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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?)

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

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    $\begingroup$ What don't you understand from the Wikipedia page on this subject? $\endgroup$ – David G. Stork May 8 at 20:25
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    $\begingroup$ 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 $2\pi$. $\endgroup$ – Count Iblis May 8 at 20:33
  • $\begingroup$ Do you understand $\arcsin$ or $\arccos$? $\endgroup$ – qbert May 8 at 21:29
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    $\begingroup$ 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. $\endgroup$ – reuns May 8 at 22:50
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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 $z+2i\pi \mathbb{Z}$.

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