Picard's great theorem

Intuition [Beginner]

An essential singularity is a point where a function behaves as wildly as possible. Near such a point, the function has no limit — not finite, not infinite. It oscillates uncontrollably.

Picard's great theorem makes this precise: near an essential singularity, a function takes every complex value, with at most one exception, and it does so infinitely often.

Compare with the three types of isolated singularities. A removable singularity is no real problem — the function can be filled in. A pole means the function blows up to infinity. An essential singularity is chaos: the function takes every value infinitely many times in any neighbourhood, missing at most one.

The function $f(z)=e^{1/z}$ has an essential singularity at $z=0$. As $z$ spirals toward $0$, $e^{1/z}$ spirals through every complex value (except $0$) infinitely often.

Visual [Beginner]

A zoom into the origin for the function $e^{1/z}$. Concentric circles of decreasing radius show the function cycling through all nonzero complex values repeatedly. One point at $0$ is marked as the omitted value.

Near an essential singularity, the function image is essentially the entire complex plane, repeated densely at every scale.

Worked example [Beginner]

Consider $f(z)=e^{1/z}$ near $z=0$.

For any nonzero value $w$, write $w=e^a$ for some $a\in \mathbb{C}$ (possible since $e^z$ is surjective onto $\mathbb{C}\setminus \{0\}$). Then $f(z)=w$ when $1/z=a+2\pi ik$ for any integer $k$, giving $z=1/(a+2\pi ik)$.

As $k\to \infty$, these solutions $z_k\to 0$. So in every neighbourhood of $0$, the function takes the value $w$ infinitely often. The only exception is $w=0$, which $e^{1/z}$ never attains.

This is exactly what Picard's great theorem predicts: one omitted value (zero), and every other value is hit infinitely often near the singularity.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Definition (Essential singularity). Let $f$ be holomorphic on a punctured disk $0<|z-z_0|<R$. The point $z_0$ is an essential singularity of $f$ if the Laurent expansion $f(z)={\sum }_{n=-\infty }^{\infty }{c}_n(z-z_0{)}^n$ has infinitely many nonzero coefficients $c_n$ with $n<0$.

Equivalently, by the Casorati-Weierstrass theorem, $z_0$ is essential if and only if $f$ has no limit (finite or infinite) as $z\to z_0$.

Definition (Normal family). A family $\mathcal{F}$ of holomorphic functions on a domain $\Omega$ is normal if every sequence in $\mathcal{F}$ has a subsequence that converges uniformly on compact subsets of $\Omega$ (either to a holomorphic function or to $\infty$).

By Montel's theorem ^{[Montel 1912]}, a family omitting three fixed values is normal. This is the key tool for the modern proof of Picard's great theorem.

Key theorem with proof [Intermediate+]

Theorem (Picard's great theorem, 1879). Let $f$ be holomorphic on $0<|z-z_0|<R$ with an essential singularity at $z_0$. Then for every $w\in \hat{\mathbb{C}}$ with at most one exception, $f(z)=w$ has infinitely many solutions in $0<|z-z_0|<R$.

Proof. Suppose for contradiction that $f$ omits two finite values $a\ne b$ in some punctured neighbourhood $0<|z-z_0|<r$. Without loss, $a=0$ and $b=1$.

Step 1: Translate to Schottky. For each $n\ge 1$, define $f_n(\zeta )=f(z_0+\zeta /n)$ on $0<|\zeta |<rn$. Then $\{f_n\}$ is a family of holomorphic functions on $|\zeta |<rn$, each omitting $0$ and $1$.

Step 2: Apply Montel. The functions $g_n(\zeta )=f_n(\zeta )$ on the fixed disk $|\zeta |<r$ form a family of holomorphic functions omitting $0$ and $1$. By Montel's theorem (a family omitting three values is normal; two values plus compactness gives normality after composing with the modular function), $\{g_n\}$ is normal.

Step 3: Extract and analyse. Extract a subsequence $g_{n_k}\to g$ uniformly on compact subsets. If $g$ is holomorphic, then $f(z_0+\zeta /n_k)\to g(\zeta )$, meaning $f$ has a removable singularity at $z_0$ (setting $f(z_0)=g(0)$), contradicting the hypothesis. If $g\equiv \infty$, then $f$ has a pole at $z_0$, again contradicting the essential-singularity hypothesis. If $g$ has an essential singularity at $0$, then the family was not normal — a contradiction to Montel.

Step 4: Conclusion. The assumption that $f$ omits two values leads to the conclusion that $z_0$ is removable or a pole. Since $z_0$ is essential, $f$ omits at most one value. $\square$

The infinite-solutions part follows: if $f(z)=w$ had only finitely many solutions, then $(f(z)-w{)}^{-1}$ would be bounded near $z_0$, giving a removable singularity and contradicting the essential-singularity hypothesis.

Bridge. The proof generalises the normal-families technique from 06.01.14; this is exactly the Montel machinery applied to the geometry of omitted values. The foundational reason Picard's great theorem holds is that normal families and essential singularities are incompatible — normal families force convergence, while essential singularities force divergence. This pattern appears again in the theory of meromorphic functions where Nevanlinna's Second Main Theorem quantifies the same tension. The bridge is that value-omission constrains function growth, and growth constraints force singularity types.

