Complex Logarithm - Branches of The Complex Logarithm

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) that is continuous on all of ? Unfortunately, the answer is no. To see why, imagine tracking such a logarithm function along the unit circle, by evaluating L at eiθ as θ increases from 0 to 2π. For simplicity, suppose that the starting value L(1) is 0. Then for L(z) to be continuous, L(eiθ) must agree with iθ as θ increases (the difference is a continuous function of θ taking values in the discrete set ). In particular, L(e2πi) = 2πi, but e2πi = 1, so this contradicts L(1) = 0.

To obtain a continuous logarithm defined on complex numbers, it is hence necessary to restrict the domain to a smaller subset 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 should be an open set. Also, it is reasonable to assume that U is connected, since otherwise the function on different components of U would be unrelated to each other. All this motivates the following definition:

A branch of log z is a continuous function L(z) defined on a connected open subset U of the complex plane such that L(z) is a logarithm of z for each z in U.

For example, the principal value defines a branch on the open set where it is continuous, which is the set obtained by removing 0 and all negative real numbers from the complex plane.

Another example: The Mercator series

\log(1+u)=\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} u^n = u - \frac{u^2}{2} + \frac{u^3}{3} - \cdots \,

converges locally uniformly for |u| < 1, so setting z = 1+u defines a branch of log z on the open disk of radius 1 centered at 1. (Actually, this is just a restriction of 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" 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" to have a precise unambiguous meaning.

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