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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?

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Here are the relevant passages of Gauss' letter:

... What should one think of $\int \phi xdx$ for $x = a + bi$? Obviously ... $x$ has to go by infinitely small increments (each one of the form $\alpha + \beta i$) from the value where the integral shall be $0$ to $x = a + bi$, and then we sum up all $\phi xdx$. ... The continuous transition from one value of $x$ to another $a + bi$ is done by means of a line and therefore is possible in infinitely many ways. I claim now that the integral $\int \phi xdx$ after two different transitions always takes the same value if within the area between the lines representing these transitions nowhere $\phi x = \infty$. This is a beautiful theorem the not very difficult proof of which I will give at an appropriate occasion. It is connected with other nice truths concerning the expansion of series. Therefore I demand to evade points where obviously the original basic notion of $\phi x$ becomes unclear and leads to contradictions. By the way, it is clear how a function generated by $\int \phi xdx$ for one and the same $x$ can have several values, namely in that during the transition we can not or once or several times go around a point where $\phi x = \infty$. Defining, for example, $log x$ by $\int (1/x)dx$, to start at $x=1$, then one gets to $logx$ either without including the point $x=0$ or by going around one or several times. Every time the constant $+2\pi i$ or $-2\pi i$ is added. This makes clear that every number has many logarithms. If $\phi x$ can never become infinite for a finite $x$ then the integral is always single-valued.

This makes clear that Gauss was in possession of the "not difficult to prove" theorem and that the logarithm is only an example for an application, not the basis of his rule.

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  • $\begingroup$ O.K I appreciate your answer to the first part of my question , but what about the second part of it? - what do the other functions and power series mean? and did Gauss knew how to develope a complex function into a Laurent series (did he know how to calculate the coefficient $a_1$?)? i have no doubt that Gauss understood a lot about the multivalued nature of complex functions around singularities, it can be inferred from the second (unpublished) part of his treatise on the hypergeometric function. $\endgroup$ – user2554 Jun 19 '17 at 12:14
  • $\begingroup$ By the way, if you know how to read german, should it be a problem for you to translate the entire letter? $\endgroup$ – user2554 Jun 19 '17 at 12:16
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For the completeness of the discussion in this post i must add a link to another relevant question -Meaning of passages by Gauss on the "convergence of expansions (in infinite series) of the (elliptical) equation of the center"? . The answer to this question, if interpreted appropriately, indicates that already in 1805 Gauss had deep insights on the meaning and applicability of integrals in the complex plane; ideas which he used to solve the astronomical problem of finding explicit expression for the coefficients of the fourier series of the "equation of the centre" - the difference between mean and true anomaly of a planet in a keplerian orbit.

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