entire function

An entire function is a holomorphic function with no exceptions and no boundaries — one that is complex differentiable at every single point of the complex plane, with no singularities anywhere. It is, so to speak, holomorphic 'all the way out to infinity' on the finite plane. The simplest examples are the polynomials, but the real stars are the transcendental ones like the exponential, sine and cosine, whose behavior at large distances is far more interesting.

Formally, f is entire if it is holomorphic on all of C. Equivalently, f equals a single power series sum a_n z^n that converges for every complex z, that is, with infinite radius of convergence. The constant function, every polynomial, e^z, sin z, cos z, and combinations and compositions thereof are entire.

Entire functions are governed by powerful global theorems precisely because there is nowhere for them to misbehave. Liouville's theorem says a bounded entire function is constant — so any non-constant entire function must grow without bound somewhere. The growth rate as |z| -> infinity (the 'order' of the function) controls the distribution of its zeros via deep results of Hadamard and Weierstrass. A caveat worth stating: 'entire' refers only to the finite plane; functions like e^z have a genuinely wild essential singularity at infinity, which is why e^z is not a polynomial.