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In complex analysis, Picard's great theorem and Picard's little theorem are related theorems about the range of an analytic function. They are named after Émile Picard.

The theorems

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Domain coloring plot of the function exp(1⁄z), centered on the essential singularity at z = 0. The hue of a point z represents the argument of exp(1⁄z), the luminance represents its absolute value. This plot shows that arbitrarily close to the singularity, all non-zero values are attained.

Little Picard Theorem: If a function is entire and non-constant, then the set of values that assumes is either the whole complex plane or the plane minus a single point.

Sketch of Proof: Picard's original proof was based on properties of the modular lambda function, usually denoted by , and which performs, using modern terminology, the holomorphic universal covering of the twice punctured plane by the unit disc. This function is explicitly constructed in the theory of elliptic functions. If omits two values, then lifting along the universal covering map sends the plane into the unit disc via a holomorphic function, which implies that is constant by Liouville's theorem.

This theorem is a significant strengthening of Liouville's theorem which states that the image of an entire non-constant function must be unbounded. Many different proofs of Picard's theorem were later found and Schottky's theorem is a quantitative version of it. In the case where the values of are missing a single point, this point is called a lacunary value of the function.

Great Picard's Theorem: If an analytic function has an essential singularity at a point , then on any punctured neighborhood of takes on all possible complex values, with at most a single exception, infinitely often.

This is a substantial strengthening of the Casorati–Weierstrass theorem, which only guarantees that the range of is dense in the complex plane. A result of the Great Picard Theorem is that any entire, non-polynomial function attains all possible complex values infinitely often, with at most one exception.

The "single exception" is needed in both theorems, as demonstrated here:

  • ez is an entire non-constant function that is never 0,
  • has an essential singularity at 0, but still never attains 0 as a value.

Proof

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Little Picard Theorem

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Suppose is an entire function that omits two values and . Then is also entire and we may assume without loss of generality that and .

Because is simply connected and the range of omits , f has a holomorphic logarithm. Let be an entire function such that . Then the range of omits all integers. By a similar argument using the quadratic formula, there is an entire function such that . Then the range of omits all complex numbers of the form , where is an integer and is a nonnegative integer.

By Landau's theorem, if , then for all , the range of contains a disk of radius . But from above, any sufficiently large disk contains at least one number that the range of h omits. Therefore for all . By the fundamental theorem of calculus, is constant, so is constant.

Great Picard Theorem

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Generalization and current research

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Great Picard's theorem is true in a slightly more general form that also applies to meromorphic functions:

Great Picard's Theorem (meromorphic version): If M is a Riemann surface, w a point on M, P1(C) = C ∪ {∞} denotes the Riemann sphere and f : M\{w} → P1(C) is a holomorphic function with essential singularity at w, then on any open subset of M containing w, the function f(z) attains all but at most two points of P1(C) infinitely often.

Example: The function f(z) = 1/(1 − e1/z) is meromorphic on C* = C - {0}, the complex plane with the origin deleted. It has an essential singularity at z = 0 and attains the value ∞ infinitely often in any neighborhood of 0; however it does not attain the values 0 or 1.

With this generalization, Little Picard Theorem follows from Great Picard Theorem because an entire function is either a polynomial or it has an essential singularity at infinity. As with the little theorem, the (at most two) points that are not attained are lacunary values of the function.

The following conjecture is related to "Great Picard's Theorem":[1]

Conjecture: Let {U1, ..., Un} be a collection of open connected subsets of C that cover the punctured unit disk D \ {0}. Suppose that on each Uj there is an injective holomorphic function fj, such that dfj = dfk on each intersection Uj ∩ Uk. Then the differentials glue together to a meromorphic 1-form on D.

It is clear that the differentials glue together to a holomorphic 1-form g dz on D \ {0}. In the special case where the residue of g at 0 is zero the conjecture follows from the "Great Picard's Theorem".

