From Liouville to a much louder claim
By now you have a whole machine for measuring an entire function: order and type say how fast it grows, Jensen's formula ties that growth to where the zeros sit, and Hadamard's theorem rebuilds a finite-order function from its zeros and a polynomial exponent. All of that is about magnitude — how big f gets. Picard's theorems ask a stranger question: not how big, but which values does f take? The answer turns out to be far more rigid than anyone would guess from the real line.
Recall the warm-up you already know. Liouville's theorem says a bounded entire function must be constant — equivalently, a non-constant entire function cannot avoid a whole disk of values, since avoiding a disk would keep it away from a region and you could bound a reciprocal. That already feels generous: e^z, for instance, is non-constant and entire, and indeed its values fill the plane except for one point. But Liouville only forbids missing an entire neighbourhood. Could a clever function miss, say, two isolated points and nothing more?
Charles Picard's answer, in 1879, was a flat no — and proving it required ideas well beyond Liouville. The little Picard theorem states: a non-constant entire function omits at most one value of the complex plane. Two missed values are impossible. One missed value is the absolute ceiling, and it really can happen. The whole drama of this guide lives in that gap between one and two.
The one value you are allowed to miss
The single missed value is not a loophole the theorem grudgingly admits; it is real and easy to see. Take f(z) = e^z. It is entire and non-constant. Its values are everything of the form e^x (cos y + i sin y), which traces out every modulus e^x in the open interval (0, infinity) and every argument. That is the entire plane except 0 — because the exponential is never zero. So e^z misses exactly the value 0 and hits absolutely everything else, often infinitely many times. The exceptional value here is 0; for e^z + 5 it would shift to 5.
A value that an entire function never takes is called an exceptional value (or a Picard exceptional value). Little Picard says: at most one exists. Notice how this slots above your earlier results. Liouville forbade missing a disk; the open mapping theorem forbade missing any interior point of the image at all once you know f is non-constant on a neighbourhood; Picard sharpens this to a global statement about the whole plane — the image of a non-constant entire function is either all of the plane or the plane minus a single point.
Why two missed values is impossible
Here is the idea behind the proof, in the form that reveals the most. Suppose, for contradiction, that a non-constant entire function f misses two distinct values; after composing with a Mobius map of the target we may as well say it misses 0 and 1. Then f maps the whole plane into the twice-punctured plane C minus {0, 1}. The trick is that this small punctured region carries a hidden hyperbolic geometry: it has a hyperbolic metric in which it is, by the uniformization theorem, covered by the unit disk. So f lifts to a map from the simply connected plane into the unit disk.
But a holomorphic map from the entire plane into the unit disk is bounded, so by Liouville it is constant — hence the original f is constant too, contradiction. That is the whole skeleton: missing two values traps the function inside a hyperbolic target, hyperbolicity bounds it, and Liouville kills it. The modular function lambda that uniformizes C minus {0, 1} is the classical workhorse here, but you do not need its formulas to feel the shape of the argument. The single allowed puncture (as for e^z, missing only 0) leaves the target merely the once-punctured plane, which is not hyperbolic, so the trap never springs.
Great Picard: the wildness near an essential singularity
Little Picard is a global statement about entire functions. The great Picard theorem is its local, sharper twin, and it is the one that should make your hair stand up. It is about behaviour near an essential singularity. Recall the ladder of isolated singularities you built in the Laurent rung: a removable singularity where f stays tame, a pole where |f| marches off to infinity, and the essential case where neither happens.
Earlier you met the Casorati-Weierstrass theorem: near an essential singularity, f comes arbitrarily close to every complex value. That is already eerie — the values are dense. Great Picard upgrades 'close to' into 'equal to, infinitely often': in every neighbourhood of an essential singularity, f takes every complex value, with at most one exception, infinitely many times. Not approached — attained, and attained without end. The single permitted exception is the same kind of escape hatch as in little Picard.
Make it concrete with the canonical monster: f(z) = e^(1/z) near z = 0. As z spirals toward the origin, 1/z races off to infinity, and e^(1/z) does something no real-variable function ever does. Pick any non-zero target w; you can solve e^(1/z) = w exactly, infinitely many times, with solutions z piling up at 0. The lone exception is again 0 — e^(1/z) is never zero. So in any tiny disk around the origin, this function hits 7, and -2, and i, and a million, each infinitely often, while never once equalling 0.
near an essential singularity z_0 : f takes EVERY value w in C (with at most one exception) and takes each such w infinitely often, in EVERY neighbourhood of z_0, however small. example: e^(1/z) at z_0 = 0 -> omits only 0
How the two Picards fit together — and what lies beyond
The two theorems are one idea seen at two scales, and the link is the point at infinity. A non-constant entire function that is not a polynomial has an essential singularity at infinity — feed it the substitution w = 1/z and the wildness of e^(1/z) at 0 becomes the wildness of e^z at infinity. Apply great Picard at that essential singularity: in every neighbourhood of infinity (that is, outside every large disk) the function takes all values but at most one infinitely often. Shrink the excluded disk and you recover exactly little Picard's verdict on the whole plane. Little Picard is great Picard, read at infinity.
One honest caveat keeps these statements from overreaching. Polynomials are entire and surjective — a degree-n polynomial hits every value exactly n times — so the 'at most one exception' clause is never used by them; they have an ordinary pole at infinity, not an essential singularity, so great Picard simply does not apply there. The drama belongs to transcendental entire functions, the ones that are not polynomials. For those, infinity is genuinely essential, and Picard's full force is felt.
- Decide if f is a polynomial. If yes, it is surjective and Picard adds nothing new — stop here.
- If f is transcendental entire, little Picard says its image is the whole plane minus at most one exceptional value — find that value (it is the one e^g never reaches, if any).
- To probe behaviour near a point where f blows up wildly, identify it as an essential singularity, then invoke great Picard: every value but one, infinitely often, in every neighbourhood.
Picard tells you the missed values number zero or one, but it stays silent on how often the other values are hit, or how that frequency relates to growth. That quantitative refinement is the subject of the final guide in this rung: Borel's theorem already sharpens Picard by tying the density of solutions of f = w to the order, and Nevanlinna theory then builds a whole accounting system — a 'first and second fundamental theorem' — in which Picard's one exception becomes a strict, countable budget. Picard is the qualitative thunderclap; Nevanlinna is the ledger that explains the weather.