Holomorphic
/ˌhɑləˈmɔrfɪk/adjective
Of a complex function, differentiable at every point of a region.
The Greek roots
Literally: “whole form”
The story of the word
Charles Briot and Jean-Claude Bouquet introduced holomorphe and méromorphe in their work on elliptic functions in the 1870s, and the pair still operate together: holomorphic means well behaved throughout a region, meromorphic means well behaved except at isolated poles. The condition is far stronger than it looks, since a function differentiable once in the complex plane is automatically differentiable forever. That same ὅλος hides inside catholic, from κατὰ ὅλου, on the whole.
- How it travelled
- Ancient Greek → French mathematical coinage → English
- First recorded
- 19th century
Same family
In a sentence
If the function is holomorphic on the disc, the integral around any closed loop inside it is zero.
Polynomials are holomorphic everywhere; the logarithm is not.