Notes

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References

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Picard theorem

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In , the Picard theorems refer to two closely related results concerning the range and value distribution of analytic functions, named after the French mathematician (1856–1941). Picard's Little Theorem states that any non-constant —that is, a defined on the entire —attains every complex value infinitely many times, with at most one possible exception. Picard's Great Theorem extends this principle to the local behavior near singularities, asserting that if a function is in a punctured neighborhood of an isolated , then in every neighborhood of that singularity, the function assumes every complex value infinitely often, again with at most one possible exception. Picard established the Little Theorem in 1879 as part of his early work on entire functions, employing advanced techniques involving modular functions inspired by the research of and . This result sharpened earlier insights from the Casorati-Weierstrass theorem on essential singularities and provided a profound limitation on the possible omissions in the range of entire functions, such as the exponential function eze^z, which omits only 0 but attains all other values infinitely often. The Great Theorem, proved shortly thereafter, generalized the Little Theorem by addressing the wild oscillatory behavior near essential singularities, demonstrating that such points force the function to densely cover the except possibly at one point. These theorems are foundational to the study of value distribution theory in , influencing later developments such as , which quantifies how often analytic functions attain values. They highlight the richness of non-constant analytic functions, contrasting with the more restricted behavior of polynomials, and have applications in proving results like the via contradiction arguments involving entire functions. Proofs of the theorems typically rely on the modular function or auxiliary constructions like the Nevanlinna characteristic, underscoring their deep connections to and uniformization.

Background Concepts

Entire Functions

In , an is a that is defined and analytic at every point in the finite C\mathbb{C}. These functions possess a global expansion centered at any point z0∈Cz_0 \in \mathbb{C}, with an infinite , allowing the series to converge everywhere in the plane. A fundamental property of entire functions is given by Liouville's theorem, which states that if an entire function is bounded, then it must be constant. This theorem highlights the rigidity of entire functions: unlike holomorphic functions on bounded domains, which can vary freely within bounds, entire functions cannot remain bounded unless they are constant, underscoring their behavior over the unbounded complex plane. Classic examples of entire functions include polynomials of any degree, such as f(z)=z2+3z+1f(z) = z^2 + 3z + 1, which are holomorphic everywhere due to their finite sums of powers. The exponential function eze^z, defined by its power series ∑n=0∞znn!\sum_{n=0}^\infty \frac{z^n}{n!}, converges for all z∈Cz \in \mathbb{C} and is thus entire. Similarly, the sine and cosine functions, expressed as sin⁡z=eiz−e−iz2i\sin z = \frac{e^{iz} - e^{-iz}}{2i} and cos⁡z=eiz+e−iz2\cos z = \frac{e^{iz} + e^{-iz}}{2}, are linear combinations of entire functions and hence entire themselves, with power series that also have infinite radius of convergence. Entire functions have no singularities in the finite plane, but their at infinity can exhibit essential singularities for non-polynomial cases, such as eze^z. The provides insight into such points, asserting that near an , the image of any punctured neighborhood under the function is dense in the .

Isolated Singularities

In , an of a ff occurs at a point z0∈Cz_0 \in \mathbb{C} where ff fails to be holomorphic at z0z_0, but is holomorphic in some punctured disk 0<∣z−z0∣<r0 < |z - z_0| < r surrounding it. To analyze the local near such a point, the Laurent series expansion is employed, which extends the Taylor series by incorporating negative powers of (z−z0)(z - z_0): f(z)=∑n=−∞∞an(z−z0)n,f(z) = \sum_{n=-\infty}^{\infty} a_n (z - z_0)^n, valid in an annulus around z0z_0, with coefficients ana_n computed via contour integrals over circles enclosing the singularity. Isolated singularities are classified according to the principal part of the Laurent series, consisting of the terms with negative exponents. A singularity is removable if all coefficients an=0a_n = 0 for n<0n < 0, allowing ff to be redefined at z0z_0 as f(z0)=a0f(z_0) = a_0 to become holomorphic there. It is a pole of finite order mm if a−m≠0a_{-m} \neq 0 and an=0a_n = 0 for all n<−mn < -m, in which case ∣f(z)∣→∞|f(z)| \to \infty as z→z0z \to z_0. The singularity is essential if infinitely many negative powers have nonzero coefficients, leading to highly irregular behavior near z0z_0. Émile Picard played a pivotal role in advancing the understanding of essential singularities through his investigations into the value distribution of holomorphic functions near such points, laying groundwork for deeper results on their dense and repetitive range behavior. A classic example is the function f(z)=e1/zf(z) = e^{1/z}, which exhibits an essential singularity at z=0z = 0 because its Laurent series around 0 contains infinitely many negative powers. Near this point, f(z)f(z) becomes unbounded along the positive real axis (where Re⁡(1/z)→+∞\operatorname{Re}(1/z) \to +\infty) while approaching 0 along the negative real axis, illustrating the wild oscillations and dense value attainment characteristic of essential singularities. The Casorati–Weierstrass theorem provides a precise characterization of this density for essential singularities: if ff is holomorphic in the punctured disk Dr(z0)∖{z0}D_r(z_0) \setminus \{z_0\} with an essential singularity at z0z_0, then the image f(Dr(z0)∖{z0})f(D_r(z_0) \setminus \{z_0\}) is dense in C\mathbb{C}. In other words, for every w∈Cw \in \mathbb{C} and every ε>0\varepsilon > 0, there exists z∈Dr(z0)∖{z0}z \in D_r(z_0) \setminus \{z_0\} such that ∣f(z)−w∣<ε|f(z) - w| < \varepsilon. This result underscores the "chaotic" nature of essential singularities, contrasting sharply with the more predictable behavior at removable singularities or poles.

