As the author of the books states;
Suppose that D is simply connected and that $0 \notin D $. Choose $z_0\in D$, fix a value of $\log z_0$ and set $ f\left(z\right)=\intop_{z_{0}}^{z}\frac{d\zeta}{\zeta}+\log z_{0} $.
Then $ f $ is an analytic branch of $\log z$ in D.
Now, what I dont get is the meaning of "fix a value of $\log z$"
, can I choose any complex number? what is this $\log z_0$ ? It seems like the author defining $\log $ but using another $\log $ in the definition.
In the proof of this theorem, in order to show that $ f $ is an inverse of the exponent he write:
$ g\left(z\right)=ze^{-f\left(z\right)} $
so $ g'\left(z\right)=0 $ and then he concludes that $ g $ is a constant
And then he says $ g\left(z_{0}\right)=z_{0}e^{-f\left(z_{0}\right)}=1 $
Which means he is using $ e^{\log z_{0}}=z_{0} $ which I dont get, because what is this $\log z_0$? if it's just an arbitrary complelx number why would it satisfy this equation?
(Im trying to understand the meaning of $\log z_0 $ for complex $z_0 $ since this is the very first chapter when the author explains what $\log $ is in the complex plane, but the usage of another $\log $ in the definition confuses me.
Thanks in advance.
Best Answer
For $z_0 \ne 0$ the equation $e^{w_0} = z_0$ has infinitely many solutions. “Fix a value of $\log z_0$” means: Choose one solution $w_0$ of that equation (it does not matter which one) and call it $\log z_0$.
You could also say: Let $w_0$ be any solution of $e^{w_0} = z_0$, and define $f$ as $f(z) = w_0 + \int_{z_0}^z \frac{d\zeta}{\zeta}$.
Then $f(z_0) = w_0$ and $g(z_0) = z_0e^{-w_0} = 1$. It follows that $g(z) \equiv 1$ and therefore $e^{f(z)} = z$ for all $z \in D$.