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Briot and/or Cauchy are often said to have written the first papers on holomorphic functions, explicitly discussing them as such and their special properties.

Which papers are these? When and where were these published?

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    $\begingroup$ If you can find a copy at a library (or do what I did a few weeks ago, which was to buy a copy, although it set me back a couple of months in book buying due to its cost), you'll want to look at Bottazzini/Gray's recent book Hidden Harmony - Geometric Fantasies: The Rise of Complex Function Theory. $\endgroup$ – Dave L Renfro Apr 28 '15 at 21:31
  • $\begingroup$ I requested it from a university library. Sounds interesting. $\endgroup$ – Guido Jorg Apr 28 '15 at 23:05
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According to F.Klein's "Lectures on the development of mathematics in the 19th century" Cauchy created this theory (by studying the convergence of series in the complex plane) in his Cours d'Analyse from 1821 ( Series 2, Volume 3 of his collected works). A direct predecessor (but with a lack of rigour seems to be Lagrange) and also Klein declares that these things were known at this time - but seemingly as "folklore".

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  • $\begingroup$ Incidentally, had theCours on my to-read list for years, since Truesdell in a book wrote it's worth reading... Now I recall, a book by Painleve from 1900 dated the theory to Cauchy, but in 1840s... Only one way to find out. I'll update this comment once I go through it in the next couple of days. $\endgroup$ – Guido Jorg Apr 28 '15 at 23:29
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The theory of analytic (holomorphic) functions was indeed created by Cauchy. Briot-Bouquet book was also very influential, in particular they introduced the modern terminology ("meromorpic functions" for sure, perhaps also the term "holomorphic", and such things as poles, removable and essential singularities etc.)

Their book is Theorie des fonctions doublement periodiques et en particulier, des fonctions elliptiques, 1859 (first edition). Available free on Internet.

Remark. In the old books they used to distinguish analytic, holomorphic, regular and monogenous functions. Nowadays this is all the same. Some people like analytic, others holomorphic, while regular and monogenous are out of date.

Cauchy's papers on holomorphic functions start in 1814 and continue to the 1840-th. Wikipedia says that he "created this theory single handed" and most mathematicians would agree. You can see some references on Cauchy's early work in Wikipedia, and I suppose his collected papers are available online (He was only slightly less prolific than Euler:-)

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