The Theorems

Little Picard Theorem

Picard's Little Theorem states that if ff is a non-constant entire function, then the equation f(z)=wf(z) = w has infinitely many solutions for every complex number ww, with at most one exception. Constant functions omit all values except one and are excluded from the non-constant case. An example is the exponential function eze^z, which never attains 0 but takes every other value infinitely often.

Great Picard Theorem

Picard's Great Theorem states that if ff has an isolated essential singularity at z0z_0, then in every punctured neighborhood of z0z_0, ff assumes every complex value, with at most one possible exception, infinitely often. This extends the Little Theorem to local behavior near essential singularities and strengthens the Casorati–Weierstrass theorem, which only guarantees density of the image in C\mathbb{C}. Detailed proofs of both theorems are provided in the Proof Outlines section.

Proof Outlines

Little Picard Theorem

A direct application of Liouville's theorem does not work for functions omitting two finite values, as the auxiliary function 1/(f(z) - a) is entire but generally unbounded. The standard proof proceeds by uniformization: Without loss of generality, assume f omits 0 and 1, so f: ℂ → ℂ \ {0, 1} is holomorphic. The domain ℂ \ {0, 1} admits a universal covering map π: 𝔻 → ℂ \ {0, 1}, where 𝔻 is the unit disk and π is holomorphic. There exists a holomorphic lift ĝ: ℂ → 𝔻 such that f = π ∘ ĝ. Since ĝ is entire and bounded (|ĝ(z)| < 1), Liouville's theorem implies ĝ is constant, hence f is constant. Picard's original 1879 proof used the elliptic modular function λ(τ), a hauptmodul for the modular group, to construct such a covering explicitly. The constant case is trivial, as constants omit all but one value.

Great Picard Theorem

The Great Picard Theorem asserts that if a function f is holomorphic in a punctured neighborhood of an isolated essential singularity at z₀, then in every such neighborhood, f assumes every complex value infinitely often, with at most one possible exception. A standard proof uses normal families. Assume f omits two distinct values a and b in 0 < |z - z₀| < r; translate and scale so z₀ = 0, a = 0, b = 1. Consider the family F = {f_n(z) = f(z/n) : n ∈ ℕ}, each holomorphic on the punctured unit disk 0 < |z| < 1 omitting 0 and 1. By Schottky's theorem, functions holomorphic on 𝔻 omitting 0 and 1 with |f(0)| ≤ K are bounded by some M(K, ρ) on |z| ≤ ρ < 1. The family F restricted to annuli or compact subsets away from 0 is locally bounded, hence normal by . Any convergent subsequence f_{n_k} → g uniformly on compact subsets of 0 < |z| < 1, where g is holomorphic on 𝔻 \ {0} omitting 0 and 1. The normality implies g extends holomorphically to all of 𝔻 (removable singularity at 0). However, since f_{n_k} → f uniformly on compacta in the punctured disk, this extension implies f has a removable singularity at 0, contradicting its essential nature. Thus, f cannot omit two values. To show infinite attainment, suppose f - c (c ≠ exception) has finitely many zeros near z₀; then h(z) = (f(z) - c)/p(z) (p polynomial for zeros) omits 0 and the exception, leading to the above contradiction. This strengthens Casorati–Weierstrass via , ensuring dense coverage except possibly one point. An alternative proof uses universal covers: The punctured disk covers the right half-plane via w = -log z; lifting f yields a map to the cover of ℂ \ {0,1}, composed with the modular function λ(τ) to get an entire function, whose non-constancy contradicts Little Picard. A modern geometric proof by Ahlfors uses quasiconformal mappings and the Schwarz lemma on Riemann surfaces to show incompatibility with the essential singularity's metric properties.

