# Principal branch of complex logarithm

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?

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 $$2\pi$$

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).
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 $$2\pi$$.