Complex logarithm
Logarithms of holomorphic functions

If f is a holomorphic function on a connected open subset U {\displaystyle U} of C {\displaystyle \mathbb {C} }, then a branch of log f {\displaystyle \log f} on U {\displaystyle U} is a continuous function g {\displaystyle g} on U {\displaystyle U} such that e g (z) = f (z) {\displaystyle e^{g(z)}=f(z)} for all z {\displaystyle z} in U {\displaystyle U}. Such a function g {\displaystyle g} is necessarily holomorphic with g ′ (z) = f ′ (z) / f (z) {\displaystyle g'(z)=f'(z)/f(z)} for all z {\displaystyle z} in U {\displaystyle U}.

If U {\displaystyle U} is a simply connected open subset of C {\displaystyle \mathbb {C} }, and f {\displaystyle f} is a nowhere-vanishing holomorphic function on U {\displaystyle U}, then a branch of log f {\displaystyle \log f} defined on U {\displaystyle U} can be constructed by choosing a starting point a in U {\displaystyle U}, choosing a logarithm b {\displaystyle b} of f (a) {\displaystyle f(a)}, and defining

for each z {\displaystyle z} in U {\displaystyle U}.
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