the little Picard theorem
/ Picard -> pee-KAR /
How much of the complex plane must a non-constant entire function manage to hit? You might guess it could systematically avoid a whole region — a disk, a line, infinitely many values. The little Picard theorem delivers a startling no. A non-constant entire function takes every complex value, with at most one single exception. Out of the entire infinite plane of possible outputs, an entire function can miss at most one point. This is a vast strengthening of the fact that a non-constant polynomial hits every value (it has a root of p(z) = w for every w); even the wildest transcendental entire function can only ever skip one target.
Precisely: if f is entire and there are two distinct complex values that f never equals, then f is constant. Equivalently, the range of a non-constant entire function is either the whole plane or the whole plane minus exactly one point. The exponential is the extremal example that shows one exception is genuinely possible: e^z is entire and non-constant, yet e^z is never 0 — the value 0 is omitted, and it is the only value omitted (e^z takes every nonzero value, infinitely often). You cannot do better than e^z; no entire function omits two values. The proof is deep, classically routed through the modular function or through Hadamard's factorization plus a clever argument.
Little Picard sits at the summit of growth-and-value theory because it shows that being holomorphic on the whole plane is an extraordinarily strong constraint on the values taken, not just on the smoothness. It dramatically sharpens earlier results: Liouville says a bounded entire function is constant (it omits a whole disk of values), and Casorati-Weierstrass says values get dense near a bad singularity; Picard says they are not merely dense but actually attained, all but one. The honest limit of the statement: 'at most one' is sharp — the single omitted value is real and you must allow for it; and the theorem is about entire functions, the local version near an essential singularity being the great Picard theorem.
The exponential e^z is the textbook witness. It is entire, non-constant, and omits exactly the value 0 — it never equals 0 because e^z = e^x (cos y + i sin y) has modulus e^x > 0. For every other complex w (including all negatives, which are e^(log|w| + i arg w)) the equation e^z = w has solutions, in fact infinitely many spaced 2 pi i apart. One value missed, all others hit infinitely often: little Picard is tight.
e^z misses only 0 and hits everything else infinitely often — the extremal case of the one allowed exception.
At most one omitted value, not zero: the single exception is real and necessary, as e^z shows. The theorem also needs f to be entire and non-constant; a mere polynomial trivially omits no values, but the surprise is that even transcendental entire functions can omit only one.