My question refers to Gauss's 1811 letter to Friedrich Bessel, which contains a statement of Cauchy integral theorem. I have access to the letter, but I'm unable to read it. I know that Gauss gives the example of the multivalued nature of the line integral of $1/x$ around $x = 0$, and gives the correct reside of $2\pi i$. This is a result of a straightforward integration along infinitesimal arcs (which are described as infinitesimal complex mumbers) which gives a differential of $e^{{-i\theta}}\cdot e^{{i(\theta + \pi/2)}}\,d\theta = i\,d\theta$. But this could also be understood from the fact that the integral of $1/x$ is $\log x$, which is of course a multivalued function. I don't understand enough the subject of complex analysis to infer what these facts imply about Gauss's understanding of the resides theorem.
The point is that the example which Gauss gives can be viewed another way; he gives the integral of $1/x$ because the key to evaluating resides of line integrals around singularities is the coefficient $a_1$ of the $-1$ power term in the Laurent series of the complex function. Therefore the question that remains is whether Gauss knew Laurent series? I saw in his letter to bessel the function $(e^x - 1)/x$ as well as certain power series , but what do these power series really mean?