Why does the integer $k$ depend on $z$ for the branch of the logarithm

I know that a branch of the logarithm is a continuous function on a region $G$ in $\mathbb{C}$, $$f:G\rightarrow \mathbb{C}$$ that satisfies $z = e^{f(z)}$ for all $z\in G$.

I was reading through my notes and I came across the following piece of theory:

If $f$ and $g$ are branches of the logarithm, then for each $z$ there exists $k_z\in \mathbb{Z}$ such that $$f(z) = g(z) +2\pi i k_z.$$

However I don't understand why this $k_z$ depends on $z$. I know the different branches of the logarithm are given by, $$h(z) = \ln(r e^{i(\theta+2\pi k)}) = \ln(r)+i(\theta+2\pi k)$$ where each branch has a fixed $k$.

I am trying to reason about it like this:

Let $z_1,z_2 \in G$. Suppose in polar form that $z_1 = r_1e^{i\theta_1}$ while $z_2 = r_2e^{i(\theta_2)}$.

Now fix the branches $f,g$. So that we have $k_1,k_2 \in \mathbb{Z}$ are fixed integers and $$f(z) = ln(r) + i(\theta +2\pi k_1)$$ $$g(z) = ln(r) + i(\theta +2\pi k_2)$$

Now $$f(z_1)-g(z_1) = 2\pi i (k_1-k_2),$$ $$f(z_2) - g(z_2) = 2\pi i (k_1-k_2)$$ as well. So there is still no dependence on $z$.

Which leads me to ask, where does this dependence on the $z$ come from?

As an aside, I am aware that we there is a $k$ that works for all $z$, I am just trying to understand the case where $k$ depends on $z$.

Any two branches of $\log z$ which are defined in the same region $G\subset{\mathbb C}$ differ by a constant $2 \pi ik$, $k\in{\mathbb Z}$, througout $G$. If, however, $f$ and $g$ are defined in different regions, e.g. $f$ in the right halfplane $H$ and $g$ in the region $G:={\mathbb C}\setminus \bar\gamma$, where $\bar\gamma$ is the logarithmic spiral $r=e^\phi$, $-\infty<\phi<\infty$, together with the origin, then all you can say is that any point $z\in H\cap G$ has a neighborhood $U_z$ in which $f$ and $g$ differ by a constant $2\pi ik_z$.
• Thanks for the response. What does $\bar{\gamma}$ denote? Aug 28 '16 at 9:31