# Branches of log [closed]

I don't really understand what branches of log are in complex analysis. The definition I have is that 'a branch of log $$z$$ in G is a continuous function $$l$$ in G such that, for each $$z$$ in G, the value $$l(z)$$ is a logarithm of z.

I know this definition, but somehow still have a hard time understanding the concept. I may be overthinking it... I thought a branch of log z was essentially an identical log of z, just rotated around the plane by a constant times 2π. So just the same log z, but on top or underneath the original one (if you graphed it using $$\theta$$ as the z axis). Is this correct? Does this mean that there is only one branch in each [0,2π] interval? (or is this interval open?)

## closed as unclear what you're asking by A. Goodier, Lord Shark the Unknown, Leucippus, Lee David Chung Lin, CesareoMay 9 at 11:42

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• What don't you understand from the Wikipedia page on this subject? – David G. Stork May 8 at 20:25
• It is equivalent to defining the polar coordinate $\theta$ in $z = r\exp(i\theta)$. The most general definition involves an arbitrary non-self intersecting oriented curve that starts at the origin and moves away to infinity that defines the branch cut across which $\theta$ will jump by $2\pi$. – Count Iblis May 8 at 20:33
• Do you understand $\arcsin$ or $\arccos$? – qbert May 8 at 21:29
• The common branches are for $t \in (a,a+1), r \in (0,\infty), L_a(r e^{2i\pi t}) = r+2i\pi t$. But there are many more : for $t \in (0,\infty)$ and $r \in (2^t,2^{t+1})$ let $\ell(r e^{2i \pi t}) = r+2i\pi t$, this is a weird branch of $\log$ whose domain of analyticity and continuity is a spiral. – reuns May 8 at 22:50

We seek an inverse $$f(z)$$ to the exponential function. So we need $$f \circ \exp(z)=z$$. Since $$\exp{z}= \exp{(z+2i\pi)}$$ this identity cannot be simultaneously satisfied by $$z$$ and $$z'$$ if $$z-z'$$ is a multiple of $$2\pi i$$. Hence we have to assume that the domain contains only one representative of each equivalence class $$z+2i\pi \mathbb{Z}$$.