History and Development

Émile Picard's Contributions

Charles Émile Picard (1856–1941) was a prominent French mathematician whose work profoundly influenced complex analysis, differential equations, and algebraic geometry. Born in Paris, he studied at the École Normale Supérieure, graduating in 1877, and began his academic career as a lecturer at the University of Toulouse in 1879 before returning to Paris as a maître de conférences at the Collège de France and the Sorbonne in 1881; he was appointed full professor of differential and integral calculus at the Sorbonne in 1885, a position he held until his retirement in 1926. Picard's groundbreaking contributions to the theorems bearing his name began in 1879 with his paper "Sur une propriété des fonctions entières," published in the Comptes rendus hebdomadaires des séances de l'Académie des Sciences, where he proved the Little Picard theorem: every non-constant entire function assumes every complex value infinitely often, except possibly one. This result built directly on Karl Weierstrass's foundational work on elliptic functions and modular forms, utilizing Hermite's modular functions to establish the theorem through arguments involving function growth and value distribution. In the same year, Picard extended his investigations in complex function theory, proving the Great Picard theorem as a local analogue, stating that near an essential singularity, a holomorphic function assumes every complex value infinitely often, except possibly one, in any neighborhood of the singularity.

Evolution and Naming

Following Émile Picard's original formulations in 1879, the theorems underwent significant refinement and distinction in nomenclature during the early 20th century. Initially referred to collectively as "Picard's theorem," the results were later differentiated as the "Little Picard Theorem"—concerning the global behavior of entire functions omitting at most one value—and the "Great Picard Theorem"—addressing the local behavior near essential singularities, omitting at most one value but taking all others infinitely often. This naming convention, emphasizing the relative strength of the local infinite repetition in the Great theorem versus the global omission in the Little theorem, emerged among mathematicians in the early 20th century. A pivotal milestone was the development of elementary proofs for the Little Picard Theorem in the early 1920s, avoiding advanced tools like modular functions and relying instead on basic properties of entire and meromorphic functions, which made the result more accessible and highlighted its foundational role in value distribution. The theorems' evolution accelerated with Rolf Nevanlinna's introduction of value distribution theory in the 1920s, which quantified how meromorphic functions assume values, building directly on Picard's qualitative assertions to provide asymptotic estimates for the frequency of value attainment. Nevanlinna's Second Main Theorem (1925), for instance, generalized the Little Picard Theorem by bounding the number of omitted values through characteristic functions, establishing a framework that treated Picard's results as limiting cases of broader distribution principles. An intermediate development appeared in André Bloch's 1925 theorem, which quantified the radius of univalence for holomorphic functions normalized at the origin, linking local mapping properties to global value omission ideas and serving as a bridge between Picard's global theorem and finer local behaviors near singularities. Further progress came with Lars Ahlfors' 1929 proof of the Great Picard Theorem, employing quasiconformal mappings and covering surface theory to demonstrate the theorem's validity near essential singularities through geometric distortion estimates, marking a shift toward topological and quasiconformal methods in complex analysis.

