Undholomorphic

Undholomorphic is a term used in certain branches of mathematics, particularly in complex analysis and differential geometry, to describe functions or forms that exhibit a specific type of regularity or degeneracy. It is essentially the negation of being holomorphic, which refers to functions that are complex differentiable. A function that is not holomorphic is, therefore, undholomorphic.

The concept of undholomorphicity can arise in various contexts. For instance, a complex-valued function of a complex variable that fails the Cauchy-Riemann equations at a point is undholomorphic at that point. In the study of differential forms, an undholomorphic form is one that does not satisfy certain differential conditions related to holomorphicity, such as being of a specific type (e.g., a (p,0) form on a complex manifold).

The study of undholomorphic functions or forms is often concerned with understanding the nature of their singularities or deviations from holomorphic behavior. Analyzing these deviations can reveal deeper geometric or analytic properties of the space or object under consideration. While "holomorphic" signifies a desirable and well-behaved property, "undholomorphic" points to a departure from this ideal, requiring further investigation to characterize the specific nature of this departure.