Does $\log z$ have a laurent series about $0$?

Question

I think not because $\log z$ isn't analytic on any neighborhood of 0. Is this correct? thanks

Answer 1

I would use Tristan Needham's approach found in his book "Visual Complex Analysis" to define the formal Laurent series of the complex logarithm as $$\forall n\in\mathbb{Z}, \ c_n = \frac{1}{2\pi i}\oint_0 \frac{\log(z)}{z^n}\frac{dz}{z}$$ and the formula would read $$\log(x)=\sum_{n\in\mathbb{Z}} c_n x^n.$$

Whether this contour integral works or not for any $n\in\mathbb{Z}$ seems to be solved nowadays, regarthless it exists for $n=-1$. We would need to find these explicit values $c_n.$

Nevertheless, everyone says there does not exists a Laurent series expansion for logarithm, for example:

Now what about log(z)? That one has singular points called "logarithmic branch points" of infinite order since circuits around the origin or infinity around the function never return to the starting value. These too are not poles since they also are not issolated and so therefore do not have Laurent series expansions about their singular points.

Source https://www.physicsforums.com/threads/log-z-singularity-series-and-pole.601750/