a glimpse of the great Picard theorem
/ pee-KAR /
Casorati-Weierstrass already says a function near an essential singularity comes arbitrarily close to every value. Charles Emile Picard proved something far more astonishing: it does not merely come close — it actually HITS almost every value, and hits each one infinitely often. The wildness at an essential singularity is even greater than the dense-image picture suggested.
The great Picard theorem states: in every punctured neighbourhood of an essential singularity z_0, the function f takes every complex value — with at most ONE single exception — infinitely many times. So out of the entire complex plane, there is at most one 'missing value' that f might dodge near z_0; every other value is attained, again and again, in any disk however small. The lone exception is genuinely possible and necessary: for e^(1/z) near 0 the missing value is 0 (the exponential is never zero), and indeed e^(1/z) attains every nonzero complex number infinitely often near 0.
We only glimpse this here because its proof is hard — it goes well beyond Laurent series, drawing on the modular function or Montel's theorem on normal families — and because the value-distribution story belongs more fully to the theory of entire functions (its sibling, the little Picard theorem, says a non-constant entire function omits at most one value). What matters for classifying singularities is the headline: an essential singularity is so violent that, up to a single possible exception, the function realizes the WHOLE complex plane infinitely often in any neighbourhood. There is no more dramatic way for a holomorphic function to misbehave.
e^(1/z) near 0 illustrates both halves: the one exceptional value it never takes is 0, while every other complex number — 1, -7, 3 + 4i, you name it — is attained infinitely often in any punctured disk 0 < |z| < delta.
The single allowed exception (0) and 'everything else infinitely often' both happen for e^(1/z).
The exception is at most one value, and it really can occur, so do not overstate Picard as 'attains every value' — for e^(1/z) the value 0 is genuinely omitted; and this glimpse is not the proof, which is far deeper than the rest of this field.