Defininition of the complex integral $\int_{z_0}^{z_1}f(z) \, dz$.

In Bak and Newman they define the integral of a complex function if the bounds of integration are real. Then they define the complex path integral for a path $C$ in the complex plane. Then, in a fairly well-hidden remark in the middle of a proof, define

$$\int_0^z f(z)\ dz$$

to denote the path integral from $0$ to the real part of $z$ and then from the real part of $z$ to $z$. They never give a full definition of

$$\int_{z_0}^{z_1} f(z)\ dz$$

although the suggestion about what this should be is clear enough.

But then they define on a simply connected domain not containing $0,$

$$\log z = \int_{z_0}^z \frac{d\zeta}{\zeta} + \log z_0$$

for an arbitrary choice of $\log z_0$. I believe it should be possible to compute $\log (i-1)$ for the principle branch, and yet this would seem to require integrating $\int_1^{-1}\frac{dx}{x}$ which does not converge.

Do you think the integral is supposed to mean "integrate along any path in the domain"?

(Note this isn't quite how they define the complex log, as they use analyticity and being the inverse of the exponential. However, they immediately prove a related theorem and claim that the above so-to-speak generates log functions, and so in a sense defines log functions.)

If they're bothering to specify a simply-connected domain not containing $0$, it would make no sense to use an integral over a path that contains $0$. Yes, I think it should be an integral over a path in the domain from $z_0$ to $z$.