<Dd> 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 . </Dd> <P> 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 . </P> <P> Another example: The Mercator series </P> <Dl> <Dd> <Dl> <Dd> <Dl> <Dd> ln ⁡ (1 + u) = ∑ n = 1 ∞ (− 1) n + 1 n u n = u − u 2 2 + u 3 3 − ⋯ (\ displaystyle \ ln (1 + u) = \ sum _ (n = 1) ^ (\ infty) (\ frac ((- 1) ^ (n + 1)) (n)) u ^ (n) = u - (\ frac (u ^ (2)) (2)) + (\ frac (u ^ (3)) (3)) - \ cdots \,) </Dd> </Dl> </Dd> </Dl> </Dd> </Dl>

Absolute value function to arguments of logarithm functions