Value Distribution: The Picard Theorems

How many values can a holomorphic function miss? Liouville says a bounded entire function is constant; Picard's theorems sharpen this astonishingly: a nonconstant entire function omits at most one value, and near an essential singularity it takes every value but at most one infinitely often. This is the summit of the elementary theory and the doorway to Nevanlinna theory. It builds on laurent-residues.md (Casorati–Weierstrass), normal-families.md, and the modular function of elliptic-functions.md.

From Casorati–Weierstrass to Picard

Near an essential singularity, Casorati–Weierstrass says the image is dense. Picard replaces "dense" with "everything but a point":

Little Picard theorem. A non-constant entire function $f:\mathbb{C}\to \mathbb{C}$ omits at most one complex value. (E.g. $e^z$ omits only $0$.)

Great Picard theorem. In every punctured neighbourhood of an essential singularity, $f$ attains every complex value, with at most one exception, infinitely often. (E.g. $e^{1/z}$ near $0$ omits only $0$.)

The great theorem contains the little one (view an entire non-polynomial as having an essential singularity at $\infty$) and dramatically strengthens Casorati–Weierstrass. It also refines the classification of singularities: if a function omits two values near an isolated singularity, that singularity is a pole or removable — never essential.

The modular route

The classical proof turns on the modular $\lambda$-function, the holomorphic covering map $$\lambda :\mathbb{H}⟶\mathbb{C}\setminus \{0,1\}.$$ Suppose an entire $f$ omitted two values; normalize them to $0$ and $1$, so $f:\mathbb{C}\to \mathbb{C}\setminus \{0,1\}$. Since $\mathbb{C}$ is simply connected, $f$ lifts through the covering to a map $\tilde{f}:\mathbb{C}\to \mathbb{H}$; composing with a Cayley map into the disc gives a bounded entire function, constant by Liouville — so $f$ is constant. The great theorem follows by feeding the same construction into Montel's theorem in its deep form:

Montel (fundamental normality criterion). A family of holomorphic functions on a domain that omits two fixed values is normal.

The covering-space non-existence of maps into $\mathbb{C}\setminus \{0,1\}$ is what makes "two omitted values" so restrictive — the hyperbolic geometry of the thrice-punctured sphere (schwarz-lemma.md) at work.

Nevanlinna theory (preview)

Value distribution theory quantifies Picard: instead of "omitted or not", it weighs how often each value is taken. For a meromorphic $f$ one forms the characteristic $T(r,f)$ (a growth measure) and the counting and proximity functions $N(r,a)$, $m(r,a)$ for each target $a$.

First main theorem. $m(r,a)+N(r,a)=T(r,f)+O(1)$ — every value is taken "equally often" on average (a quantitative argument principle).

Second main theorem. For distinct targets $a_1,\dots ,a_q$, ${\sum }_{k}m(r,a_k)\lesssim 2T(r,f)$, so only finitely many values can be deficient (taken too rarely) — a strong quantitative form of Picard's "at most one exception".

Nevanlinna theory is the modern home of these questions and connects onward to Diophantine approximation and complex dynamics.

References

• Ahlfors, Complex Analysis, Ch. 8 §3 (the Picard theorems).
• Stein & Shakarchi, Complex Analysis, Ch. 8 (Montel, Picard).
• Nevanlinna, Analytic Functions; Hayman, Meromorphic Functions.