# can we define logarithm function $\log$ on $\mathbb C((T))$

Let $\mathbb C[[T]]$ be the ring of formal power series in one formal variable $T$, and $\mathbb C((T))$ be its fraction field. At first we can definitely define the exponential function $\exp:\mathbb C((T))\to \mathbb C((T))$ by $$f\mapsto \exp(f):= \sum_{k=0}^{\infty}\frac{f^k}{k!}$$ However it seems there are troubles in defining $\log$. For example, for any $a$ we can naively 'put' $\log(f)=\log(a)+\log(1+\frac{f-a}{a})$ and in a similar way we may use the Taylor expansion to the last term.

Note that a meaningful definition of $\log$ should satisfy $\log \circ \exp =id$ or $\exp \circ \log=id$.

So, it seems that the definition depends on $a$, right or not? Let me put the question simpler: is it possible to define $\log(T)$ in $\mathbb C((T))$?

In fact, you can't define the formal exponential even in $$\Bbb C[[T]]$$. If $$f(T) = a_0 + a_1T + \cdots$$ with $$a_0\ne 0$$, the 0-th term of $$\exp(f)$$ is the infinite sum $$\sum_{k=0}^{\infty}\frac{a_0^k}{k!}.$$
• But we are working with $\mathbb C$, and this is just $e^{a_0}\in \mathbb C$, right? – Hang Jul 23 '18 at 17:44