displaystyle \log z} if no confusion can result. Different branches can give different values for the logarithm of a particular complex number, however, so a branch must be fixed in advance (or else the principal branch must be understood) in order for " log z {\displaystyle \log z} " to have a precise unambiguous meaning.