quality:corruption, information:classification, structural:summarization, stylistic:tone
Constructing branches via integration

The function ln (x) {\displaystyle \ln(x)} for real x > 0 {\displaystyle x>0} can be constructed by the formula ln (x) = ∫ 1 x d u u. {\displaystyle \ln(x)=\int _{1}^{x}{\frac {du}{u}}.} If the range of integration started at a positive number a {\displaystyle a} other than 1, the formula would have to be ln (x) = ln (a) + ∫ a x d u u {\displaystyle \ln(x)=\ln(a)+\int _{a}^{x}{\frac {du}{u}}} instead.

In developing the analogue for the complex logarithm, there is an additional complication: the definition of the complex integral requires a choice of path. Fortunately, if the integrand is holomorphic, then the value of the integral is unchanged by deforming the path (while holding the endpoints fixed), and in a simply connected region U {\displaystyle U} (a region with "no holes"), any path from a {\displaystyle a} to z {\displaystyle z} inside U {\displaystyle U} can be continuously deformed inside U {\displaystyle U} into any other. All this leads to the following: