the great Picard theorem
/ Picard -> pee-KAR /
Near an essential singularity — a point where a holomorphic function goes genuinely wild, like z = 0 for e^(1/z) — how chaotic can the values really get? The Casorati-Weierstrass theorem already says they come arbitrarily close to every complex number. The great Picard theorem goes far further: in every tiny punctured neighborhood of an essential singularity, the function actually attains every complex value, with at most one exception, and attains each of them infinitely often. Not just 'comes close' — actually hits, over and over, in any shrinking disk around the bad point.
Precisely: suppose f is holomorphic on a punctured disk 0 < |z - z_0| < delta and has an essential singularity at z_0. Then for every complex value w, with at most one single exceptional w, the equation f(z) = w has infinitely many solutions z accumulating at z_0. So pick any radius, however small; on that punctured disk f already takes (infinitely often) every value but at most one. The exponential near infinity is the model: e^(1/z) has an essential singularity at 0, and in any neighborhood of 0 it takes every nonzero value infinitely many times while never taking the value 0. The little Picard theorem is the global shadow of this local statement — an entire function with an essential singularity at infinity inherits the same one-value restriction.
Great Picard is the definitive measure of how violent an essential singularity is, and it cleanly separates the three kinds of isolated singularity: at a removable singularity f stays bounded; at a pole f tends to infinity (taking large values but missing a neighborhood of finitely many); at an essential singularity f is so wild it hits essentially everything infinitely often. The proof is genuinely hard — it rests on Montel's theorem about normal families and the modular function, far beyond elementary methods. The exact caveat to respect: 'at most one' exception again, and 'infinitely often,' and the singularity must be essential — a pole does not behave this way, since near a pole f simply runs off to infinity and omits a whole neighborhood of values.
Look at f(z) = e^(1/z) near z = 0, an essential singularity. Solve e^(1/z) = w for any w not 0: take 1/z = log|w| + i(arg w + 2 pi k) for every integer k, giving infinitely many solutions z = 1 / (log|w| + i(arg w + 2 pi k)) that pile up at 0. Every nonzero w is attained infinitely often in any disk around 0; only w = 0 is missed. That is great Picard made completely explicit.
In any neighborhood of its essential singularity, e^(1/z) attains every nonzero value infinitely often.
The wildness is special to essential singularities — at a pole f just tends to infinity and omits a whole neighborhood of values, so great Picard does not apply there. And 'at most one' exception is sharp, witnessed again by the omitted value 0 of e^(1/z).