Generalizations and Extensions

In One Complex Variable

Nevanlinna theory represents a profound quantitative generalization of the within the framework of value distribution for meromorphic functions in one complex variable. Developed by Rolf Nevanlinna, it introduces the Nevanlinna characteristic T(r,f)T(r, f), which measures the average growth of ∣f(z)∣|f(z)| along circles of radius rr, along with the proximity function m(r,a)=12π∫02πlog⁡+1∣f(reiθ)−a∣dθm(r, a) = \frac{1}{2\pi} \int_0^{2\pi} \log^+ \frac{1}{|f(re^{i\theta}) - a|} d\theta and the counting function N(r,a)N(r, a), which counts aa-points with multiplicity inside ∣z∣<r|z| < r. The first main theorem equates T(r,f)=m(r,a)+N(r,a)+O(1)T(r, f) = m(r, a) + N(r, a) + O(1) for any a∈C^a \in \hat{\mathbb{C}}, providing a balance between how often ff approximates aa and how often it attains aa. The second main theorem extends this by quantifying the distribution of values across multiple points, stating that for distinct a1,…,aq∈C^a_1, \dots, a_q \in \hat{\mathbb{C}} with q>2q > 2, (q−2)T(r,f)=∑ν=1qN‾(r,f,aν)+O(log⁡T(r,f)+log⁡r),(q-2) T(r, f) = \sum_{\nu=1}^q \overline{N}(r, f, a_\nu) + O(\log T(r, f) + \log r), holding as r→∞r \to \infty outside a subset of finite logarithmic measure, where N‾\overline{N} is the truncated counting function. For functions of finite order, up to a small error term S(r,f)=o(T(r,f))S(r, f) = o(T(r, f)). This implies deficiency relations ∑δ(ai)≤2\sum \delta(a_i) \leq 2, where δ(a)=lim inf⁡r→∞m(r,a)T(r,f)\delta(a) = \liminf_{r \to \infty} \frac{m(r, a)}{T(r, f)} measures the "deficiency" of exceptional values, directly generalizing Picard's observation that at most two values can be omitted by non-constant meromorphic functions. A cornerstone of Nevanlinna's approach is the lemma on logarithmic derivatives, which bounds the proximity function for f′/ff'/f: for a ff of finite order ρ\rho, m(r,f′f)=O(log⁡T(r,f))m\left(r, \frac{f'}{f}\right) = O\left(\log T(r, f)\right) outside a set of finite logarithmic measure. This lemma enables estimates on the spacing of , facilitating proofs of the second main theorem and revealing how exceptional values relate to the function's growth. Bloch's theorem complements Picard's results by addressing the range of on disk, linking local behavior to global covering properties. For a holomorphic function ff on the unit disk D\mathbb{D} with f(0)=0f(0) = 0 and f′(0)=1f'(0) = 1, there exists a univalent (schlicht) holomorphic map from some onto a domain containing a disk of radius equal to the Bloch constant B>0B > 0, approximately $0.433.Thisguaranteesthat. This guarantees that f(\mathbb{D})$ contains a disk of fixed positive radius, extending the idea of dense value distribution in Picard's theorems to bounded domains and influencing normal families in . The Denjoy–Carleman–Ahlfors theorem provides bounds on asymptotic values for of finite order, connecting to Picard's exceptional values through the analysis of tracts where the function approaches infinity or finite limits. For a transcendental ff of order ρ\rho, the number of distinct asymptotic values is at most 2ρ2\rho, with equality achieved for functions like eze^z. Asymptotic values serve as "exceptions" in the radial direction, and the theorem limits their number, thereby refining the exceptional set in Picard's great theorem for functions omitting neighborhoods of these values. A specific application of the great Picard theorem arises for holomorphic functions in the unit disk with the unit circle as a natural boundary, such as lacunary ∑znk\sum z^{n_k} where nk+1/nk≥λ>1n_{k+1}/n_k \geq \lambda > 1. Near any boundary point, the function exhibits an in the sense of radial limits, taking every complex value except possibly one infinitely often in every neighborhood intersecting the disk, mirroring the behavior at an isolated . This ties into the study of Julia sets for transcendental entire functions, where the boundary of the escaping set acts analogously, with dense orbits under iteration implying wild value distribution consistent with great Picard's density.

