Picard's little theorem

Intuition [Beginner]

An entire function is a function that has a valid power series everywhere in the complex plane. Polynomials like $z^2+3z+1$ are entire. The exponential function $e^z$ is entire.

A surprising fact: an entire function that never takes the value $0$ must be $e^{g(z)}$ for some entire function $g$. This is because $e^z$ never hits zero, and any function with the same property factors through the exponential.

Picard's little theorem goes further. It says that if an entire function misses two complex values, it must be constant. The exponential $e^z$ misses only zero — that is allowed. But missing a second value as well forces the function to be unchanging.

Compare this with the real case. The real function $f(x)=e^x$ never takes the value $-5$, but is not constant. In the complex plane, one omitted value is the most any non-constant entire function can manage.

Visual [Beginner]

A diagram of the complex plane on the left, with the entire function $f$ drawn as an arrow mapping into the target plane on the right. The target plane has two marked points (say $0$ and $1$) crossed out. The arrow is dotted because $f$ never reaches those points. A second arrow labelled "constant function" maps the entire domain to a single point.

The power of Picard's theorem: a single constraint on missed values tells you the function is the simplest possible — a constant.

Worked example [Beginner]

Consider the entire function $f(z)=e^z$. What values does it miss?

Every value $w$ that $e^z$ takes satisfies $w\ne 0$, because $e^z=r{e}^{i\theta }$ with $r>0$. So $e^z$ misses exactly one value: $0$.

Does this violate Picard's theorem? No. Picard says an entire function that misses two values must be constant. Missing one value is fine — $e^z$ is a counterexample showing that one is the sharp bound.

Now consider $g(z)=e^{e^z}$. This is entire (a composition of entire functions). It also misses only zero, because $e^{e^z}$ is again an exponential and never vanishes. It does not miss a second value.

What about $h(z)=e^z+1$? This misses only $1$ (shifted zero). Again, one value, so Picard allows it.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Definition (Entire function). A function $f:\mathbb{C}\to \mathbb{C}$ is entire if it is holomorphic on all of $\mathbb{C}$, equivalently if its power series centred at the origin has infinite radius of convergence.

Definition (Omitted value). A value $w\in \mathbb{C}$ is omitted by $f$ if $f(z)\ne w$ for all $z\in \mathbb{C}$.

Picard's theorem constrains the set of omitted values of a non-constant entire function to have cardinality at most one.

The proof strategy proceeds through an intermediate result:

Theorem (Schottky, 1904). Let $f$ be holomorphic on the open unit disk $\mathbb{D}$ and suppose $f$ omits $0$ and $1$. Then $|f(z)|\le \exp (A(1-|z|{)}^{-1/2})$ for a universal constant $A>0$ and all $z\in \mathbb{D}$.

Schottky's theorem gives a quantitative growth bound for functions omitting two values. From there, Picard's theorem follows by a rescaling argument.

Key theorem with proof [Intermediate+]

Theorem (Picard's little theorem, 1879). If $f:\mathbb{C}\to \mathbb{C}$ is entire and omits two distinct values $a,b\in \mathbb{C}$, then $f$ is constant.

Proof. Without loss of generality $a=0$ and $b=1$ (compose $f$ with the affine map $z\mapsto (z-a)/(b-a)$; the composition is still entire and omits $0$ and $1$).

Assume $f$ is non-constant. Then $f$ is not identically zero, so by the isolated zeros theorem, $f$ has at most countably many zeros. Since $f$ omits $0$ entirely, $f$ has no zeros.

Step 1: Reduction to Schottky. For any $z_0\in \mathbb{C}$ and any $R>0$, define $g(\zeta )=f(z_0+R\zeta )$ on $|\zeta |<1$. Then $g$ is holomorphic on the unit disk and omits $0$ and $1$. By Schottky's theorem:

$$|g(\zeta )|\le \exp (A(1-|\zeta |{)}^{-1/2}).$$

Evaluating at $\zeta =0$: $|f(z_0)|\le \exp (A)$, so $|f(z_0)|$ is bounded by a constant independent of $z_0$.

Step 2: Apply Liouville. Since $|f(z)|\le \exp (A)$ for all $z\in \mathbb{C}$, the entire function $f$ is bounded. By Liouville's theorem 06.01.01, $f$ is constant. This contradicts the assumption. $\square$

The proof builds toward Schottky's theorem as the engine. The bridge is: Picard's result is a Liouville argument, but the growth control comes from the deeper combinatorial-geometric content of Schottky's bound on functions that omit two values.

Bridge. This argument generalises the classical Liouville theorem by injecting a growth estimate via Schottky's theorem; the same pattern appears again in 06.01.21 where Picard's great theorem handles essential singularities. The foundational reason Picard works is that omitting two values constrains the monodromy of a lifted map into the universal cover of $\mathbb{C}\setminus \{0,1\}$, identifying the growth bound with the geometry of the modular function $\lambda$. Putting these together, Picard's theorem is the simplest case of a family of value-distribution results culminating in Nevanlinna theory.

Exercises [Intermediate+]

Advanced results [Master]

The deeper proof of Picard's theorem uses the modular function $\lambda$, the universal covering map $\mathbb{H}\to \mathbb{C}\setminus \{0,1\}$ where $\mathbb{H}$ is the upper half-plane.

If $f$ is entire and omits $0$ and $1$, then $f(\mathbb{C})\subset \mathbb{C}\setminus \{0,1\}$. By the monodromy theorem (since $\mathbb{C}$ is simply connected), the map $f$ lifts through $\lambda$: there exists an entire function $h:\mathbb{C}\to \mathbb{H}$ with $f=\lambda \circ h$.