Exercises [Intermediate+]

Advanced results [Master]

Montel's theorem (sharp form). A family $\mathcal{F}$ of meromorphic functions on a domain $\Omega$ is normal if and only if $\mathcal{F}$ omits three distinct values in $\hat{\mathbb{C}}$. This is the sharpest possible result: omitting two values does not guarantee normality (consider $e^{nz}$ on $\mathbb{C}$, which omits $0$ but no second value uniformly).

Zalcman's lemma (1975) gives a rescaling characterisation of non-normality: a family $\mathcal{F}$ is not normal at $z_0$ if and only if there exist sequences $f_n\in \mathcal{F}$, $z_n\to z_0$, and ${\rho }_n\to 0^{+}$ such that $f_n(z_n+{\rho }_n\zeta )\to g(\zeta )$ uniformly on compact subsets of $\mathbb{C}$, where $g$ is a non-constant entire function of order at most $2$. Zalcman's lemma is the modern engine behind proofs of Picard-type theorems.

Nevanlinna's Second Main Theorem ^{[Nevanlinna 1929]} provides the quantitative form: for a meromorphic function $f$ on $\mathbb{C}$ and distinct values $a_1,\dots ,a_q\in \hat{\mathbb{C}}$,

$$(q-2)T(r,f)\le \sum _{j=1}^q\overline{N}(r,a_j,f)+S(r,f)$$

where $T(r,f)$ is the Nevanlinna characteristic, $\overline{N}$ counts distinct $a_j$-points, and $S(r,f)=o(T(r,f))$ outside a set of finite measure. Setting $q=3$ and assuming two deficiencies equal $1$ recovers Picard's great theorem.

Synthesis. Picard's great theorem is the local counterpart of the little theorem 06.01.20; the central insight is that essential singularities force the maximal range of values, just as non-constancy forces the maximal range for entire functions. The Montel normal-families approach builds toward a unified framework: both Picard theorems are instances of the general principle that functions with restricted ranges are "tame" (normal, bounded, or constant). This pattern recurs across complex analysis: the Riemann mapping theorem 06.01.06 and the uniformisation theorem both reflect the same tension between range restriction and geometric rigidity. Zalcman's lemma identifies non-normality with the existence of a "limit entire function," putting these together with the Bloch theorem 06.01.29 to give the modern, streamlined proofs.

Full proof set [Master]

Proposition (Zalcman's lemma, simplified). Let $\mathcal{F}$ be a family of functions holomorphic on a domain $\Omega$, and suppose $\mathcal{F}$ is not normal at $z_0\in \Omega$. Then there exist sequences $f_n\in \mathcal{F}$, $z_n\to z_0$, and ${\rho }_n\to 0^{+}$ such that $f_n(z_n+{\rho }_n\zeta )\to g(\zeta )$ on $\mathbb{C}$, where $g$ is a non-constant entire function of order at most $2$.

Proof sketch. Non-normality at $z_0$ means there is a compact neighbourhood $K$ of $z_0$ on which the family fails to be equicontinuous. By Marty's theorem (a family is normal if and only if the spherical derivative is locally bounded), the spherical derivatives $f_n^{\#}(z_n)$ blow up for some $z_n\to z_0$. Set ${\rho }_n=1/f_n^{\#}(z_n)$ and rescale. The rescaled functions satisfy a uniform Lipschitz bound on compact sets, yielding a normal limit $g$ that is non-constant (since $g^{\#}(0)=1$) and of finite order. $\square$

Connections [Master]

Picard's little theorem 06.01.20 is the global special case: an entire function has a removable singularity or pole at infinity, so the great theorem applied at infinity recovers the little theorem as a corollary.

Normal families and Montel's theorem 06.01.14 provide the framework; Zalcman's lemma is the modern refinement that converts non-normality into the existence of a non-constant entire limit function.

Schottky's and Bloch's theorems 06.01.29 give alternative routes to Picard via quantitative growth bounds rather than normal-family arguments; the two approaches are ultimately equivalent through the Schottky--Landau theorem.

The Casorati-Weierstrass theorem is the weak precursor: near an essential singularity, the range is dense in $\mathbb{C}$; Picard strengthens "dense" to "all but at most one value, infinitely often" 06.01.27.

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}
}

@article{montel1912,
  author = {Montel, Paul},
  title = {Sur les familles de fonctions analytiques qui admettent des valeurs exceptionnelles},
  journal = {Ann. Sci. {\'E}cole Norm. Sup.},
  volume = {29},
  pages = {483--511},
  year = {1912}
}

@article{zalcman1975,
  author = {Zalcman, Lawrence},
  title = {A heuristic principle in complex function theory},
  journal = {Amer. Math. Monthly},
  volume = {82},
  pages = {813--817},
  year = {1975}
}

@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]

Picard proved his great theorem in the same 1879 paper ^{[Picard 1879]} as the little theorem. The result completed the classification of isolated singularities by showing that essential singularities are not merely "wild" but maximally range-filling.

Montel's theory of normal families (1912) ^{[Montel 1912]} revolutionised the proof strategy. The modern approach via Zalcman's lemma (1975) reduced the entire argument to a rescaling technique, making Picard's theorem accessible to graduate students.

The philosophical significance is that complex differentiability, a seemingly mild hypothesis, has devastatingly strong consequences. A single complex derivative constrains the global behaviour of a function far more than any finite number of real derivatives. Picard's theorem is one of the sharpest expressions of this principle.