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I want to know the researth results of Weierstrass in 1841, where he proved some theories about Laurent series. Can you help me find it? I'd appreciate it if you can do me a favor!:)

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    $\begingroup$ You need to give more information to go on. What theorems are these? Where did you get the year 1841? $\endgroup$
    – Somos
    Commented Apr 15 at 3:27
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    $\begingroup$ Cf. this $\endgroup$ Commented Apr 15 at 3:31
  • $\begingroup$ It comes from the annotation of a book about complex analysis@Somos $\endgroup$
    – makabaka
    Commented Apr 15 at 3:35
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    $\begingroup$ @makabaka: How do you think those comments are helpful? Can you give a precise quote of the reference? If you are unable to do that, could you at least tell us which "book about complex analysis" that would be? $\endgroup$ Commented Apr 15 at 4:52
  • $\begingroup$ The best source of information on Weierstrass's notes would be Detlef Spalt. He recently published a book on the history of analysis, so he is still active. $\endgroup$ Commented Apr 15 at 9:03

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As some have remarked, indeed the information is sparse; nevertheless, I will venture a response nonetheless:

«Zur Theorie der Potenzreihen», in Karl Weierstrass, Georg Hettner, Johannes Knoblauch e Rudolf Ernst Rothe, Mathematische Werke, Mayer & Müller, 1894, vol. 1, p. 67-74

I am familiar with this manuscript because Giuseppe Peano mentions it in the Formulario Mathematico (although it might be more appropriate to say Giovanni Vacca, as he was responsible for the historical notes). It is cited as the first known written work to use $|x|$ to denote the absolute value of $x$, and it is also stated that Weierstrass was the first to employ the concept of uniform convergence ("gleichmässig") in that manuscript.

With that said, I add for completeness that there is another manuscript from 1841 on Laurent series:

«Darstellung einer analytischen Function einer complexen Veränderlichen, deren absoluter Betrag zwischen zwei gegebenen Grenzen liegt», in Karl Weierstrass, Georg Hettner, Johannes Knoblauch e Rudolf Ernst Rothe, Mathematische Werke, Mayer & Müller, 1894, vol. 1, p. 51-66

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