Why is it called a holomorphic function?

Solution 1:

Not yet a definitive answer, but this should narrow the search.

The Wikipedia article tells us the term "holomorphic function" (fonction holomorphe in french) was coined by Briot and Bouquet.

Actually, they published together several works on elliptic functions, so we may investigate this a bit. Here are two references:

• 1858, Théorie des fonctions doublement périodiques et, en particulier, des fonctions elliptiques
• 1875, Théorie des fonctions elliptiques, deuxième édition

In both books, they introduce several kinds of functions in the begining, and then procede to study periodic and doubly periodic functions. The vocabulary has changed completely in the meantime:

• A function whose values do not depend on the path followed by the complex variable is called monodrome in the first book, monotrope in the second.
• A function which has a complex derivative is called monogène in the first book, no term is given in the second.
• A function which is both monodrome and monogène is called synectique in the first book, while a function which is both monotrope and has a complex derivative is called holomorphe (holomorphic) in the second.

This recent book on the history on noneuclidean geometry tells us here that Cauchy coined both terms monogène and holomorphe. However, Cauchy died in 1857, so it looks awkward that Briot and Bouquet changed the terms after the first edition, following Cauchy who was already dead. On the other hand, given that the terminology wasn't apparently completely stabilized, Cauchy might have given the vocabulary earlier.

See also this question on HSM.SE: First papers on holomorphic functions.