In Several Complex Variables

In several complex variables, the direct analogs of the Picard theorems fail to hold due to fundamental differences in the behavior of s compared to the one-variable case, primarily arising from Hartogs' extension theorem. This theorem states that if n≥2n \geq 2, any defined on Cn\mathbb{C}^n minus a compact set extends holomorphically across the compact set, rendering isolated singularities removable in higher dimensions. As a consequence, essential singularities, which are central to the great Picard theorem in one variable, do not occur in the same isolated form; instead, singularities are often removable or behave like poles, preventing the dense value distribution near such points that characterizes the one-variable result. For entire functions f:Cn→Cf: \mathbb{C}^n \to \mathbb{C} with n≥2n \geq 2, the little Picard theorem has no counterpart: there exist non-constant entire functions that omit two or more values, unlike in one variable where non-constant entire functions omit at most one value. This flexibility stems from the higher-dimensional domain allowing for constructions such as approximations via or explicit examples like certain transcendental functions whose images avoid prescribed finite sets, highlighting how the increased dimensionality permits greater control over value omission. Polynomials in several variables, for instance, can also exhibit restricted ranges relative to their one-variable counterparts, further illustrating that entire functions in Cn\mathbb{C}^n can omit more values without being constant. Partial analogs emerge through generalizations involving s and hyperbolicity notions. A key result is the following higher-dimensional extension: if V⊂PnV \subset \mathbb{P}^n is a of degree d≥n+3d \geq n+3 whose singularities are locally normal crossings, then any holomorphic map Cn→Pn∖V\mathbb{C}^n \to \mathbb{P}^n \setminus V is constant, providing a Picard-type rigidity for maps avoiding ample divisors. This theorem, proved using techniques, captures omission of "many" values in the form of codimension-one subvarieties. Related results, such as those concerning complements of moving hyperplanes, extend the big Picard theorem by showing that near certain singularities or in punctured neighborhoods, holomorphic maps from polydisks into Pn\mathbb{P}^n minus 2n−12n-1 moving s take all values in the complement infinitely often, except possibly along thin sets. Hyperbolicity provides another avenue for Picard-type results, contrasting with Oka theory's emphasis on flexible mappings. Borel's generalization of the little Picard theorem asserts that a XX is (Kobayashi) hyperbolic if and only if there are no non-constant holomorphic maps from C\mathbb{C} to XX, implying such maps omit open sets and strengthening the omission of two points. In several variables, this extends to entire maps from Cn\mathbb{C}^n to hyperbolic targets, where hyperbolicity ensures algebraic degeneracy or constancy for maps from quasi-projective varieties, yielding omission properties in hyperbolic domains. Oka's theorem on hyperbolicity relates inversely: Oka manifolds admit dominant rational maps from Cn\mathbb{C}^n, allowing omission of fewer restrictions and underscoring the absence of full Picard rigidity in non-hyperbolic settings. Kiernan's contributions in the further developed these ideas, proving that hyperbolically embedded submanifolds in Stein spaces satisfy big Picard-type extension theorems, where maps near boundaries omit values only if extendable holomorphically. Modern developments refine these partial analogs, particularly in value distribution for maps into hyperbolic domains, where hyperbolicity implies Picard-type omission of exceptional hypersurfaces or values along entire curves. For instance, in negatively curved Kähler manifolds or complements of ample divisors, holomorphic maps from Cn\mathbb{C}^n exhibit boundedness or , omitting large sets unless constant. These results, building on extensions, emphasize that while full Picard theorems elude higher dimensions, targeted generalizations via geometry and dynamics provide robust substitutes for understanding value distribution.

Current Research Directions

Recent research in transcendental dynamics has focused on the behavior of Fatou components for transcendental entire functions, where the Great Picard theorem provides key insights into the dense distribution of values within these components. For instance, studies in the Eremenko-Lyubich class of transcendental functions have utilized the theorem to analyze asymptotic behaviors and connectivity properties of Fatou sets, showing that certain wandering domains exhibit infinite value attainment near essential singularities. Similarly, investigations into periodic boundary points of simply connected Fatou components for transcendental maps have extended these ideas, confirming that the theorem's implications hold for slowly growing functions with bounded components. Open problems persist regarding the exact constants in the Bloch-Landau theorems, which are intimately linked to radii through their role in proving the Little Picard theorem via covering arguments. The Bloch constant BB, representing the supremum of radii for univalent images under normalized holomorphic functions, remains undetermined despite improved lower bounds exceeding 3/4≈0.433\sqrt{3}/4 \approx 0.433
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