Entiere

Entière, in mathematics, is the term used in French (and in related contexts) to describe a function that is holomorphic on the entire complex plane. In English mathematics, the standard term is “entire function.” Equivalently, a function f: C → C is entire if it is complex differentiable at every point of the complex plane, and this implies that around any a ∈ C there exists a power series expansion f(z) = ∑_{n=0}^∞ c_n (z − a)^n with infinite radius of convergence.

Examples of entire functions include polynomials, the exponential function e^z, sine and cosine functions sin z and cos z, and many other transcendental functions. Entire functions are analytic everywhere and hence infinitely differentiable.

Key properties include that the zeros of a non-identically-zero entire function are isolated. Liouville’s theorem states that any bounded entire function must be constant. The behavior of an entire function at infinity is described by whether it is a polynomial (infinity is a pole) or a non-polynomial (infinity is an essential singularity). Entire functions can be studied by their growth (order and type) and by factorization results such as Hadamard’s factorization.

Terminology and usage: the term entiere is the feminine form used when describing a feminine noun, such as “fonction entière” in French; in English texts, the term “entire function” is standard. The concept is central in complex analysis, providing a clean setting in which holomorphic behavior is understood globally on the complex plane.