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Consider the complex plane $\mathbb{C}$ with the following cut along positive imaginary axis: $$G=\mathbb{C}-\{z\in \mathbb{C}: \operatorname{Re}z=0, \operatorname{Im}z\geq 0\}.$$

We see that $G$ is an open and connected set in $\mathbb{C}$. For any $z\in G$ we can write $z=re^{i\theta}$ with $r>0$ and $\theta\in (-3\pi/2,\pi/2)$ and define the complex logarithm by the following formula:

$$\operatorname{Log} z:=\log r+i\theta.$$

Honestly to say, I do not understand the meaning of complex logarithm and its branches in great depth. But is my construction OK?

P.S. I would be very grateful if anyone can explain how the branches of logarithm can be constructed. I read about it in my textbook. But can anyone explain it in simple language?

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Keep always in mind that the logarithm of $y$ is just the number $x$ such that $e^x=y$

On the real numberline, this is quite straightforward. If $y$ is positive, there is a unique $x \in \mathbb{R}$ such that $e^x=y$, therefore $log(y)=x$ If y is zero or negative, there is no such number $x$, thus $log(y)$ in undefined.

On the complex plane, however, for any given $z \neq 0$, there are infinitely many complex numbers $w$ that satisfy $e^w=z$, since the expression $\log z = \log r + i\theta$ allows for \theta to increase in multiples of $2\pi$ with $z$ remaining the same. The multiple branches of the logarithm come from here

In short, the complex logarithm is not really a function in the sense we often think of functions as mappings that assign one unique value to each member of the domain set. It is a function if we restrict ourselves to bands of the complex plane with height $2\pi$

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Your construction is fine.

The idea of branches of the logarithm is that if you start at some point $z$ and let the point wonder along some path in the domain, the angle $\theta$ will vary continuously. If it arrives back at the original point, then you want the value of the angle $\theta$ not to have changed (i.e., you want it to be equal to the original value).

The angle can only be different if the path has completely circled the origin. If you make a branch cut starting at the origin and running off to infinity (no need for it to be a straight ray -- it could even be a spiral (!)), then it will be impossible for the path to encircle the origin when it returns to the original point. Even if it winds around a spiral branch cut and does circle many times, it will eventually have to "unwind" for a net "non-encircling" by the time it returns to the original point.

Various branches of the logarithm are defined by choosing various initial values of the angle $\theta$, which will all differ by an integral multiple of $2\pi$.

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  • $\begingroup$ Your answer looks quite unclear. Could give more details and examples? $\endgroup$
    – ZFR
    Mar 18 '19 at 14:03
  • $\begingroup$ @ZFR : OP specifically asked for an explanation in simple language. That's what I gave. $\endgroup$
    – MPW
    Mar 18 '19 at 14:06

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