Now $h$ maps $\mathbb{C}$ into $\mathbb{H}$, so $|h|$ is bounded below (the imaginary part of $h$ is positive). The function $g=1/(h-i)$ maps $\mathbb{C}$ into a bounded subset of $\mathbb{C}$ and is entire. By Liouville's theorem, $g$ is constant, hence $h$ is constant, hence $f$ is constant.

This modular-function proof is more conceptual than the Schottky route: it replaces the quantitative growth bound with a topological lifting argument. The price is the need for the monodromy theorem and the existence of the modular function.

Nevanlinna theory ^{[Nevanlinna 1929]} extends Picard's theorem from "how many values are missed" to "how often does $f$ come close to each value." The First Main Theorem gives a precise balance between the growth of $f$ and its proximity to any value $a$. The Second Main Theorem then shows that the deficiency $\delta (a)=1-m(r,a)/T(r)$ satisfies ${\sum }_{a\in \mathbb{C}}\delta (a)\le 2$, which contains Picard's theorem as the special case where two deficiencies equal $1$.

Synthesis. Picard's little theorem is the gateway to value-distribution theory; it appears again in 06.01.21 where the great theorem extends the conclusion from entire functions to meromorphic functions near essential singularities. The central insight is that the universal cover of a twice-punctured plane carries hyperbolic geometry, and maps from the flat complex plane into a hyperbolic surface must degenerate. This pattern recurs throughout complex analysis: curvature constraints on the target force rigidity on the source. The modular function $\lambda$ is dual to the $j$-invariant of elliptic curves, connecting complex analysis to arithmetic geometry — the bridge is that value-distribution constraints and Diophantine constraints (Vojta's dictionary) share the same geometric origin.

Full proof set [Master]

Proposition (Schottky's theorem, simplified form). There exists a constant $C>0$ such that if $f$ is holomorphic on $\mathbb{D}$ and omits $0$ and $1$, then $\log |f(z)|\le C\cdot (1-|z|{)}^{-1}$ for all $|z|<1$.

Proof sketch. Write $f=-e^{i\pi g}$ where $g$ is a holomorphic branch of $\frac{1}{\pi i}\log f$, constructed via the monodromy theorem since $f$ omits $0$. Then $g$ omits integer values (since $f\ne 1$). Applying the same construction to $g$, one obtains a map into a strip of bounded width. Iterating this procedure produces a bounded holomorphic function whose growth controls $f$. The exponent $-1/2$ in the sharp form requires a more refined argument via the hyperbolic metric on $\mathbb{C}\setminus \{0,1\}$. $\square$

Connections [Master]

Picard's great theorem 06.01.21 strengthens the conclusion from entire functions to functions near essential singularities: in every neighbourhood of an essential singularity, the function takes every value infinitely often with at most one exception.

Schottky's theorem provides the quantitative engine behind Picard's proof; a full treatment of Schottky's bound with the sharp exponent connects to the hyperbolic metric and Ahlfors's theory of covering surfaces 06.01.29.

Nevanlinna theory generalises Picard's theorem to a quantitative balance between growth and value distribution; the Second Main Theorem $\sum \delta (a)\le 2$ recovers Picard as a corollary ^{[Nevanlinna 1929]}.

The modular-function proof lifts $f$ through the universal cover $\lambda :\mathbb{H}\to \mathbb{C}\setminus \{0,1\}$; this connects to the theory of covering spaces 06.01.04 and the Riemann mapping theorem 06.01.06.

Bibliography [Master]

@article{picard1879,
  author = {Picard, {\'E}mile},
  title = {Sur une propri{\'e}t{\'e} des fonctions enti{\`e}res},
  journal = {Comptes Rendus Acad. Sci. Paris},
  volume = {89},
  pages = {663--665},
  year = {1879}
}

@book{stein-shakarchi-complex,
  author = {Stein, Elias M. and Shakarchi, Rami},
  title = {Complex Analysis},
  series = {Princeton Lectures in Analysis},
  volume = {II},
  publisher = {Princeton University Press},
  year = {2003}
}

@book{ahlfors-complex,
  author = {Ahlfors, Lars V.},
  title = {Complex Analysis},
  edition = {3},
  publisher = {McGraw-Hill},
  year = {1979}
}

@book{conway-complex,
  author = {Conway, John B.},
  title = {Functions of One Complex Variable},
  edition = {2},
  publisher = {Springer},
  year = {1978}
}

@article{nevanlinna1929,
  author = {Nevanlinna, Rolf},
  title = {Le th{\'e}or{\`e}me de {P}icard-{B}orel et la th{\'e}orie des fonctions m{\'e}romorphes},
  journal = {Acta Math.},
  volume = {52},
  pages = {131--143},
  year = {1929}
}

Historical & philosophical context [Master]

Charles Emile Picard proved his little theorem in 1879 ^{[Picard 1879]} at the age of 25, while still a doctoral student. The result shocked the mathematical community: it showed that entire functions, no matter how wild, are constrained by a devastatingly simple property — they can miss at most one value. Picard's original proof used the elliptic modular function and was considered deep and mysterious.

The simpler proof via Schottky's theorem (1904) and Bloch's theorem (1925) made the result more accessible. Lars Ahlfors's theory of covering surfaces (1935) gave the most geometric interpretation, earning him one of the first Fields Medals. Nevanlinna's theory (1925--1929) placed Picard's theorem within a vast quantitative framework that remains central to complex analysis.

The theorem has a striking parallel in number theory: Roth's theorem on Diophantine approximation and Vojta's dictionary both reflect the same principle that maps from "flat" spaces to "hyperbolic" spaces must be degenerate. This analogy, first articulated by Serge Lang and Paul Vojta, remains one of the deepest bridges between analysis and arithmetic.