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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 $\mathbb C((T))$?

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Too long for a comment:

At first we can definitely define the exponential function...

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 $$\sum_{k=0}^{\infty}\frac{a_0^k}{k!}.$$

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  • $\begingroup$ But we are working with $\mathbb C$, and this is just $e^{a_0}\in \mathbb C$, right? $\endgroup$ – Hang Jul 23 '18 at 17:44
  • $\begingroup$ @Hang, formal series means always finite sums. $\endgroup$ – Martín-Blas Pérez Pinilla Jul 24 '18 at 8:34

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