Picard's Theorems and Value Distribution

Picard's theorems describe the remarkable value distribution of entire and meromorphic functions, showing that they take "almost all" values, with very few exceptions.

Picard's Little Theorem

Theorem9.5Picard's little theorem

A non-constant entire function takes every complex value with at most one exception.

ExampleIllustrations of Picard's little theorem

$e^z$ omits $0$ (and takes every other value infinitely often). This is the maximal case: exactly one omitted value.
A non-constant polynomial takes every value (no exceptions — the FTA).
$\sin z$ takes every value infinitely often (no exceptions).
$e^z + z$ takes every value (no exceptions).

RemarkConnection to Liouville

Picard's little theorem vastly strengthens Liouville's theorem. Liouville says: omitting a disk of values forces constancy. Picard says: omitting just two points forces constancy. The proof uses the theory of the modular function $\lambda(\tau)$ or, alternatively, the Bloch-Landau theory.

Picard's Great Theorem

Theorem9.6Picard's great theorem

Let $f$ have an essential singularity at $z_0$ . Then in every punctured neighborhood of $z_0$ , $f$ takes every complex value infinitely often with at most one exception.

RemarkComparison with Casorati-Weierstrass

The Casorati-Weierstrass theorem says the image of a punctured neighborhood is dense in $\mathbb{C}$ . Picard's great theorem is an enormous strengthening: the image is all of $\mathbb{C}$ except possibly one point, and each value is taken infinitely often.

ExampleEssential singularity of $e^{1/z}$

Near $z = 0$ , $e^{1/z}$ takes every value except $0$ infinitely often. For any $w \neq 0$ , the equation $e^{1/z} = w$ has solutions $z_k = 1/(\ln|w| + i\arg w + 2\pi ik) \to 0$ as $k \to \infty$ . The exceptional value $w = 0$ is never taken.

Nevanlinna Theory (Overview)

Definition9.6Nevanlinna characteristic

For a meromorphic function $f$ on $\mathbb{C}$ , the Nevanlinna characteristic is

$T(r, f) = m(r, f) + N(r, f)$

where $m(r, f) = \frac{1}{2\pi}\int_0^{2\pi}\log^+|f(re^{i\theta})|\,d\theta$ is the proximity function and $N(r, f) = \int_0^r \frac{n(t, \infty)}{t}\,dt$ is the counting function ( $n(t, \infty)$ counts poles in $|z| \leq t$ ).

Theorem9.7First Fundamental Theorem of Nevanlinna

For any meromorphic function $f$ and any $a \in \mathbb{C}$ :

$T(r, f) = N(r, 1/(f-a)) + m(r, 1/(f-a)) + O(1).$

This says: $f$ takes the value $a$ (measured by $N$ ) with approximately the same "total weight" $T(r,f)$ for all $a$ .

Theorem9.8Second Fundamental Theorem of Nevanlinna

For distinct $a_1, \ldots, a_q \in \hat{\mathbb{C}}$ :

$\sum_{j=1}^q m(r, 1/(f-a_j)) \leq 2T(r,f) + S(r,f)$

where $S(r,f) = o(T(r,f))$ outside an exceptional set. Equivalently,

$(q - 2)T(r,f) \leq \sum_{j=1}^q \overline{N}(r, 1/(f-a_j)) + S(r,f).$

This implies Picard's theorem: if $f$ omits three values, then $T(r,f) = O(S(r,f))$ , forcing $f$ to be constant.

The Defect Relation

RemarkDeficiency

The deficiency of $a$ is $\delta(a, f) = 1 - \limsup \frac{N(r,1/(f-a))}{T(r,f)} \in [0, 1]$ . A value with $\delta(a) > 0$ is taken "less often than expected." The defect relation $\sum_a \delta(a, f) \leq 2$ (from the Second Fundamental Theorem) quantifies Picard's theorem: at most countably many values can have positive deficiency, and their deficiencies sum to at most $2$ .